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A374904
Numbers whose divisors have an integer mean number of divisors.
5
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
1,2
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.
LINKS
EXAMPLE
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.
MATHEMATICA
f[p_, e_] := (e + 2)/2; q[1] = True; q[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[300], q]
PROG
(PARI) is(n) = denominator(vecprod(apply(x -> x/2 +1, factor(n)[, 2]))) == 1;
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Jul 23 2024
STATUS
approved