A046764 Sum of the 4th powers of the divisors of n is divisible by n.

1, 34, 84, 156, 364, 492, 1092, 3444, 5617, 6396, 11234, 22468, 33628, 44772, 67404, 100884, 157276, 190978, 292084, 435708, 437164, 471828, 549687, 569772, 709937, 742612, 763912, 876252, 986076, 1099374, 1118480, 1289484, 1311492, 1419874

Offset: 1

Author

Labos Elemer

Keywords

• nonn

Comments

Compare with multiply perfect numbers, A007691. Here Sum[ divisors ] is replaced by Sum[ 4th powers of divisors ].

Problem 11090 proves that this sequence is infinite. - T. D. Noe, Apr 18 2006

Examples

			n=84, Sigma[ 4,84 ] = Sum(d^4) = 53771172 = 640133*84 = 640133*n;
n=5617, Sigma[ 4,5617 ] = 995446331475844 = 5617*17722083332, a multiple of n.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000 (terms 1..200 from T. D. Noe)
• Florian Luca and John Ferdinands, Problem 11090: Sometimes n divides sigma_k(n), Amer. Math. Monthly 113:4 (2006), pp. 372-373.

Crossrefs

Cf. A001159, A007691.

Programs

Mathematica

Do[If[Mod[DivisorSigma[4, n], n]==0, Print[n]], {n, 1, 2*10^6}]
Select[Range[1500000],Divisible[DivisorSigma[4,#],#]&] (* Harvey P. Dale, Jun 25 2014 *)

PARI

is(n)=sigma(n, 4)%n==0 \\ Charles R Greathouse IV, Feb 04 2013

Formula

Mod[ Sigma [ 4, n ], n ]=0.

Extensions

More terms from Robert G. Wilson v, Jun 09 2000