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A331666 Refactorable numbers (A033950) that are simultaneously arithmetic (A003601) and harmonic (A001599). 0
1, 672, 30240, 23569920, 45532800, 164989440, 447828480, 623397600, 1381161600, 1862023680, 2144862720, 3134799360, 3831421440, 13584130560, 14182439040, 16569653760, 21943595520, 22933532160, 34482792960, 35032757760, 40752391680, 53621568000, 56481384960 (list; graph; refs; listen; history; text; internal format)
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, ...).

Intersection of A033950 and A007340.

Multiply-perfect numbers from this sequence are in A047728.

LINKS

Table of n, a(n) for n=1..23.

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))]

CROSSREFS

Cf. A000005, A000203, A001599, A003601, A007340, A033950, 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

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Last modified November 28 16:04 EST 2021. Contains 349413 sequences. (Running on oeis4.)