A374904 Numbers whose divisors have an integer mean number of divisors.

1, 4, 9, 12, 16, 18, 20, 25, 28, 36, 44, 45, 49, 50, 52, 63, 64, 68, 72, 75, 76, 81, 92, 98, 99, 100, 108, 116, 117, 121, 124, 144, 147, 148, 153, 164, 169, 171, 172, 175, 180, 188, 192, 196, 200, 207, 212, 225, 236, 242, 244, 245, 252, 256, 261, 268, 275, 279

Offset: 1

Author

Amiram Eldar, Jul 23 2024

Keywords

• nonn
• easy

Comments

Numbers k such that A000005(k) | A007425(k).

Numbers k such that A374903(k) = 1.

If k is a term then all the numbers with the same prime signature as k are terms. The least terms of each prime signature are in A374905.

If {e_i} are the exponents in the prime factorization of k, then k is a term if and only if Product_{i} (e_i/2 + 1) is an integer.

1 is the only squarefree (A005117) term.

All the squares are terms.

Examples

			4 is a term since it has 3 divisors, 1, 2 and 4, their numbers of divisors are 1, 2 and 3, and their mean is (1 + 2 + 3)/3 = 2 which is an integer.
		

Links

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

Crossrefs

Cf. A000005, A005117, A007425, A374902, A374903.

Programs

Mathematica

f[p_, e_] := (e + 2)/2; q[1] = True; q[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[300], q]

PARI

is(n) = denominator(vecprod(apply(x -> x/2 +1, factor(n)[, 2]))) == 1;