A029960 -id:A029960 - 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

A249155 Palindromic in bases 6 and 15.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 14, 80, 160, 301, 602, 693, 994, 1295, 1627, 1777, 2365, 2666, 5296, 5776, 6256, 17360, 34720, 51301, 52201, 105092, 155493, 209284, 587846, 735644, 7904800, 11495701, 80005507, 80469907, 83165017, 89731777, 90196177

Offset: 1

Ray Chandler, Oct 27 2014

Intersection of A029953 and A029960.

			301 is a term since 301 = 1221 base 6 and 301 = 151 base 15.

Ray Chandler and Chai Wah Wu, Table of n, a(n) for n = 1..71 (terms < 6^28). First 65 terms from Ray Chandler.
Attila Bérczes and Volker Ziegler, On Simultaneous Palindromes, arXiv:1403.0787 [math.NT], 2014.

Cf. A007632, A060792, A249156, A249157, A249158.

palQ[n_Integer, base_Integer] := Block[{idn = IntegerDigits[n, base]}, idn == Reverse[idn]]; Select[Range[10^6] - 1, palQ[#, 6] && palQ[#, 15] &]

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): # generator of palindromes in base b of length <= 2*l
    if l > 0:
        yield 0
        for x in range(1, l+1):
            for y in range(b**(x-1), b**x):
                s = digits(y, b)
                yield int(s+s[-2::-1], b)
            for y in range(b**(x-1), b**x):
                s = digits(y, b)
                yield int(s+s[::-1], b)
A249155_list = [n for n in palQgen(8, 6) if palQ(n, 15)] # Chai Wah Wu, Nov 29 2014

A297286 Numbers whose base-15 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, 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, 632, 647

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 from A029960 after the zero first for 3391 = 1011_15, which is not in A029960 but in this sequence. - R. J. Mathar, Jan 23 2018

			647 in base-15:  2,13,2 having DV = 11, UV = 11, so that 647 is in the sequence.

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

Cf. A297330, A297285, A297287.

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 = 15; 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]   (* A297285 *)
Take[Flatten[Position[w, 0]], 120]    (* A297286 *)
Take[Flatten[Position[w, 1]], 120]    (* A297287 *)

A046252 Cubes which are palindromes in base 15.

Original entry on oeis.org

0, 1, 8, 64, 512, 4096, 11543176, 13997521, 738763264, 5910106112, 38477541376, 47280848896, 2462562648064, 19700501184512, 129754026714376, 131491746445051, 157604009476096, 8304257709720064, 66434061677760512

Offset: 1

Patrick De Geest, May 15 1998

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

Intersection of A029960 and A000578.

Cf. A046251.

p15Q[n_]:=Module[{d=IntegerDigits[n,15]},d==Reverse[d]]; Select[Range[ 0,410000]^3,p15Q] (* Harvey P. Dale, Mar 28 2020 *)

a(n) = A046251(n)^3. - Andrew Howroyd, Aug 10 2024

Offset corrected by Andrew Howroyd, Aug 10 2024

A043274 Sum of the digits of the n-th base 15 palindrome.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18

Offset: 1

Clark Kimberling

A029960 (base 15 palindromes)

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A029957 Numbers that are palindromic in base 12.

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A249155 Palindromic in bases 6 and 15.

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A297286 Numbers whose base-15 digits have equal down-variation and up-variation; see Comments.

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A046252 Cubes which are palindromes in base 15.

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A043274 Sum of the digits of the n-th base 15 palindrome.

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