A336687 Numbers m such that tau(sigma(m)) and sigma(tau(m)) both divide m, where tau(m) is the number of divisors function (A000005) and sigma(m) is the sum of divisors function (A000203).

1, 3, 4, 12, 64, 84, 140, 144, 162, 192, 336, 360, 420, 468, 480, 576, 600, 644, 720, 780, 1008, 1344, 1512, 1584, 1600, 1740, 1872, 2160, 2240, 2448, 2592, 2736, 2880, 2884, 3136, 3240, 3888, 4032, 4158, 4228, 4464, 4608, 4800, 5040, 5115, 5184, 5328, 5670, 6060, 6192, 6336

Offset

1,2

Comments

Conjecture: The only m such that m = tau(sigma(m))*sigma(tau(m)) are in {1,468,3240}. Verified for m up to 1*10^9. - Ivan N. Ianakiev, Aug 06 2020

Links

Table of n, a(n) for n=1..51.

Example

For 84: tau(84) = 12 and sigma(12) = 28 with 84/28 = 3. Also, sigma(84) = 224 and tau(224) = 12 with 84/12 = 7. Hence, 84 is a term.

Maple

with(numtheory):
filter:= m-> irem(m, tau(sigma(m)))=0 and irem(m, sigma(tau(m)))=0:
select(filter, [$1..7000])[];

Mathematica

Select[Range[6400], And @@ Divisible[#, {DivisorSigma[0, DivisorSigma[1, #]], DivisorSigma[1, DivisorSigma[0, #]]}] &] (* Amiram Eldar, Jul 31 2020 *)

Prog

(PARI) isok(m) = !(m % numdiv(sigma(m))) && !(m % sigma(numdiv(m))); \\ Michel Marcus, Aug 02 2020

Crossrefs

Cf. A000005, A000203, A062068, A062069.

Intersection of A336612 and A336613.

Sequence in context: A299809 A109771 A052626 * A298115 A122903 A059792

Adjacent sequences: A336684 A336685 A336686 * A336688 A336689 A336690

Keyword

nonn

Author

Bernard Schott, Jul 31 2020

Status

approved