A037301 -id:A037301 - cp's OEIS frontend

A037310 Numbers whose base-3 and base-5 expansions have the same digit sum.

1, 2, 6, 7, 8, 10, 11, 15, 16, 17, 20, 30, 31, 32, 40, 41, 55, 56, 60, 61, 62, 65, 70, 71, 78, 79, 100, 101, 105, 106, 107, 129, 135, 136, 137, 141, 142, 143, 145, 146, 153, 154, 159, 170, 177, 178, 179, 180, 181, 182, 186, 187, 188

Offset: 1

Clark Kimberling

Also, numbers n such that the exponent of the largest power of 3 dividing n! is exactly twice the exponent of the largest power of 5 dividing n!. - Ivan Neretin, May 21 2015

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A037301 (similar, based upon 2 and 3).

Cf. A053735, A053824.

select(t -> convert(convert(t,base,3),`+`)=convert(convert(t,base,5),`+`), [$1..1000]); # Robert Israel, May 21 2015

Select[Range[200],Total[IntegerDigits[#,3]]==Total[IntegerDigits[#,5]]&] (* Harvey P. Dale, Jun 06 2016 *)

is(n)=sumdigits(n,3)==sumdigits(n,5) \\ Charles R Greathouse IV, May 21 2015

A053735(a(n)) = A053824(a(n)). - Robert Israel, May 21 2015

A212283 First a(n) > 1 whose sum of digits is the same in base 2 as in base n.

Original entry on oeis.org

2, 6, 4, 6, 12, 21, 8, 10, 20, 12, 14, 172, 30, 46, 16, 18, 36, 20, 22, 126, 46, 24, 26, 126, 28, 30, 58, 60, 120, 126, 32, 34, 68, 36, 38, 185, 78, 40, 42, 126, 44, 46, 90, 92, 138, 48, 50, 246, 52, 54, 106, 108, 56, 58, 114, 60, 62, 120, 182, 126, 188, 378

Offset: 2

Stanislav Sykora, May 08 2012

Theoretically, there might exist an n for which there is no solution, in which case a(n) would be set to 0 by convention; however, no such case was found so far. Problem: does it exist?

			Example: a(13) = 172 because 172 is the first number >1 such that its expansions in base 2 (10101100) and in base 13 (103) have the same sum of digits, namely 4.

Stanislav Sykora, Table of n, a(n) for n = 2..10000

Cf. A037301, A212222.

sdn[n_]:=Module[{a=2},While[Total[IntegerDigits[a,2]]!=Total[ IntegerDigits[ a,n]], a++];a]; Array[sdn,70,2] (* Harvey P. Dale, May 29 2013 *)

A345296 Integers whose sum of digits in base b is the same for every prime b up to 17.

Original entry on oeis.org

0, 1, 70911040973874056146188543, 77332999599545910254098143, 139857575920160383360253101

Offset: 1

Thomas König, Jun 13 2021

This is a subset of A335839 for bases 2,3,5,11,13, which is a subset of A212222 for bases 2, 3, 5, 7, 11, which is a subset of A135127 for bases 2, 3, 5, 7, which is a subset of A135121 for bases 2, 3, 5, which is a subset of A037301 for bases 2, 3. The third term also occurs in A212223.

Based on a computer search, the next term is believed to be larger than 2.1e28. - Thomas König, Dec 08 2024

			77332999599545910254098143 = 11111111110111111001100100111011111011111110101111110010001010111101111101011011011111_2 =
1022220111022022121010102021222111100222120112011112120_3 = 10124120314223101043140143200022120033_5 = 3300561310042202241132326120022_7 = 7940063801000011830000282_11 = 1B101304100834600A304201_13 = 120802053643008116067_17. In these bases, the sum of digits is 63, so 77332999599545910254098143 is a term.

Cf. A212223, A212222, A135127, A135121, A037301, A335839.

a(5) from Thomas König, Dec 08 2024

A375258 Array read by antidiagonals: T(k,n) is the least positive integer whose sum of base-2 digits is k and sum of base-3 digits is n, or -1 if there is none.

Original entry on oeis.org

1, 2, 3, -1, 6, 81, 8, 5, 28, 27, -1, 20, 7, 30, 2187, 128, 17, 14, 15, 244, 243, -1, 68, 25, 46, 31, 246, -1, 512, 8193, 26, 23, 94, 63, 6570, 19683, -1, 80, 131, 78, 47, 126, 247, 2430, 59049, 2048, 1025, 134, 53, 62, 95, 254, 255, 19926, 531441, -1, 2050, 161, 212, 79, 222, 127, 766, 2431

Offset: 1

Robert Israel, Aug 07 2024

T(k,n) is the least positive integer x, if it exists, such that A000120(x) = k and A053735(x) = n.
T(k,n) == n (mod 2) unless T(k,n) = -1, since A053735(x) == x (mod 2). In particular, T(1, n) = -1 if n >= 3 is odd.
Dimitrov and Howe prove that for n > 25, the sum of binary digits of 3^n is > 22.  In particular, this implies T(7,1) = T(12,1) = T(21,1) = -1, since none of the first 25 powers of 3 work.

			Array starts
     1,     2,    -1,     8,    -1,   128,    -1,   512, ...
     3,     6,     5,    20,    17,    68,  8193,    80, ...
    81,    28,     7,    14,    25,    26,   131,   134, ...
    27,    30,    15,    46,    23,    78,    53,   212, ...
  2187,   244,    31,    94,    47,    62,    79,   158, ...
   243,   246,    63,   126,    95,   222,   125,   238, ...
    -1,  6570,   247,   254,   127,   382,   223,   446, ...
 19683,  2430,   255,   766,   507,   510,   383,   958, ...

Robert Israel, Table of n, a(n) for n = 1..253 (first 22 antidiagonals)
Vassil S. Dimitrov and Everett W. Howe, Powers of 3 with few nonzero bits and a conjecture of Erdős, arXiv:2105.06440 [math.NT], 2021.

Cf. A000120, A011754, A037301, A053735, A375257.

T:= Matrix(8,8,-1):
for x from 1 to 10^5 do
  k:= convert(convert(x,base,2),`+`);
  n:= convert(convert(x,base,3),`+`);
  if k <= 8 and n <= 8 and T[k,n] = -1 then T[k,n]:= x; fi
od:
T;

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A037310 Numbers whose base-3 and base-5 expansions have the same digit sum.

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A212283 First a(n) > 1 whose sum of digits is the same in base 2 as in base n.

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A345296 Integers whose sum of digits in base b is the same for every prime b up to 17.

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A375258 Array read by antidiagonals: T(k,n) is the least positive integer whose sum of base-2 digits is k and sum of base-3 digits is n, or -1 if there is none.

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