A336613 -id:A336613 - cp's OEIS frontend

A336612 Numbers m such that sigma(tau(m)) divides 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, 14, 21, 30, 35, 64, 77, 84, 91, 105, 119, 133, 135, 140, 144, 161, 162, 165, 192, 195, 203, 217, 224, 255, 259, 285, 287, 301, 308, 329, 336, 343, 345, 360, 364, 371, 375, 392, 413, 420, 427, 435, 465, 468, 469, 476, 480, 497, 511, 532, 540, 553, 555, 576

Offset: 1

Bernard Schott, Jul 27 2020

nonn

Every 7*p with p prime <> 7 is a term because 7*p / sigma(tau(7*p)) = p (see example).

			35 = 7 * 5, tau(35) = 4, sigma(tau(35)) = sigma(4) = 4 + 2 + 1 = 7 and 35/7 = 5 hence 35 is a term.

Cf. A000005, A000203, A062069.

Cf. A336613 (tau(sigma(m)) divides m).

with(numtheory) filter:= m -> m/sigma(tau(m)) = floor(m/sigma(tau(m))) : select(filter, [$1..600]);

Select[Range[600], Divisible[#, DivisorSigma[1, DivisorSigma[0, #]]] &] (* Amiram Eldar, Jul 27 2020 *)

isok(m) = !(m % sigma(numdiv(m))); \\ Michel Marcus, Jul 29 2020

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).

Original entry on oeis.org

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

Bernard Schott, Jul 31 2020

nonn

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

			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.

Cf. A000005, A000203, A062068, A062069.

Intersection of A336612 and A336613.

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

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

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

cp's OEIS Frontend

A336612 Numbers m such that sigma(tau(m)) divides m, where tau(m) is the number of divisors function (A000005) and sigma(m) is the sum of divisors function (A000203).

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