A331666 Refactorable numbers (A033950) that are simultaneously arithmetic (A003601) and harmonic (A001599).

1, 672, 30240, 23569920, 45532800, 164989440, 447828480, 623397600, 1381161600, 1862023680, 2144862720, 3134799360, 3831421440, 13584130560, 14182439040, 16569653760, 21943595520, 22933532160, 34482792960, 35032757760, 40752391680, 53621568000, 56481384960

Offset

1,2

Comments

Numbers m such that all values of sigma(m)/tau(m), m/tau(m) and m * tau(m)/sigma(m) are any integers (f, g, and h respectively).

Corresponding values of numbers f, g and h: (1, 84, 1260, 294624, 474300, 1178496, 2946240, 3298400, 5754840, 11784960, ...); (1, 28, 315, 73656, 118575, 257796, 699732, 721525, 1198925, 2909412, 1675674, ...); (1, 8, 24, 80, 96, 140, 152, 189, 240, 158, 260, 266, 220, 380, 384, 296, 392, ...).

Multiply-perfect numbers from this sequence are in A047728.

Links

Amiram Eldar, Table of n, a(n) for n = 1..118 (terms below 10^14)

Example

For m = 672, f = sigma(m)/tau(m) = 2016/24 = 84; g = m/tau(m) = 672/24 = 28; h = m * tau(m)/sigma(m) = 672*24/2016 = 8.

Mathematica

Select[Range[3*10^7], Divisible[#, (d = DivisorSigma[0, #])] && Divisible[(s = DivisorSigma[1, #]), d] && Divisible[#*d, s] &] (* Amiram Eldar, Jan 24 2020 *)

Prog

(Magma) [m: m in [1..10^6] | IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m)) and  IsIntegral(m / NumberOfDivisors(m)) and IsIntegral(m * NumberOfDivisors(m) / SumOfDivisors(m))]
(PARI) is(k) = {my(f = factor(k), s = sigma(f), d = numdiv(f)); !(k % d) && !(s % d) && !((k * d) % s) ; } \\ Amiram Eldar, May 09 2024

Crossrefs

Intersection of A033950 and A007340.

Cf. A000005, A000203, A001599, A003601, A047728.

Sequence in context: A234481 A234476 A340864 * A245782 A047728 A297123

Adjacent sequences: A331663 A331664 A331665 * A331667 A331668 A331669

Keyword

nonn

Author

Jaroslav Krizek, Jan 23 2020

Status

approved