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A049692
Numbers k such that the square of the number of divisors of k does not divide the sum of the divisors of k.
2
2, 4, 5, 6, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 20, 22, 24, 25, 26, 27, 28, 29, 30, 32, 34, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 60, 61, 63, 64, 65, 66, 68, 70, 72, 73, 74, 75, 76, 78, 80, 81, 82, 84, 85, 86
OFFSET
1,1
COMMENTS
The asymptotic density of this sequence is 1/2 (Bateman et al., 1981). - Amiram Eldar, Apr 25 2024
REFERENCES
József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter III, p. 119, section III.51.
LINKS
Paul T. Bateman, Paul Erdős, Carl Pomerance and E. G. Straus, The arithmetic mean of the divisors of an integer, in Marvin I. Knopp (ed.), Analytic Number Theory, Proceedings of a Conference Held at Temple University, Philadelphia, May 12-15, 1980, Lecture Notes in Mathematics, Vol. 899, Springer, Berlin - New York, 1981, pp. 197-220, alternative link.
MAPLE
with(numtheory): t := [ ]: f := [ ]: for n from 1 to 500 do if sigma(n) mod sigma[ 0 ](n)^2 = 0 then t := [ op(t), n ] else f := [ op(f), n ]; fi; od: f;
MATHEMATICA
Select[Range[1, 100], !Divisible[DivisorSigma[1, #], DivisorSigma[0, #]^2] &] (* Amiram Eldar, Apr 25 2024 *)
PROG
(PARI) is(n) = {my(f = factor(n)); sigma(f) % numdiv(f)^2 > 0; } \\ Amiram Eldar, Apr 25 2024
CROSSREFS
Cf. A000005, A000203, A003601, A049642, A049645 (complement).
Sequence in context: A081694 A355890 A145395 * A047839 A285601 A139255
KEYWORD
nonn
STATUS
approved