A187771 Numbers whose sum of divisors is the cube of the sum of its prime divisors.

245180, 612408, 639198, 1698862, 1721182, 5154168, 7824284, 15817596, 20441848, 25969788, 27688078, 28404862, 35860609, 67149432, 77378782, 91397838, 96462862, 179302264, 191550135, 289772221, 306901244, 311657084, 392802179, 441839706, 572673855, 652117774, 988918364

Offset

1,1

Comments

This sequence and A187824 and A187761 are winners in the contest held at the 2013 AMS/MAA Joint Mathematics Meetings. - T. D. Noe, Jan 14 2013

The identity sigma(k) = (sopf(k))^m only occurs for m = 3 (this sequence) in the given range, however it is likely that it also occurs for other powers m in higher numbers.

The smallest k such that sigma(k) = sopf(k)^m, for m=4,5,6 are 1056331752 (A221262), 213556659624 (A221263) and 45770980141656, respectively. - Giovanni Resta, Jan 07 2013

Prime divisors are taken without multiplicity. - Harvey P. Dale, Dec 17 2016

References

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.

Links

Donovan Johnson and Robert Gerbicz, Table of n, a(n) for n = 1..1105 (first 100 terms from Donovan Johnson)

W. Sierpinski, Number Of Divisors And Their Sum, Elementary theory of numbers, Warszawa, 1964.

Formula

a(n) = k if sigma(k) = (sopf(k))^3, where sigma(k) = A000203(k) and sopf(k) = A008472(k).

Example

a(13) = 35860609 = 41 * 71 * 97 * 127, then sigma(35860609) = 37933056 = (41 + 71 + 97 + 127)^3.

Mathematica

d[n_]:= If[Plus@@Divisors[n]-Power[Plus@@Select[Divisors[n], PrimeQ], 3]==0, n]; Select[Range[2, 10^9], #==d[#]&]
Select[Range[2, 10^9], DivisorSigma[1, #]==Total[FactorInteger[#][[All, 1]]]^3&] (* Harvey P. Dale, Dec 17 2016 *)

Prog

(PARI) is(n)=my(f=factor(n)); sum(i=1, #f~, f[i, 1])^3==sigma(n) \\ Charles R Greathouse IV, Jun 29 2013

Crossrefs

Cf. A000203, A008472, A020477, A070222.

Cf. A221262 (sigma(k)=sopf(k)^4), A221263 (sigma(k)=sopf(k)^5).

Sequence in context: A157761 A349086 A347890 * A233632 A251856 A146544

Adjacent sequences: A187768 A187769 A187770 * A187772 A187773 A187774

Keyword

nonn,nice

Author

Manuel Valdivia, Jan 04 2013

Status

approved