A245778 Numbers n such that k(n) = n/tau(n) - sigma(n)/n is an integer.

1, 672, 4680, 30240, 435708, 23569920, 45532800, 4138364160, 14182439040, 53798734080, 153003540480, 403031236608, 518666803200

Offset: 1

Author

Jaroslav Krizek, Aug 01 2014

Keywords

• nonn
• more

Comments

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

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

Conjecture: subsequence of A216793.

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

a(14) > 10^13. - Giovanni Resta, Jul 13 2015

The numbers 13661860101120 and 740344994887680 are also terms. - Giovanni Resta, Nov 14 2019

Examples

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

Crossrefs

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

Programs

Magma

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

Maple

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

PARI

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

Formula

A245777(a(n)) = 1.

Extensions

a(8)-a(13) from Giovanni Resta, Jul 13 2015