A336613 Numbers m such that tau(sigma(m)) divides m, where tau(m) is the number of divisors function (A000005) and sigma(m) is the sum of divisors function (A000203).

1, 2, 3, 4, 6, 8, 12, 16, 24, 36, 48, 64, 72, 80, 81, 84, 100, 112, 120, 128, 140, 144, 156, 160, 162, 168, 192, 198, 200, 208, 210, 216, 240, 256, 270, 288, 300, 320, 324, 336, 357, 360, 368, 384, 390, 420, 432, 448, 464, 468, 480, 512, 560, 576, 592, 600, 624, 630

Offset: 1

Author

Bernard Schott, Jul 29 2020

Keywords

• nonn

Comments

Two subsets of terms:

1) If 2^p - 1 is a Mersenne prime (p is in A000043 and 2^p-1 is in A000668), then m = 2^(p-1) is a term that belongs to A019279: the even superperfect numbers (2, 4, 16, 64, 4096, ...). Proof: sigma(m) = 1+2+...+2^(p-1) = 2^p - 1 that is a Mersenne prime so tau(2^p-1) = 2 that divides m = 2^(p-1); indeed, m/tau(sigma(m)) = 2^(p-2).

2) If m = 2^(p-1) is a term as above, then 3*m is another term (see example) with 3*m/tau(sigma(3*m)) = 2^(p-2).

Examples

			48 = 2^4 * 3, so, sigma(48) = sigma(2^4) * sigma(3) = (2^5 - 1) * (1+3) = 31 * 4 = 124; then, tau(2^2 * 31) = tau(4) * tau(31) = 3 * 2 = 6, and  48/6 = 8 = 2^3, hence 48 is a term.
		

Crossrefs

Cf. A000005, A000203, A062068.

Cf. A019279 (subsequence), A336612 (sigma(tau(m)) divides m).

Programs

Maple

with(numtheory) filter:= m -> m/tau(sigma(m)) = floor(m/tau(sigma(m))) : select(filter, [$1..650]);

Mathematica

Select[Range[630], Divisible[#, DivisorSigma[0, DivisorSigma[1, #]]] &] (* Amiram Eldar, Jul 30 2020 *)

PARI

isok(m) = !(m % numdiv(sigma(m))); \\ Michel Marcus, Jul 30 2020