A331666 - OEIS

%I #17 May 09 2024 02:58:12

%S 1,672,30240,23569920,45532800,164989440,447828480,623397600,

%T 1381161600,1862023680,2144862720,3134799360,3831421440,13584130560,

%U 14182439040,16569653760,21943595520,22933532160,34482792960,35032757760,40752391680,53621568000,56481384960

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

%C 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).

%C 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, ...).

%C Multiply-perfect numbers from this sequence are in A047728.

%H Amiram Eldar, <a href="/A331666/b331666.txt">Table of n, a(n) for n = 1..118</a> (terms below 10^14)

%e 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.

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

%o (Magma) [m: m in [1..10^6] | IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m)) and  IsIntegral(m / NumberOfDivisors(m)) and IsIntegral(m * NumberOfDivisors(m) / SumOfDivisors(m))]

%o (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

%Y Intersection of A033950 and A007340.

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

%K nonn

%O 1,2

%A _Jaroslav Krizek_, Jan 23 2020