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A002113 Palindromes in base 10.
(Formerly M0484 N0178) 		+10
797
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, 242, 252, 262, 272, 282, 292, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 404, 414, 424, 434, 444, 454, 464, 474, 484, 494, 505, 515 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
n is a palindrome (i.e., a(k) = n for some k) if and only if n = A004086(n). - Reinhard Zumkeller, Mar 10 2002
It seems that if n*reversal(n) is in the sequence then n = 3 or all digits of n are less than 3. - Farideh Firoozbakht, Nov 02 2014
The position of a palindrome within the sequence can be determined almost without calculation: If the palindrome has an even number of digits, prepend a 1 to the front half of the palindrome's digits. If the number of digits is odd, prepend the value of front digit + 1 to the digits from position 2 ... central digit. Examples: 98766789 = a(19876), 515 = a(61), 8206028 = a(9206), 9230329 = a(10230). - Hugo Pfoertner, Aug 14 2015
This sequence is an additive basis of order at most 49, see Banks link. - Charles R Greathouse IV, Aug 23 2015
The order has been reduced from 49 to 3; see the Cilleruelo-Luca and Cilleruelo-Luca-Baxter links. - Jonathan Sondow, Nov 27 2017
See A262038 for the "next palindrome" and A261423 for the "preceding palindrome" functions. - M. F. Hasler, Sep 09 2015
The number of palindromes with d digits is 10 if d = 1, and otherwise it is 9 * 10^(floor((d - 1)/2)). - N. J. A. Sloane, Dec 06 2015
Sequence A033665 tells how many iterations of the Reverse-then-add function A056964 are needed to reach a palindrome; numbers for which this will never happen are Lychrel numbers (A088753) or rather Kin numbers (A023108). - M. F. Hasler, Apr 13 2019
REFERENCES
Karl G. Kröber, "Palindrome, Perioden und Chaoten: 66 Streifzüge durch die palindromischen Gefilde" (1997, Deutsch-Taschenbücher; Bd. 99) ISBN 3-8171-1522-9.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 71.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, List of first 19999 palindromes: Table of n, a(n) for n = 1..19999
Hunki Baek, Sejeong Bang, Dongseok Kim, and Jaeun Lee, A bijection between aperiodic palindromes and connected circulant graphs, arXiv:1412.2426 [math.CO], 2014.
William D. Banks, Derrick N. Hart, and Mayumi Sakata, Almost all palindromes are composite, Math. Res. Lett., Vol. 11, No. 5-6 (2004), pp. 853-868.
William D. Banks, Every natural number is the sum of forty-nine palindromes, arXiv:1508.04721 [math.NT], 2015; Integers, 16 (2016), article A3.
Javier Cilleruelo, Florian Luca and Lewis Baxter, Every positive integer is a sum of three palindromes, Mathematics of Computation, Vol. 87, No. 314 (2018), pp. 3023-3055, arXiv preprint, arXiv:1602.06208 [math.NT], 2017.
Patrick De Geest, World of Numbers.
Kritkhajohn Onphaeng, Tammatada Khemaratchatakumthorn, Phakhinkon Napp Phunphayap, and Prapanpong Pongsriiam, Exact Formulas for the Number of Palindromes in Certain Arithmetic Progressions, Journal of Integer Sequences, Vol. 27 (2024), Article 24.4.8. See p. 2.
Phakhinkon Phunphayap and Prapanpong Pongsriiam, Reciprocal sum of palindromes, arXiv:1803.00161 [math.CA], 2018.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Prapanpong Pongsriiam and Kittipong Subwattanachai, Exact Formulas for the Number of Palindromes up to a Given Positive Integer, Intl. J. of Math. Comp. Sci. (2019) 14:1, 27-46.
E. A. Schmidt, Positive Integer Palindromes. [Cached copy at the Wayback Machine]
Eric Weisstein's World of Mathematics, Palindromic Number.
Wikipedia, Palindromic number.
Index entries for sequences that are an additive basis, order 3.
Index entries for sequences related to palindromes
Index entries for "core" sequences
FORMULA
A136522(a(n)) = 1.
A178788(a(n)) = 0 for n > 9. - Reinhard Zumkeller, Jun 30 2010
A064834(a(n)) = 0. - Reinhard Zumkeller, Sep 18 2013
a(n+1) = A262038(a(n)+1). - M. F. Hasler, Sep 09 2015
Sum_{n>=2} 1/a(n) = A118031. - Amiram Eldar, Oct 17 2020
MAPLE
read transforms; t0:=[]; for n from 0 to 2000 do if digrev(n) = n then t0:=[op(t0), n]; fi; od: t0;
# Alternatively, to get all palindromes with <= N digits in the list "Res":
N:=5;
Res:= $0..9:
for d from 2 to N do
if d::even then
m:= d/2;
Res:= Res, seq(n*10^m + digrev(n), n=10^(m-1)..10^m-1);
else
m:= (d-1)/2;
Res:= Res, seq(seq(n*10^(m+1)+y*10^m+digrev(n), y=0..9), n=10^(m-1)..10^m-1);
fi
od: Res:=[Res]: # Robert Israel, Aug 10 2014
# A variant: Gets all base-10 palindromes with exactly d digits, in the list "Res"
d:=4:
if d=1 then Res:= [$0..9]:
elif d::even then
m:= d/2:
Res:= [seq(n*10^m + digrev(n), n=10^(m-1)..10^m-1)]:
else
m:= (d-1)/2:
Res:= [seq(seq(n*10^(m+1)+y*10^m+digrev(n), y=0..9), n=10^(m-1)..10^m-1)]:
fi:
Res; # N. J. A. Sloane, Oct 18 2015
isA002113 := proc(n)
simplify(digrev(n) = n) ;
end proc: # R. J. Mathar, Sep 09 2015
MATHEMATICA
palQ[n_Integer, base_Integer] := Module[{idn = IntegerDigits[n, base]}, idn == Reverse[idn]]; (* then to generate any base-b sequence for 1 < b < 37, replace the 10 in the following instruction with b: *) Select[Range[0, 1000], palQ[#, 10] &]
base10Pals = {0}; r = 2; Do[Do[AppendTo[base10Pals, n * 10^(IntegerLength[n] - 1) + FromDigits@Rest@Reverse@IntegerDigits[n]], {n, 10^(e - 1), 10^e - 1}]; Do[AppendTo[base10Pals, n * 10^IntegerLength[n] + FromDigits@Reverse@IntegerDigits[n]], {n, 10^(e - 1), 10^e - 1}], {e, r}]; base10Pals (* Arkadiusz Wesolowski, May 04 2012 *)
nthPalindromeBase[n_, b_] := Block[{q = n + 1 - b^Floor[Log[b, n + 1 - b^Floor[Log[b, n/b]]]], c = Sum[Floor[Floor[n/((b + 1) b^(k - 1) - 1)]/(Floor[n/((b + 1) b^(k - 1) - 1)] - 1/b)] - Floor[Floor[n/(2 b^k - 1)]/(Floor[n/(2 b^k - 1)] - 1/ b)], {k, Floor[Log[b, n]]}]}, Mod[q, b] (b + 1)^c * b^Floor[Log[b, q]] + Sum[Floor[Mod[q, b^(k + 1)]/b^k] b^(Floor[Log[b, q]] - k) (b^(2 k + c) + 1), {k, Floor[Log[b, q]]}]] (* after the work of Eric A. Schmidt, works for all integer bases b > 2 *)
Array[nthPalindromeBase[#, 10] &, 61, 0] (* please note that Schmidt uses a different, a more natural and intuitive offset, that of a(1) = 1. - Robert G. Wilson v, Sep 22 2014 and modified Nov 28 2014 *)
Select[Range[10^3], PalindromeQ] (* Michael De Vlieger, Nov 27 2017 *)
PROG
(PARI) is_A002113(n)=Vecrev(n=digits(n))==n \\ M. F. Hasler, Nov 17 2008, updated Apr 26 2014, Jun 19 2018
(PARI) is(n)=n=digits(n); for(i=1, #n\2, if(n[i]!=n[#n+1-i], return(0))); 1 \\ Charles R Greathouse IV, Jan 04 2013
(PARI) a(n)={my(d, i, r); r=vector(#digits(n-10^(#digits(n\11)))+#digits(n\11)); n=n-10^(#digits(n\11)); d=digits(n); for(i=1, #d, r[i]=d[i]; r[#r+1-i]=d[i]); sum(i=1, #r, 10^(#r-i)*r[i])} \\ David A. Corneth, Jun 06 2014
(PARI) \\ recursive--feed an element a(n) and it gives a(n+1)
nxt(n)=my(d=digits(n)); i=(#d+1)\2; while(i&&d[i]==9, d[i]=0; d[#d+1-i]=0; i--); if(i, d[i]++; d[#d+1-i]=d[i], d=vector(#d+1); d[1]=d[#d]=1); sum(i=1, #d, 10^(#d-i)*d[i]) \\ David A. Corneth, Jun 06 2014
(PARI) \\ feed a(n), returns n.
inv(n)={my(d=digits(n)); q=ceil(#d/2); sum(i=1, q, 10^(q-i)*d[i])+10^floor(#d/2)} \\ David A. Corneth, Jun 18 2014
(PARI) inv_A002113(P)={P\(P=10^(logint(P+!P, 10)\/2))+P} \\ index n of palindrome P = a(n), much faster than above: no sum is needed. - M. F. Hasler, Sep 09 2018
(PARI) A002113(n, L=logint(n, 10))=(n-=L=10^max(L-(n<11*10^(L-1)), 0))*L+fromdigits(Vecrev(digits(if(n<L, n, n\10)))) \\ M. F. Hasler, Sep 11 2018
(Python) # edited by M. F. Hasler, Jun 19 2018
def A002113_list(nMax):
mlist=[]
for n in range(nMax+1):
mstr=str(n)
if mstr==mstr[::-1]:
mlist.append(n)
return mlist # Bill McEachen, Dec 17 2010
(Python)
from itertools import chain
A002113 = sorted(chain(map(lambda x:int(str(x)+str(x)[::-1]), range(1, 10**3)), map(lambda x:int(str(x)+str(x)[-2::-1]), range(10**3)))) # Chai Wah Wu, Aug 09 2014
(Python)
from itertools import chain, count
A002113 = chain(k for k in count(0) if str(k) == str(k)[::-1])
print([next(A002113) for k in range(60)]) # Jan P. Hartkopf, Apr 10 2021
(Python) is_A002113 = lambda n: (s:=str(n))[::-1]==s # M. F. Hasler, May 23 2024
(Python)
from math import log10
def A002113(n):
if n < 2: return 0
P = 10**floor(log10(n//2)); M = 11*P
s = str(n - (P if n < M else M-P))
return int(s + s[-2 if n < M else -1::-1]) # M. F. Hasler, Jun 06 2024
(Haskell)
a002113 n = a002113_list !! (n-1)
a002113_list = filter ((== 1) . a136522) [1..] -- Reinhard Zumkeller, Oct 09 2011
(Haskell)
import Data.List.Ordered (union)
a002113_list = union a056524_list a056525_list -- Reinhard Zumkeller, Jul 29 2015, Dec 28 2011
(Magma) [n: n in [0..600] | Intseq(n, 10) eq Reverse(Intseq(n, 10))]; // Vincenzo Librandi, Nov 03 2014
(SageMath)
[n for n in (0..515) if Word(n.digits()).is_palindrome()] # Peter Luschny, Sep 13 2018
(GAP) Filtered([0..550], n->ListOfDigits(n)=Reversed(ListOfDigits(n))); # Muniru A Asiru, Mar 08 2019
(Scala) def palQ(n: Int, b: Int = 10): Boolean = n - Integer.parseInt(n.toString.reverse) == 0
(0 to 999).filter(palQ(_)) // Alonso del Arte, Nov 10 2019
CROSSREFS
Palindromes in bases 2 through 11: A006995 and A057148, A014190 and A118594, A014192 and A118595, A029952 and A118596, A029953 and A118597, A029954 and A118598, A029803 and A118599, A029955 and A118600, this sequence, A029956. Also A262065 (base 60), A262069 (subsequence).
Palindromic primes: A002385. Palindromic nonprimes: A032350.
Palindromic-pi: A136687.
Cf. A029742 (complement), A086862 (first differences).
Palindromic floor function: A261423, also A261424. Palindromic ceiling: A262038.
Union of A056524 and A056525.
Cf. A004086 (read n backwards), A064834, A118031, A136522 (characteristic function), A178788.
Ways to write n as a sum of three palindromes: A261132, A261422.
Minimal number of palindromes that add to n using greedy algorithm: A088601.
Minimal number of palindromes that add to n: A261675.
Subsequence of A061917 and A221221.
Subsequence: A110745.
KEYWORD
nonn,base,easy,nice,core
AUTHOR
N. J. A. Sloane
STATUS
approved
A029958 Numbers that are palindromic in base 13. +10
7
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 28, 42, 56, 70, 84, 98, 112, 126, 140, 154, 168, 170, 183, 196, 209, 222, 235, 248, 261, 274, 287, 300, 313, 326, 340, 353, 366, 379, 392, 405, 418, 431, 444, 457, 470, 483, 496, 510, 523, 536, 549, 562 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Cilleruelo, Luca, & Baxter prove that this sequence is an additive basis of order (exactly) 3. - Charles R Greathouse IV, May 04 2020
LINKS
John Cerkan, Table of n, a(n) for n = 1..10000
Javier Cilleruelo, Florian Luca and Lewis Baxter, Every positive integer is a sum of three palindromes, Mathematics of Computation, Vol. 87, No. 314 (2018), pp. 3023-3055, arXiv preprint, arXiv:1602.06208 [math.NT], 2017.
Patrick De Geest, Palindromic numbers beyond base 10.
Phakhinkon Phunphayap and Prapanpong Pongsriiam, Estimates for the Reciprocal Sum of b-adic Palindromes, 2019.
Index entries for sequences that are an additive basis, order 3.
FORMULA
Sum_{n>=2} 1/a(n) = 3.55686013... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020
MATHEMATICA
f[n_, b_]:=Module[{i=IntegerDigits[n, b]}, i==Reverse[i]]; lst={}; Do[If[f[n, 13], AppendTo[lst, n]], {n, 7!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jul 08 2009 *)
Select[Range[0, 600], IntegerDigits[#, 13]==Reverse[IntegerDigits[#, 13]]&] (* Harvey P. Dale, Nov 16 2022 *)
PROG
(PARI) isok(n) = my(d=digits(n, 13)); d == Vecrev(d); \\ Michel Marcus, May 13 2017
(Python)
from sympy import integer_log
from gmpy2 import digits
def A029958(n):
if n == 1: return 0
y = 13*(x:=13**integer_log(n>>1, 13)[0])
return int((c:=n-x)*x+int(digits(c, 13)[-2::-1]or'0', 13) if n<x+y else (c:=n-y)*y+int(digits(c, 13)[-1::-1]or'0', 13)) # Chai Wah Wu, Jun 14 2024
CROSSREFS
Palindromes in bases 2 through 12: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113, A029956, A029957.
KEYWORD
nonn,base,easy
AUTHOR
Patrick De Geest
STATUS
approved
A029959 Numbers that are palindromic in base 14. +10
5
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, 195, 197, 211, 225, 239, 253, 267, 281, 295, 309, 323, 337, 351, 365, 379, 394, 408, 422, 436, 450, 464, 478, 492, 506, 520, 534, 548, 562, 576, 591 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Cilleruelo, Luca, & Baxter prove that this sequence is an additive basis of order (exactly) 3. - Charles R Greathouse IV, May 04 2020
LINKS
John Cerkan, Table of n, a(n) for n = 1..10000
Javier Cilleruelo, Florian Luca and Lewis Baxter, Every positive integer is a sum of three palindromes, Mathematics of Computation, Vol. 87, No. 314 (2018), pp. 3023-3055, arXiv preprint, arXiv:1602.06208 [math.NT], 2017.
Patrick De Geest, Palindromic numbers beyond base 10.
Phakhinkon Phunphayap and Prapanpong Pongsriiam, Estimates for the Reciprocal Sum of b-adic Palindromes, 2019.
Index entries for sequences that are an additive basis, order 3.
FORMULA
Sum_{n>=2} 1/a(n) = 3.6112482... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020
EXAMPLE
195 is DD in base 14.
196 is 100 in base 14, so it's not in the sequence.
197 is 101 in base 14.
MATHEMATICA
palQ[n_, b_:10] := Module[{idn = IntegerDigits[n, b]}, idn == Reverse[idn]]; Select[ Range[0, 600], palQ[#, 14] &] (* Harvey P. Dale, Aug 03 2014 *)
PROG
(PARI) isok(n) = Pol(d=digits(n, 14)) == Polrev(d); \\ Michel Marcus, Mar 12 2017
(Python)
from sympy import integer_log
from gmpy2 import digits
def A029959(n):
if n == 1: return 0
y = 14*(x:=14**integer_log(n>>1, 14)[0])
return int((c:=n-x)*x+int(digits(c, 14)[-2::-1]or'0', 14) if n<x+y else (c:=n-y)*y+int(digits(c, 14)[-1::-1]or'0', 14)) # Chai Wah Wu, Jun 14 2024
CROSSREFS
Palindromes in bases 2 through 13: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113, A029956, A029957, A029958.
KEYWORD
nonn,base,easy
AUTHOR
Patrick De Geest
STATUS
approved
A029960 Numbers that are palindromic in base 15. +10
5
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 226, 241, 256, 271, 286, 301, 316, 331, 346, 361, 376, 391, 406, 421, 436, 452, 467, 482, 497, 512, 527, 542, 557, 572, 587, 602, 617 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Cilleruelo, Luca, & Baxter prove that this sequence is an additive basis of order (exactly) 3. - Charles R Greathouse IV, May 04 2020
LINKS
John Cerkan, Table of n, a(n) for n = 1..10000
Javier Cilleruelo, Florian Luca and Lewis Baxter, Every positive integer is a sum of three palindromes, Mathematics of Computation, Vol. 87, No. 314 (2018), pp. 3023-3055, arXiv preprint, arXiv:1602.06208 [math.NT], 2017.
Patrick De Geest, Palindromic numbers beyond base 10.
Phakhinkon Phunphayap and Prapanpong Pongsriiam, Estimates for the Reciprocal Sum of b-adic Palindromes, 2019.
Index entries for sequences that are an additive basis, order 3.
FORMULA
Sum_{n>=2} 1/a(n) = 3.66254285... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020
MATHEMATICA
f[n_, b_]:=Module[{i=IntegerDigits[n, b]}, i==Reverse[i]]; lst={}; Do[If[f[n, 15], AppendTo[lst, n]], {n, 7!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jul 08 2009 *)
Select[Range@ 620, PalindromeQ@ IntegerDigits[#, 15] &] (* Michael De Vlieger, May 13 2017, Version 10.3 *)
PROG
(PARI) isok(n) = my(d=digits(n, 15)); d == Vecrev(d); \\ Michel Marcus, May 14 2017
(Python)
from sympy import integer_log
from gmpy2 import digits
def A029960(n):
if n == 1: return 0
y = 15*(x:=15**integer_log(n>>1, 15)[0])
return int((c:=n-x)*x+int(digits(c, 15)[-2::-1]or'0', 15) if n<x+y else (c:=n-y)*y+int(digits(c, 15)[-1::-1]or'0', 15)) # Chai Wah Wu, Jun 14 2024
CROSSREFS
Palindromes in bases 2 through 14: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113, A029956, A029957, A029958, A029959.
KEYWORD
nonn,base,easy
AUTHOR
Patrick De Geest
STATUS
approved
A262065 Numbers that are palindromes in base-60 representation. +10
5
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 122, 183, 244, 305, 366 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Chai Wah Wu, Table of n, a(n) for n = 1..10000 (corrected b-file originally from Reinhard Zumkeller)
Eric Weisstein's World of Mathematics, Palindromic Number
Eric Weisstein's World of Mathematics, Sexagesimal
Wikipedia, Palindromic number
Wikipedia, Sexagesimal
Index entries for sequences related to palindromes
EXAMPLE
. n | a(n) | base 60 n | a(n) | base 60
. -----+------+----------- ------+-------+--------------
. 100 | 2440 | [40, 40] 1000 | 56415 | [15, 40, 15]
. 101 | 2501 | [41, 41] 1001 | 56475 | [15, 41, 15]
. 102 | 2562 | [42, 42] 1002 | 56535 | [15, 42, 15]
. 103 | 2623 | [43, 43] 1003 | 56595 | [15, 43, 15]
. 104 | 2684 | [44, 44] 1004 | 56655 | [15, 44, 15]
. 105 | 2745 | [45, 45] 1005 | 56715 | [15, 45, 15]
. 106 | 2806 | [46, 46] 1006 | 56775 | [15, 46, 15]
. 107 | 2867 | [47, 47] 1007 | 56835 | [15, 47, 15]
. 108 | 2928 | [48, 48] 1008 | 56895 | [15, 48, 15]
. 109 | 2989 | [49, 49] 1009 | 56955 | [15, 49, 15]
. 110 | 3050 | [50, 50] 1010 | 57015 | [15, 50, 15]
. 111 | 3111 | [51, 51] 1011 | 57075 | [15, 51, 15]
. 112 | 3172 | [52, 52] 1012 | 57135 | [15, 52, 15]
. 113 | 3233 | [53, 53] 1013 | 57195 | [15, 53, 15]
. 114 | 3294 | [54, 54] 1014 | 57255 | [15, 54, 15]
. 115 | 3355 | [55, 55] 1015 | 57315 | [15, 55, 15]
. 116 | 3416 | [56, 56] 1016 | 57375 | [15, 56, 15]
. 117 | 3477 | [57, 57] 1017 | 57435 | [15, 57, 15]
. 118 | 3538 | [58, 58] 1018 | 57495 | [15, 58, 15]
. 119 | 3599 | [59, 59] 1019 | 57555 | [15, 59, 15]
. 120 | 3601 | [1, 0, 1] 1020 | 57616 | [16, 0, 16]
. 121 | 3661 | [1, 1, 1] 1021 | 57676 | [16, 1, 16]
. 122 | 3721 | [1, 2, 1] 1022 | 57736 | [16, 2, 16]
. 123 | 3781 | [1, 3, 1] 1023 | 57796 | [16, 3, 16]
. 124 | 3841 | [1, 4, 1] 1024 | 57856 | [16, 4, 16]
. 125 | 3901 | [1, 5, 1] 1025 | 57916 | [16, 5, 16] .
MATHEMATICA
f[n_, b_]:=Module[{i=IntegerDigits[n, b]}, i==Reverse[i]]; lst={}; Do[If[f[n, 60], AppendTo[lst, n]], {n, 400}]; lst (* Vincenzo Librandi, Aug 24 2016 *)
pal60Q[n_]:=Module[{idn60=IntegerDigits[n, 60]}, idn60==Reverse[idn60]]; Select[Range[0, 400], pal60Q] (* Harvey P. Dale, Nov 04 2017 *)
PROG
(Haskell)
import Data.List.Ordered (union)
a262065 n = a262065_list !! (n-1)
a262065_list = union us vs where
us = [val60 $ bs ++ reverse bs | bs <- bss]
vs = [0..59] ++ [val60 $ bs ++ cs ++ reverse bs |
bs <- tail bss, cs <- take 60 bss]
bss = iterate s [0] where
s [] = [1]; s (59:ds) = 0 : s ds; s (d:ds) = (d + 1) : ds
val60 = foldr (\b v -> 60 * v + b) 0
(Magma) [n: n in [0..600] | Intseq(n, 60) eq Reverse(Intseq(n, 60))]; // Vincenzo Librandi, Aug 24 2016
(PARI) isok(m) = my(d=digits(m, 60)); d == Vecrev(d); \\ Michel Marcus, Jan 22 2022
(Python)
from sympy import integer_log
from gmpy2 import digits, mpz
def A262065(n):
if n == 1: return 0
y = 60*(x:=60**integer_log(n>>1, 60)[0])
return int((c:=n-x)*x+mpz(digits(c, 60)[-2::-1]or'0', 60) if n<x+y else (c:=n-y)*y+mpz(digits(c, 60)[::-1]or'0', 60)) # Chai Wah Wu, Jun 13-14 2024
CROSSREFS
Cf. A262079 (first differences).
Intersection with A002113: A262069.
Corresponding sequences for bases 2 through 12: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113, A029956, A029957.
KEYWORD
nonn,base,look
AUTHOR
Reinhard Zumkeller, Sep 10 2015
STATUS
approved
A249157 Palindromic in bases 11 and 13. +10
4
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 84, 366, 510, 732, 876, 1020, 1098, 1242, 1464, 10248, 30252, 31110, 62220, 103704, 146541, 3382050, 3698730, 4391268, 225622530, 272466250, 413186676, 713998530, 801837204, 848770222, 912265732 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Intersection of A029956 and A029958.
LINKS
Ray Chandler, Table of n, a(n) for n = 1..69 (terms < 10^18)
Attila Bérczes and Volker Ziegler, On Simultaneous Palindromes, arXiv:1403.0787 [math.NT]
EXAMPLE
366 is a term since 366 = 303 base 11 and 366 = 222 base 13.
MATHEMATICA
palQ[n_Integer, base_Integer]:=Block[{idn=IntegerDigits[n, base]}, idn==Reverse[idn]]; Select[Range[10^6]-1, palQ[#, 11]&&palQ[#, 13]&]
PROG
(Python)
from gmpy2 import digits
def palQ(n, b): # check if n is a palindrome in base b
....s = digits(n, b)
....return s == s[::-1]
def palQgen(l, b): # unordered generator of palindromes in base b of length <= 2*l
....if l > 0:
........yield 0
........for x in range(1, b**l):
............s = digits(x, b)
............yield int(s+s[-2::-1], b)
............yield int(s+s[::-1], b)
A249157_list = sorted([n for n in palQgen(6, 11) if palQ(n, 13)]) # Chai Wah Wu, Nov 25 2014
CROSSREFS
Cf. A007632, A060792, A249155, A249156, A249158.
KEYWORD
nonn,base
AUTHOR
Ray Chandler, Oct 27 2014
STATUS
approved
A297274 Numbers whose base-11 digits have equal down-variation and up-variation; see Comments. +10
4
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 122, 133, 144, 155, 166, 177, 188, 199, 210, 221, 232, 244, 255, 266, 277, 288, 299, 310, 321, 332, 343, 354, 366, 377, 388, 399, 410, 421, 432, 443, 454, 465, 476, 488, 499, 510, 521 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Suppose that n has base-b digits b(m), b(m-1), ..., b(0). The base-b down-variation of n is the sum DV(n,b) of all d(i)-d(i-1) for which d(i) > d(i-1); the base-b up-variation of n is the sum UV(n,b) of all d(k-1)-d(k) for which d(k) < d(k-1). The total base-b variation of n is the sum TV(n,b) = DV(n,b) + UV(n,b). See the guide at A297330.
Differs after the zero from A029956 first at 1343 = 1011_11, which is not a palindrome in base 11 but has DV(1343,11) = UV(1343,11) =1. - R. J. Mathar, Jan 23 2018
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..10000
EXAMPLE
521 in base-11: 4,3,4, having DV = 1, UV = 1, so that 521 is in the sequence.
MATHEMATICA
g[n_, b_] := Map[Total, GatherBy[Differences[IntegerDigits[n, b]], Sign]];
x[n_, b_] := Select[g[n, b], # < 0 &]; y[n_, b_] := Select[g[n, b], # > 0 &];
b = 11; z = 2000; p = Table[x[n, b], {n, 1, z}]; q = Table[y[n, b], {n, 1, z}];
w = Sign[Flatten[p /. {} -> {0}] + Flatten[q /. {} -> {0}]];
Take[Flatten[Position[w, -1]], 120] (* A297273 *)
Take[Flatten[Position[w, 0]], 120] (* A297274 *)
Take[Flatten[Position[w, 1]], 120] (* A297275 *)
CROSSREFS
Cf. A297330, A297273, A297275.
KEYWORD
nonn,base,easy
AUTHOR
Clark Kimberling, Jan 16 2018
STATUS
approved
A043270 Sum of the digits of the n-th base 11 palindrome. +10
0
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 10, 11, 12, 13, 14, 15, 16 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Table of n, a(n) for n=1..72.
CROSSREFS
A029956 (base 11 palindromes)
KEYWORD
nonn,base
AUTHOR
Clark Kimberling
STATUS
approved
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