A078541 Refactorable numbers x, such that quotient x/A000005(x) equals a power of 2.

1, 2, 8, 12, 36, 80, 96, 128, 288, 448, 2560, 6144, 11264, 18432, 32768, 53248, 57344, 245760, 737280, 1114112, 2621440, 4980736, 22020096, 23068672, 25165824, 66060288, 75497472, 96468992, 436207616, 939524096, 1258291200, 1811939328, 2147483648, 3774873600

Offset: 1

Author

Labos Elemer, Dec 04 2002

Keywords

• nonn

Comments

This sequence is infinite since it contains all the terms of A058891. Also, 3*2^A083329(k) is a term for all k >= 1. - Amiram Eldar, Jan 27 2025

Examples

			a(6)=80: tau(80)=10, quotient=80/10=8=2^3; certain powers of 2 do not appear as quotient, like 64, 1024, 16384.
		

Links

• Donovan Johnson, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A000005, A033950, A036763, A051278, A051279, A051280, A058891, A083329.

Programs

Mathematica

Do[s=n/DivisorSigma[0, n]; If[IntegerQ[Log[2, s]], Print[{n, s, n/s}]], {n, 1, 1000000000}]

PARI

isok(k) = my(r = k/numdiv(k)); denominator(r) == 1 && r >> valuation(r, 2) == 1; \\ Amiram Eldar, Jan 27 2025

Formula

a(n)/tau(a(n))=2^s with some s, tau()=A000005().

Extensions

a(29)-a(34) from Donovan Johnson, Jun 04 2011