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.

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

Author

Robert Israel, Aug 07 2024

Keywords

• sign
• base
• tabl

Comments

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.

Examples

			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, ...
		

Links

• 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.

Crossrefs

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

Programs

Maple

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;