A078177 Composite numbers with an integer arithmetic mean of all prime factors.

4, 8, 9, 15, 16, 20, 21, 25, 27, 32, 33, 35, 39, 42, 44, 49, 50, 51, 55, 57, 60, 64, 65, 68, 69, 77, 78, 81, 85, 87, 91, 92, 93, 95, 105, 110, 111, 112, 114, 115, 116, 119, 121, 123, 125, 128, 129, 133, 140, 141, 143, 145, 155, 156, 159, 161, 164, 169, 170, 177, 180

Offset: 1

Author

Reinhard Zumkeller, Nov 20 2002

Keywords

• nonn

Comments

That is, composite numbers such that the arithmetic mean of their prime factors (counted with multiplicity) is an integer.

Examples

			60 = 2*2*3*5: (2+2+3+5)/4 = 3, therefore 60 is a term.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A078175, A001222, A100118, A046363, A133620, A133621.

Cf. A133880, A133890, A133900, A133910, A133911, A046346, A134331, A134332, A134333, A134334, A134344, A134376.

Programs

Mathematica

Select[Range[200], CompositeQ[#] && IntegerQ[Mean[Flatten[Table[#[[1]], #[[2]]]& /@ FactorInteger[#]]]]&] (* Jean-François Alcover, Aug 03 2018 *)

PARI

lista(nn) = {forcomposite(n=1, nn, my(f = factor(n)); if (! (sum(k=1, #f~, f[k,1]*f[k,2]) % vecsum(f[,2])), print1(n, ", ")););} \\ Michel Marcus, Feb 22 2016

Formula

A001414(a(n)) == 0 modulo A001222(a(n)).

Extensions

Edited by N. J. A. Sloane, May 30 2008 at the suggestion of R. J. Mathar