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Numbers n such that k(n) = n/tau(n) - sigma(n)/n is an integer.
4

%I #19 Sep 08 2022 08:46:09

%S 1,672,4680,30240,435708,23569920,45532800,4138364160,14182439040,

%T 53798734080,153003540480,403031236608,518666803200

%N Numbers n such that k(n) = n/tau(n) - sigma(n)/n is an integer.

%C Numbers n such that A245776(n) / A245777(n) = (n / A000005(n) - A000203(n) / n) is an integer.

%C Sequence of integers k(n): 0, 25, 94, 311, 4031, 73652, 118571, …

%C Conjecture: subsequence of A216793.

%C Refactorable multiply-perfect numbers (A245782) are members of this sequence.

%C a(14) > 10^13. - _Giovanni Resta_, Jul 13 2015

%C The numbers 13661860101120 and 740344994887680 are also terms. - _Giovanni Resta_, Nov 14 2019

%F A245777(a(n)) = 1.

%e 672 is in sequence because 672 / tau(672) - sigma(672) / 672 = 672 / 24 - 2016 / 672 = 25 (integer).

%p select(n -> type(n/numtheory:-tau(n) - numtheory:-sigma(n)/n,integer), [$1..10^8]); # _Robert Israel_, Aug 03 2014

%o (Magma) [n: n in [1..100000] | (Denominator((n/(#[d: d in Divisors(n)])) - (SumOfDivisors(n)/n)) eq 1)]

%o (PARI)

%o for(n=1,10^8,s=n/numdiv(n);t=sigma(n)/n;if(floor(s-t)==s-t,print1(n,", "))) \\ _Derek Orr_, Aug 01 2014

%Y Cf. A000005, A000203, A216793, A245776, A245777, A245779, A245782.

%K nonn,more

%O 1,2

%A _Jaroslav Krizek_, Aug 01 2014

%E a(8)-a(13) from _Giovanni Resta_, Jul 13 2015