A169655 Numbers n such that 2^n 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

1,3

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.

References

V. Shevelev, Compact integers and factorials, Acta Arith., 126.3 (2007), 195-236.

Links

Table of n, a(n) for n=1..66.

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 *)

Crossrefs

Cf. A050376, A054861.

Sequence in context: A264763 A126167 A026260 * A286489 A002153 A047607

Adjacent sequences: A169652 A169653 A169654 * A169656 A169657 A169658

Keyword

nonn

Author

Vladimir Shevelev, Apr 05 2010

Extensions

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

Status

approved