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

1,1

Comments

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

Links

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

Formula

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

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

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

Sequence in context: A122785 A280450 A137055 * A326692 A336663 A329936

Adjacent sequences: A078174 A078175 A078176 * A078178 A078179 A078180

Keyword

nonn

Author

Reinhard Zumkeller, Nov 20 2002

Extensions

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

Status

approved