A126594 Floor of the average of the prime factors of n with multiplicity.

2, 3, 2, 5, 2, 7, 2, 3, 3, 11, 2, 13, 4, 4, 2, 17, 2, 19, 3, 5, 6, 23, 2, 5, 7, 3, 3, 29, 3, 31, 2, 7, 9, 6, 2, 37, 10, 8, 2, 41, 4, 43, 5, 3, 12, 47, 2, 7, 4, 10, 5, 53, 2, 8, 3, 11, 15, 59, 3, 61, 16, 4, 2, 9, 5, 67, 7, 13, 4, 71, 2, 73, 19, 4, 7, 9, 6, 79, 2, 3, 21, 83, 3, 11, 22, 16, 4, 89, 3, 10

Offset: 2

Author

Cino Hilliard, Jan 06 2007

Keywords

• easy
• nonn

Crossrefs

Cf. A067629 (rounding instead of flooring), A076690.

This is the floor of A123528/A123529.

Without multiplicity we have A363895.

For prime indices instead of factors we have A363943, triangle A363945.

Positions of first appearances are A364037.

The ceiling is A364156.

Positions of 2's are A364157, for prime indices A363949.

A051293 counts subsets with integer mean, median A000975.

A067538 counts partitions with integer mean, ranks A316413.

A078175 lists numbers with integer mean of prime factors.

Cf. A026905, A326567/A326568, A327473, A327476, A363944, A363948, A363950.

Programs

Mathematica

Table[Floor[(Plus@@Times@@@FactorInteger[n])/PrimeOmega[n]], {n, 2, 90}] (* Alonso del Arte, May 21 2012 *)

PARI

avg(n) = { local(x,j,ln) for(x=2,n,a=ifactor(x); ln=length(a); print1(floor(sum(j=1,ln,a[j])/ln)",")) } ifactor(n) = \The vector of the prime factors of n with multiplicity. { local(f,j,k,flist); flist=[]; f=Vec(factor(n)); for(j=1,length(f[1]), for(k = 1,f[2][j],flist = concat(flist,f[1][j]) ); ); return(flist) }

Formula

a(p^n)=p, p prime, n >= 1. - Philippe Deléham, Nov 23 2008

a(n) = floor(A001414(n)/A001222(n)). - Philippe Deléham, Nov 24 2008