A277553 Numbers k such that the sum of the divisors of k is divisible by the number of divisors of k, and the sum of the squares of the divisors of k is divisible by the sum of the divisors of k.

1, 20, 49, 169, 361, 500, 605, 961, 980, 1025, 1369, 1445, 1700, 1849, 2645, 3380, 3721, 4205, 4352, 4489, 4693, 5329, 6241, 7220, 8228, 8281, 8405, 9409, 9425, 10609, 11045, 11849, 11881, 12493, 12500, 14045, 14580, 14641, 15125, 16129, 17405, 17689, 18785

Offset: 1

Author

Harvey P. Dale, Oct 19 2016

Keywords

• nonn

Comments

Numbers k such that A000005(k) divides A000203(k), and A000203(k) divides A001157(k).

Examples

			1369 has 3 divisors which sum to 1407; 1407 is divisible by 3; the sum of the squares of the divisors of 1369 is 1875531 which is divisible by 1407; so 1369 is a term of the sequence.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Crossrefs

Cf. A000005, A000203, A001157.

Intersection of A003601 and A020487. See also A020486.

Programs

Mathematica

Select[Range[50000],Divisible[DivisorSigma[1,#],DivisorSigma[0,#]] && Divisible[ DivisorSigma[2,#], DivisorSigma[1,#]]&]

PARI

isok(k) = {my(f = factor(k), d = numdiv(f), s = sigma(f), s2 = sigma(f, 2)); !(s % d) && !(s2 % s);} \\ Amiram Eldar, Jan 25 2025