A357460 Numbers whose number of deficient divisors is equal to their number of nondeficient divisors.

72, 108, 120, 168, 180, 252, 420, 528, 560, 624, 1188, 1224, 1368, 1400, 1404, 1632, 1656, 1824, 1836, 1960, 1980, 2040, 2052, 2088, 2208, 2232, 2280, 2340, 2484, 2664, 2760, 2772, 2784, 2856, 2952, 2976, 3060, 3096, 3132, 3192, 3200, 3276, 3348, 3384, 3420, 3432

Offset: 1

Author

Amiram Eldar, Sep 29 2022

Keywords

• nonn

Comments

Numbers k such that A080226(k) = A341620(k).

This sequence is infinite: if p >= 17 is a prime then 72*p is a term.

The least odd term of this sequence is a(36126824) = A357461(1) = 3010132125.

Since the number of divisors of any term is even, none of the terms are squares.

The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 1, 10, 131, 1172, 12003, 120647, 1199147, 11992293, 120089446, ... . Apparently, the asymptotic density of this sequence exists and is equal to about 0.012.

Examples

			72 is a term since it has 12 divisors, 6 of them (1, 2, 3, 4, 8 and 9) are deficient and 6 (6, 12, 18, 24, 36 and 72) are not.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A000037 and A005101.

Cf. A080226, A335543, A335544, A341620, A357461, A357462.

Programs

Mathematica

q[n_] := DivisorSum[n, If[DivisorSigma[-1, #] < 2, 1, -1] &] == 0; Select[Range[3500], q]

PARI

is(n) = sumdiv(n, d, if(sigma(d, -1) < 2, 1, -1)) == 0;