

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
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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, SpringerVerlag, 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



