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A187771 Numbers such that the sum of its divisors is the cube of the sum of its prime divisors. 5
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 (list; graph; refs; listen; history; text; internal format)
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))^n only occurs for n = 3 (this sequence) in the given range, however it is likely that occurs for other powers n in higher numbers.

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

Prime divisors 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

FORMULA

a(n) = k if sigma(k) = (sopf(k))^3,  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, A221262 sigma(n)=sopf(n)^4, A221263 sigma(n)=sopf(n)^5.

Sequence in context: A083623 A186801 A157761 * A233632 A251856 A146544

Adjacent sequences:  A187768 A187769 A187770 * A187772 A187773 A187774

KEYWORD

nonn,nice

AUTHOR

Manuel Valdivia, Jan 04 2013

STATUS

approved

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Last modified December 12 16:06 EST 2018. Contains 318077 sequences. (Running on oeis4.)