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

1,2

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).

Links

Table of n, a(n) for n=1..58.

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A000005, A000203, A062068.

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

Sequence in context: A018328 A018280 A068927 * A018261 A306201 A218935

Adjacent sequences: A336610 A336611 A336612 * A336614 A336615 A336616

Keyword

nonn

Author

Bernard Schott, Jul 29 2020

Status

approved