A029952 -id:A029952 - cp's OEIS frontend

A342725 Numbers that are palindromic in base i-1.

0, 1, 13, 17, 189, 205, 257, 273, 3005, 3069, 3277, 3341, 4033, 4097, 4305, 4369, 48061, 48317, 49149, 49405, 52173, 52429, 53261, 53517, 64449, 64705, 65537, 65793, 68561, 68817, 69649, 69905, 768957, 769981, 773309, 774333, 785405, 786429, 789757, 790781, 834509

Offset: 1

Amiram Eldar, Mar 19 2021

Amiram Eldar, Table of n, a(n) for n = 1..512
Walter Penney, A "binary" system for complex numbers, Journal of the ACM, Vol. 12, No. 2 (1965), pp. 247-248.

Cf. A066321, A066323, A271472, A342726, A342727, A342728, A342729.

Similar sequences: A002113 (decimal), A006995 (binary), A014190 (base 3), A014192 (base 4), A029952 (base 5), A029953 (base 6), A029954 (base 7), A029803 (base 8), A029955 (base 9), A046807 (factorial base), A094202 (Zeckendorf), A331191 (dual Zeckendorf), A331891 (negabinary), A333423 (primorial base).

v = {{0, 0, 0, 0}, {0, 0, 0, 1}, {1, 1, 0, 0}, {1, 1, 0, 1}}; q[n_] := PalindromeQ @ FromDigits[Flatten @ v[[1 + Reverse @ Most[Mod[NestWhileList[(# - Mod[#, 4])/-4 &, n, # != 0 &], 4]]]]]; Select[Range[0, 10^4], q]

13 is a term since its base-(i-1) presentation is 100010001 which is palindromic.

A350991 Triangular numbers that are palindromes in base 5.

Original entry on oeis.org

0, 1, 3, 6, 36, 78, 378, 1953, 20706, 23436, 48828, 147696, 239778, 426426, 449826, 1220703, 2155926, 6011778, 14625936, 30517578, 74218836, 74316336, 149083278, 314290056, 351562386, 762939453, 7897542681, 9141750936, 10201418541, 19073486328, 35952613476, 38218245156

Offset: 1

Amiram Eldar, Jan 28 2022

This sequence is infinite since A000217((5^k-1)/2) is a term for all k >= 0 (Trigg, 1972).

			6 is a term since 6 = A000217(3) is a triangular number and also a palindromic number in base 5: 6 = 11_5.
36 is a term since 36 = A000217(8) is a triangular number and also a palindromic number in base 5: 36 = 121_5.

Amiram Eldar, Table of n, a(n) for n = 1..68
Charles W. Trigg, Problem 840, Problems and Solutions, Mathematics Magazine, Vol. 45, No. 4 (1972), p. 228; Triangular Palindromes, Solution to Problem 840 by E. P. Starke, ibid., Vol. 46, No. 3 (1973), p. 170.
Charles W. Trigg, Infinite sequences of palindromic triangular numbers, The Fibonacci Quarterly, Vol. 12, No. 2 (1974), pp. 209-212.
Maciej Ulas, On certain diophantine equations related to triangular and tetrahedral numbers, arXiv:0811.2477 [math.NT], 2008.

Intersection of A000217 and A029952.
The quinary version of A003098.
Cf. A003098, A125831, A350987, A350990, A350992, A350993.

t[n_] := n*(n + 1)/2; Select[t /@ Range[0, 3*10^5], PalindromeQ[IntegerDigits[#, 5]] &]

A261917 Numbers which in base 5 are the sum of two palindromes but are not palindromes themselves.

Original entry on oeis.org

5, 7, 8, 9, 10, 13, 14, 15, 16, 19, 20, 21, 22, 25, 27, 28, 29, 30, 32, 33, 34, 35, 37, 38, 39, 40, 42, 43, 44, 45, 47, 48, 49, 50, 53, 54, 55, 56, 58, 59, 60, 61, 63, 64, 65, 66, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 84, 85, 86, 87, 89, 90, 91, 92, 94, 95, 96, 97, 99

Offset: 1

N. J. A. Sloane, Sep 13 2015

N. J. A. Sloane, Table of n, a(n) for n = 1..13703

Cf. A029952 (palindromes), A261907 (base 10 analog), A261918.

A261918 Numbers which in base 5 are neither palindromes nor the sum of two palindromes.

Original entry on oeis.org

11, 17, 23, 51, 131, 141, 146, 147, 149, 151, 153, 154, 163, 164, 169, 173, 175, 177, 179, 181, 184, 185, 194, 195, 199, 200, 201, 203, 205, 206, 211, 215, 221, 225, 226, 229, 231, 236, 237, 241, 251, 259, 261, 262, 263, 266, 267, 271, 281, 287, 289, 291, 296, 297

Offset: 1

N. J. A. Sloane, Sep 13 2015

The terms less than 20000, and conjecturally all terms, are the sum of three base 5 palindromes.

N. J. A. Sloane, Table of n, a(n) for n = 1..6235

Cf. A029952, A261917; A035137 (base 10 analog).

A333423 Numbers that are palindromes in primorial base.

Original entry on oeis.org

0, 1, 3, 7, 9, 11, 31, 39, 47, 211, 217, 223, 229, 235, 243, 249, 255, 261, 267, 275, 281, 287, 293, 299, 2311, 2347, 2383, 2419, 2455, 2523, 2559, 2595, 2631, 2667, 2735, 2771, 2807, 2843, 2879, 30031, 30061, 30091, 30121, 30151, 30181, 30211, 30247, 30277, 30307

Offset: 1

Amiram Eldar, Mar 20 2020

			3 is a term since its representation in primorial base is 11 (1 * 2# + 1) which is a palindrome.
7 is a term since its representation in primorial base is 101 (1 * 3# + 0 * 2# + 1 = 6 + 1) which is a palindrome.

Amiram Eldar, Table of n, a(n) for n = 1..2500
Wikipedia, Primorial number system.
Index entries for sequences related to primorial base.

Cf. A049345, A235168.

Cf. A002113, A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A046807.

max = 6; bases = Prime @ Range[max, 1, -1]; nmax = Times @@ bases - 1; Select[Range[0, nmax], PalindromeQ @ IntegerDigits[#, MixedRadix[bases]] &]

A256085 Non-palindromic balanced numbers in base 5.

Original entry on oeis.org

140, 170, 202, 232, 266, 296, 328, 358, 392, 422, 454, 484, 518, 548, 635, 660, 685, 710, 735, 765, 790, 815, 840, 865, 877, 895, 902, 920, 927, 945, 952, 970, 977, 995, 1007, 1032, 1057, 1082, 1107, 1128, 1137, 1153, 1162, 1178, 1187, 1203, 1212, 1228, 1237, 1261, 1270, 1286, 1295, 1311, 1320, 1336, 1345, 1361, 1370

Offset: 1

M. F. Hasler, Mar 14 2015

Here a number is called balanced if the sum of digits weighted by their arithmetic distance from the "center" is zero. Since palindromes (A029952) are trivially balanced, they are excluded here.

This is the base-5 variant of the decimal version A256075 invented by Eric Angelini. See there, and the base-2 version A256082, for further information and examples.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A256082 - A256089, A256075, A256080, A029952.

filter:= proc(n) local L, m,i;
  L:= convert(n, base, 5);
  m:= (1+nops(L))/2;
add(L[i]*(i-m), i=1..nops(L))=0  and L <> ListTools:-Reverse(L)
end proc:
select(filter, [$1..10000]); # Robert Israel, Nov 04 2024

is(n,b=5,d=digits(n,b),o=(#d+1)/2)=!(vector(#d,i,i-o)*d~)&&d!=Vecrev(d)

A046234 Cubes which are palindromes in base 5.

Original entry on oeis.org

0, 1, 216, 17576, 2000376, 245314376, 30546884376, 3815429734376, 476855468984376, 59605102540234376, 7450592041021484376, 931322860717802734376, 116415328979492333984376

Offset: 1

Patrick De Geest, May 15 1998

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

Intersection of A029952 and A000578.

Cf. A046233.

pb5Q[n_]:=Module[{idn5=IntegerDigits[n,5]},idn5==Reverse[idn5]]; Select[ Range[ 0,49*10^6]^3,pb5Q] (* Harvey P. Dale, Jan 21 2018 *)

a(n) = A046233(n)^3. - Andrew Howroyd, Aug 10 2024

Offset corrected by Andrew Howroyd, Aug 10 2024

A260184 Numbers n written in base 10 that are palindromic in exactly three bases b, 2 <= b <= 10 and not simultaneously bases 2, 4 and 8.

Original entry on oeis.org

9, 10, 21, 40, 55, 80, 85, 100, 130, 154, 164, 178, 191, 203, 235, 242, 255, 257, 273, 282, 292, 300, 328, 400, 455, 585, 656, 819, 910, 2709, 6643, 8200, 14762, 32152, 53235, 74647, 428585, 532900, 1181729, 1405397, 4210945, 5259525, 27711772, 719848917, 43253138565

Offset: 1

Giovanni Resta and Robert G. Wilson v, Jul 17 2015

			273 is in the sequence because 100010001_2 = 101010_3 = 10101_4 = 2043_5 = 1133_6 = 540_7 = 421_8 = 333_9 = 273_10 and three of the bases, namely 2, 4 & 9, yield palindromes.

Cf. A214426, A214425.

(* see A214425 and set all terms as lst, then *)
gQ[n_] := Count[ palQ[n,#] & /@ {2, 4, 8}, True];
Select[ lst, gQ[#] != 3 &]

The intersection of A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955 & A002113 which yields just three members, not simultaneously bases 2, 4 and 8.

A043264 Sum of the digits of the n-th base 5 palindrome.

Original entry on oeis.org

0, 1, 2, 3, 4, 2, 4, 6, 8, 2, 3, 4, 5, 6, 4, 5, 6, 7, 8, 6, 7, 8, 9, 10, 8, 9, 10, 11, 12, 2, 4, 6, 8, 10, 4, 6, 8, 10, 12, 6, 8, 10, 12, 14, 8, 10, 12, 14, 16, 2, 3, 4, 5, 6, 4, 5, 6, 7, 8, 6, 7, 8, 9, 10, 8, 9, 10, 11, 12, 10, 11, 12, 13, 14, 4, 5, 6, 7, 8, 6

Offset: 1

Clark Kimberling

A029952 (base 5 palindromes)

cp's OEIS Frontend

A342725 Numbers that are palindromic in base i-1.

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A350991 Triangular numbers that are palindromes in base 5.

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A261917 Numbers which in base 5 are the sum of two palindromes but are not palindromes themselves.

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A261918 Numbers which in base 5 are neither palindromes nor the sum of two palindromes.

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A333423 Numbers that are palindromes in primorial base.

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A256085 Non-palindromic balanced numbers in base 5.

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A046234 Cubes which are palindromes in base 5.

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A260184 Numbers n written in base 10 that are palindromic in exactly three bases b, 2 <= b <= 10 and not simultaneously bases 2, 4 and 8.

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A043264 Sum of the digits of the n-th base 5 palindrome.

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