A029954 -id:A029954 - 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.

A350992 Triangular numbers that are palindromes in base 7.

Original entry on oeis.org

0, 1, 3, 6, 78, 171, 300, 2850, 8256, 9453, 14706, 120786, 208335, 399171, 405450, 416328, 448878, 720600, 5877306, 6046503, 6835753, 9350650, 10122750, 18431556, 19130205, 22596003, 35309406, 499169406, 934394835, 969430528, 999335571, 1059265378, 1730160900

Offset: 1

Amiram Eldar, Jan 28 2022

This sequence is infinite since A000217((7^k-1)/2) is a term for all k >= 0 (Trigg, 1974).

			78 is a term since 78 = A000217(12) is a triangular number and also a palindromic number in base 7: 78 = 141_7.
171 is a term since 171 = A000217(18) is a triangular number and also a palindromic number in base 7: 171 = 333_7.

Amiram Eldar, Table of n, a(n) for n = 1..103
Charles W. Trigg, Infinite sequences of palindromic triangular numbers, The Fibonacci Quarterly, Vol. 12, No. 2 (1974), pp. 209-212.

Intersection of A000217 and A029954.
The septenary version of A003098.
Cf. A120741, A350987, A350990, A350991, A350993.

t[n_] := n*(n + 1)/2; Select[t /@ Range[0, 3*10^5], PalindromeQ[IntegerDigits[#, 7]] &]

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]] &]

A256087 Non-palindromic balanced numbers in base 7.

Original entry on oeis.org

364, 420, 476, 492, 532, 548, 604, 640, 660, 708, 728, 764, 820, 836, 876, 892, 948, 984, 1004, 1052, 1072, 1108, 1164, 1180, 1220, 1236, 1292, 1328, 1348, 1396, 1416, 1452, 1508, 1524, 1564, 1580, 1636, 1672, 1692, 1740, 1760, 1796, 1852, 1868, 1908, 1924, 1980

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 (A029954) are trivially balanced, they are excluded here.

This is the base-7 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, A029954.

filter:= proc(n) local L, m,i;
  L:= convert(n, base, 7);
  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=7,d=digits(n,b),o=(#d+1)/2)=!(vector(#d,i,i-o)*d~)&&d!=Vecrev(d)

A043016 Base-7 palindromes that start with 2.

Original entry on oeis.org

2, 16, 100, 107, 114, 121, 128, 135, 142, 688, 744, 800, 856, 912, 968, 1024, 4804, 4853, 4902, 4951, 5000, 5049, 5098, 5154, 5203, 5252, 5301, 5350, 5399, 5448, 5504, 5553, 5602, 5651, 5700, 5749, 5798, 5854, 5903, 5952, 6001, 6050, 6099, 6148, 6204, 6253

Offset: 1

Clark Kimberling

Robert P. P. McKone, Table of n, a(n) for n = 1..800

Cf. A007093, A029954, A043060.

okQ[n_] := Module[{idn = IntegerDigits[n, 7]}, First[idn] == 2 && FromDigits[ IntegerDigits[ n, 7]] == FromDigits[Reverse[idn]]]; Select[Range[6253], okQ] // Parallelize (* Robert P. P. McKone, Aug 22 2021, after Harvey P. Dale *)

More terms from Robert P. P. McKone, Aug 22 2021

A046238 Cubes which are palindromes in base 7.

Original entry on oeis.org

0, 1, 8, 64, 512, 4096, 125000, 1000000, 8000000, 40707584, 325660672, 2605285376, 13858588808, 110868710464, 886949683712, 4748408986112, 37987271888896, 303898175111168, 1628455122125000, 13027640977000000

Offset: 1

Patrick De Geest, May 15 1998

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

Intersection of A029954 and A000578.

Cf. A046237.

Select[Range[0,250000]^3,IntegerDigits[#,7]==Reverse[IntegerDigits[#,7]]&] (* Harvey P. Dale, Oct 11 2022 *)

a(n) = A046237(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.

A043266 Sum of the digits of the n-th base 7 palindrome.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

A029954 (base 7 palindromes)

A319595 Numbers in base 10 that are palindromic in bases 3, 7, and 9.

Original entry on oeis.org

0, 1, 2, 4, 8, 40, 100, 164, 328, 400, 8200, 14762, 532900

Offset: 1

Jeremias M. Gomes, Sep 23 2018

Intersection of A014190, A029954, and A029955.

No other terms < 10^17. It is likely that there are no more terms. - Chai Wah Wu, Mar 20 2020

			400 = 112211_3 = 1111_7 = 484_9.

Cf. A014190 (base 3), A029954 (base 7), and A029955 (base 9).

[n for n in (0..100000) if Word(n.digits(3)).is_palindrome() and Word(n.digits(7)).is_palindrome() and Word(n.digits(9)).is_palindrome()]

cp's OEIS Frontend

A342725 Numbers that are palindromic in base i-1.

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A350992 Triangular numbers that are palindromes in base 7.

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A333423 Numbers that are palindromes in primorial base.

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A256087 Non-palindromic balanced numbers in base 7.

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A043016 Base-7 palindromes that start with 2.

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A046238 Cubes which are palindromes in base 7.

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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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A043266 Sum of the digits of the n-th base 7 palindrome.

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A319595 Numbers in base 10 that are palindromic in bases 3, 7, and 9.

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