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A046763 Numbers n such that the sum of the cubes of the divisors of n is divisible by n. 6

%I

%S 1,6,42,120,168,270,280,312,496,672,728,840,1080,1560,1782,1806,1890,

%T 2044,2184,2520,3472,3640,3913,4256,5880,6048,6552,6615,7224,7560,

%U 7826,8128,9120,9424,9933,10804,10920,11400,12040,12768,13230,13626,14040

%N Numbers n such that the sum of the cubes of the divisors of n is divisible by n.

%C Compare with multiply perfect numbers, A007691. Here Sum[ divisors ] is replaced by Sum[ cube of divisors ].

%C Problem 11090 proves that this sequence is infinite. - _T. D. Noe_, Apr 18 2006

%C Tomohiro Yamada found that the odd number 209195 is a member. (See the Editorial Comment after the solution to Problem 11090.) - _Jonathan Sondow_, Nov 23 2012

%H Amiram Eldar, <a href="/A046763/b046763.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)

%H Florian Luca and John Ferdinands, <a href="https://www.jstor.org/stable/27641939">Problem 11090: Sometimes n divides sigma_k(n)</a>, Amer. Math. Monthly 113:4 (2006), pp. 372-373.

%e n = 168 = a[ 5 ], Sum[ d^3 ] = 5634720 = 33540*168 = 33540*n or if n = 8128, Sigma[ 3,8128 ] = 613681507712 = 8128*75502154. Moreover 8128 is a perfect number.

%p with(numtheory);

%p A046763:=proc(q)

%p local a,i,n;

%p for n from 1 to q do

%p a:=divisors(n); if frac(add(a[i]^3,i=1..nops(a))/n)=0 then print(n); fi;

%p od; end:

%p A046763(100000); # _Paolo P. Lava_, Dec 07 2012

%t Select[Range[10^4], Divisible[DivisorSigma[3, #], #] &] (* _Amiram Eldar_, Sep 10 2019 *)

%o (PARI) is(n)=sigma(n,3)%n==0 \\ _Charles R Greathouse IV_, Feb 04 2013

%Y Cf. A001158, A007691.

%K nonn

%O 1,2

%A _Labos Elemer_

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Last modified May 30 18:07 EDT 2020. Contains 334728 sequences. (Running on oeis4.)