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%I #41 Sep 09 2021 12:25:13
%S 245180,612408,639198,1698862,1721182,5154168,7824284,15817596,
%T 20441848,25969788,27688078,28404862,35860609,67149432,77378782,
%U 91397838,96462862,179302264,191550135,289772221,306901244,311657084,392802179,441839706,572673855,652117774,988918364
%N Numbers whose sum of divisors is the cube of the sum of its prime divisors.
%C 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
%C 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.
%C 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
%C Prime divisors are taken without multiplicity. - _Harvey P. Dale_, Dec 17 2016
%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
%H Donovan Johnson and Robert Gerbicz, <a href="/A187771/b187771.txt">Table of n, a(n) for n = 1..1105</a> (first 100 terms from _Donovan Johnson_)
%H W. Sierpinski, <a href="http://matwbn.icm.edu.pl/ksiazki/mon/mon42/mon4204.pdf">Number Of Divisors And Their Sum</a>, Elementary theory of numbers, Warszawa, 1964.
%F a(n) = k if sigma(k) = (sopf(k))^3, where sigma(k) = A000203(k) and sopf(k) = A008472(k).
%e a(13) = 35860609 = 41 * 71 * 97 * 127, then sigma(35860609) = 37933056 = (41 + 71 + 97 + 127)^3.
%t d[n_]:= If[Plus@@Divisors[n]-Power[Plus@@Select[Divisors[n], PrimeQ], 3]==0, n]; Select[Range[2,10^9], #==d[#]&]
%t Select[Range[2, 10^9],DivisorSigma[1,#]==Total[FactorInteger[#][[All, 1]]]^3&] (* _Harvey P. Dale_, Dec 17 2016 *)
%o (PARI) is(n)=my(f=factor(n));sum(i=1,#f~,f[i,1])^3==sigma(n) \\ _Charles R Greathouse IV_, Jun 29 2013
%Y Cf. A000203, A008472, A020477, A070222.
%Y Cf. A221262 (sigma(k)=sopf(k)^4), A221263 (sigma(k)=sopf(k)^5).
%K nonn,nice
%O 1,1
%A _Manuel Valdivia_, Jan 04 2013
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