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A049645
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Numbers k such that the square of the number of divisors of k divides the sum of the divisors of k.
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2
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1, 3, 7, 11, 19, 21, 23, 31, 33, 35, 43, 47, 57, 59, 62, 67, 69, 71, 77, 79, 83, 91, 93, 94, 103, 105, 107, 115, 119, 127, 129, 131, 133, 139, 141, 151, 155, 158, 161, 163, 167, 177, 179, 186, 189, 191, 199, 201, 203, 209, 211, 213, 217, 223
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OFFSET
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1,2
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COMMENTS
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Bateman et al. (1981) proved that the asymptotic density of this sequence is 1/2. - Amiram Eldar, Jan 16 2020
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REFERENCES
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Richard G. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004, chapter 2, p. 76.
József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter III, section 51, page 119.
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LINKS
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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.
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FORMULA
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MAPLE
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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: t;
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MATHEMATICA
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Select[Range[1, 250], Mod[DivisorSigma[1, #], DivisorSigma[0, #]^2] == 0 &] (* G. C. Greubel, Dec 06 2017 *)
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PROG
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(PARI) isok(n) = sigma(n) % numdiv(n)^2 == 0; \\ Michel Marcus, Dec 07 2017
(Magma) [k:k in [1..230]| DivisorSigma(1, k) mod (DivisorSigma(0, k))^2 eq 0]; // Marius A. Burtea, Jan 16 2020
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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