cp's OEIS Frontend

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.

%I A375258 #30 Feb 19 2025 12:12:03
%S A375258 1,2,3,-1,6,81,8,5,28,27,-1,20,7,30,2187,128,17,14,15,244,243,-1,68,
%T A375258 25,46,31,246,-1,512,8193,26,23,94,63,6570,19683,-1,80,131,78,47,126,
%U A375258 247,2430,59049,2048,1025,134,53,62,95,254,255,19926,531441,-1,2050,161,212,79,222,127,766,2431
%N 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.
%C A375258 T(k,n) is the least positive integer x, if it exists, such that A000120(x) = k and A053735(x) = n.
%C A375258 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.
%C A375258 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.
%H A375258 Robert Israel, <a href="/A375258/b375258.txt">Table of n, a(n) for n = 1..253</a> (first 22 antidiagonals)
%H A375258 Vassil S. Dimitrov and Everett W. Howe, <a href="https://arxiv.org/abs/2105.06440">Powers of 3 with few nonzero bits and a conjecture of Erdős</a>, arXiv:2105.06440 [math.NT], 2021.
%e A375258 Array starts
%e A375258      1,     2,    -1,     8,    -1,   128,    -1,   512, ...
%e A375258      3,     6,     5,    20,    17,    68,  8193,    80, ...
%e A375258     81,    28,     7,    14,    25,    26,   131,   134, ...
%e A375258     27,    30,    15,    46,    23,    78,    53,   212, ...
%e A375258   2187,   244,    31,    94,    47,    62,    79,   158, ...
%e A375258    243,   246,    63,   126,    95,   222,   125,   238, ...
%e A375258     -1,  6570,   247,   254,   127,   382,   223,   446, ...
%e A375258  19683,  2430,   255,   766,   507,   510,   383,   958, ...
%p A375258 T:= Matrix(8,8,-1):
%p A375258 for x from 1 to 10^5 do
%p A375258   k:= convert(convert(x,base,2),`+`);
%p A375258   n:= convert(convert(x,base,3),`+`);
%p A375258   if k <= 8 and n <= 8 and T[k,n] = -1 then T[k,n]:= x; fi
%p A375258 od:
%p A375258 T;
%Y A375258 Cf. A000120, A011754, A037301, A053735, A375257.
%K A375258 sign,base,tabl
%O A375258 1,2
%A A375258 _Robert Israel_, Aug 07 2024