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

1,2

Comments

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

Links

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

Cf. A000005, A000203, A001157.

Intersection of A003601 and A020487. See also A020486.

Sequence in context: A053245 A115882 A303295 * A260093 A304832 A007589

Adjacent sequences: A277550 A277551 A277552 * A277554 A277555 A277556

Keyword

nonn

Author

Harvey P. Dale, Oct 19 2016

Status

approved