A029959 -id:A029959 - cp's OEIS frontend

A029957 Numbers that are palindromic in base 12.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 145, 157, 169, 181, 193, 205, 217, 229, 241, 253, 265, 277, 290, 302, 314, 326, 338, 350, 362, 374, 386, 398, 410, 422, 435, 447, 459, 471, 483, 495, 507, 519, 531

Offset: 1

Patrick De Geest

Cilleruelo, Luca, & Baxter prove that this sequence is an additive basis of order (exactly) 3. - Charles R Greathouse IV, May 04 2020

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.

Cf. A029958, A029959, A029960 (in bases 13..15).

f[n_,b_]:=Module[{i=IntegerDigits[n,b]},i==Reverse[i]];lst={};Do[If[f[n,12],AppendTo[lst,n]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 08 2009 *)

isok(n) = my(d=digits(n, 12)); d == Vecrev(d); \\ Michel Marcus, May 13 2017

from sympy import integer_log
from gmpy2 import digits
def A029957(n):
    if n == 1: return 0
    y = 12*(x:=12**integer_log(n>>1,12)[0])
    return int((c:=n-x)*x+int(digits(c,12)[-2::-1]or'0',12) if nChai Wah Wu, Jun 14 2024

Sum_{n>=2} 1/a(n) = 3.4989489... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020

A029960 Numbers that are palindromic in base 15.

Original entry on oeis.org

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

Offset: 1

Patrick De Geest

Cilleruelo, Luca, & Baxter prove that this sequence is an additive basis of order (exactly) 3. - Charles R Greathouse IV, May 04 2020

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.

Palindromes in bases 2 through 14: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113, A029956, A029957, A029958, A029959.

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 *)

isok(n) = my(d=digits(n, 15)); d == Vecrev(d); \\ Michel Marcus, May 14 2017

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 nChai Wah Wu, Jun 14 2024

Sum_{n>=2} 1/a(n) = 3.66254285... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020

A297283 Numbers whose base-14 digits have equal down-variation and up-variation; see Comments.

Original entry on oeis.org

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, 605, 619

Offset: 1

Clark Kimberling, Jan 17 2018

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 first from A029959 after the zero for 2759 = 1011_14, which is not a palindrome in base 14 but has UV(2759,14) = DV(2759,14) = 1. - R. J. Mathar, Jan 23 2018

			619 in base-14:  3,2,3 having DV = 1, UV = 1, so that 619 is in the sequence.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A297330, A297282, A297284.

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 = 14; 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]   (* A297282 *)
Take[Flatten[Position[w, 0]], 120]    (* A297283 *)
Take[Flatten[Position[w, 1]], 120]    (* A297284 *)

A046250 Cubes which are palindromes in base 14.

Original entry on oeis.org

0, 1, 8, 3375, 7645373, 9393931, 97972181, 20683643625, 25803133875, 56698339857713, 57570584012397, 69807297234375, 155568963323390625, 158134170512197125, 191356897019781375, 426879024292396457153

Offset: 1

Patrick De Geest, May 15 1998

Patrick De Geest, World!Of Numbers, Palindromic cubes in bases 2 to 17.

Intersection of A029959 and A000578.

Cf. A046249.

a(n) = A046249(n)^3. - Andrew Howroyd, Aug 10 2024

Offset corrected by Andrew Howroyd, Aug 10 2024

A043273 Sum of the digits of the n-th base 14 palindrome.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 8, 9, 10

Offset: 1

Clark Kimberling

A029959 (base 14 palindromes)

Table[With[{id14=IntegerDigits[n,14]},If[id14==Reverse[id14],Total[id14],Nothing]],{n,0,1000}]  (* Harvey P. Dale, Oct 10 2024 *)

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A029957 Numbers that are palindromic in base 12.

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A029960 Numbers that are palindromic in base 15.

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A297283 Numbers whose base-14 digits have equal down-variation and up-variation; see Comments.

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A046250 Cubes which are palindromes in base 14.

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A043273 Sum of the digits of the n-th base 14 palindrome.

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