A221262 -id:A221262 - cp's OEIS frontend

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

Manuel Valdivia, Jan 04 2013

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

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

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

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.

Cf. A000203, A008472, A020477, A070222.

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

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 *)

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

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

A221263 Numbers k such that sigma(k) is the fifth power of the sum of the prime divisors of k.

Original entry on oeis.org

213556659624, 359544809085, 1329797339640, 1548635130140, 1746049287480, 1810001934510, 1867318744632, 1875874796664, 1909975290390, 2040256862622, 2516452216712, 3407803953785, 6329875033944, 7792308679512, 7840198408728, 7877108796312, 8098434291288, 8241610823832

Offset: 1

Giovanni Resta, Jan 07 2013

nonn

The smallest k for which sigma(k) = sopf(k)^6 is 45770980141656. Other such values are 5245619666623908 and 5582294774581035.

			213556659624 = 2^3*3^3*7^3*11*31*79*107 and sigma(213556659624) = (2+3+7+11+31+79+107)^5.

Robert Gerbicz, Table of n, a(n) for n = 1..10000

Cf. A000203, A008472.

Cf. A187771 (sigma(k)=sopf(k)^3), A221262 (sigma(k)=sopf(k)^4).

cp's OEIS Frontend

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

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