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A169655 Numbers k such that 2^k is in A054861.

%I A169655 #31 Jan 23 2025 10:23:37
%S A169655 0,1,2,3,5,8,9,10,11,12,13,14,15,17,18,19,21,23,24,26,27,28,29,31,32,
%T A169655 33,34,35,36,37,41,42,43,45,46,47,49,53,54,55,56,58,59,60,62,64,65,67,
%U A169655 69,70,71,72,73,74,75,76,78,79,82,84,85,87,88,89,91,93
%N A169655 Numbers k such that 2^k is in A054861.
%C A169655 For a prime p, we call a number p-compact if the exponent of p in the factorization of the number is a power of two. However, if m=k!, then not all exponents of p of the form 2^t are possible. The sequence lists numbers t in possible exponents of the form 2^t of 3 in 3-compact factorials k!The question of description of the p-compact factorials is interesting since there exists only finite set of factorials compact over both 2 and an arbitrary fixed odd prime (cf. A177436). On the other hand, there exist infinitely many 2-compact factorials. However, up to now it is unknown, whether exist infinitely many p-compact factorials for a fixed odd prime p. It is expected that the answer to be in affirmative.
%H A169655 Vladimir Shevelev, <a href="https://eudml.org/doc/277854">Compact integers and factorials</a>, Acta Arithmetica 126 (2007), no. 3, 195-236.
%t A169655 A054861 := (Plus @@ Floor[#/3^Range[Length[IntegerDigits[#, 3]] - 1]] &);DeleteCases[Table[n - n Sign[2^n - A054861[2^(n + 1) + NestWhile[# + 1 &, 1, 2^n - A054861[2^(n + 1) + #] >= 0 &]  - 1]],{n, 1, 125}], 0] (* _Peter J. C. Moses_, Apr 10 2012 *)
%Y A169655 Cf. A050376, A054861.
%K A169655 nonn
%O A169655 1,3
%A A169655 _Vladimir Shevelev_, Apr 05 2010
%E A169655 More terms given by _Peter J. C. Moses_, Apr 07 2012