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

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, 33, 34, 35, 36, 37, 41, 42, 43, 45, 46, 47, 49, 53, 54, 55, 56, 58, 59, 60, 62, 64, 65, 67, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 82, 84, 85, 87, 88, 89, 91, 93

Offset: 1

Author

Vladimir Shevelev, Apr 05 2010

Keywords

• nonn

Comments

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.

Links

• Vladimir Shevelev, Compact integers and factorials, Acta Arithmetica 126 (2007), no. 3, 195-236.

Crossrefs

Cf. A050376, A054861.

Programs

Mathematica

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 *)

Extensions

More terms given by Peter J. C. Moses, Apr 07 2012