A304818 If n = Product_i p(y_i) where p(i) is the i-th prime number and y_i <= y_j for i < j, then a(n) = Sum_i y_i*i.

0, 1, 2, 3, 3, 5, 4, 6, 6, 7, 5, 9, 6, 9, 8, 10, 7, 11, 8, 12, 10, 11, 9, 14, 9, 13, 12, 15, 10, 14, 11, 15, 12, 15, 11, 17, 12, 17, 14, 18, 13, 17, 14, 18, 15, 19, 15, 20, 12, 16, 16, 21, 16, 19, 13, 22, 18, 21, 17, 21, 18, 23, 18, 21, 15, 20, 19, 24, 20, 19

Offset: 1

Author

Gus Wiseman, May 18 2018

Keywords

• nonn
• look

Comments

If n > 1 is not a prime number, we have a(n) >= A056239(n) >= Omega(n) >= omega(n) >= A071625(n) >= ... >= Omicron(n) >= omicron(n) > 1, where Omega = A001222, omega = A001221, Omicron = A304687 and omicron = A304465.

Examples

			The multiset of prime indices (see A112798) of 216 is {1,1,1,2,2,2}, which becomes {1,2,3,4,4,5,5,6,6} under A304660, so a(216) = 1+2+3+4+4+5+5+6+6 = 36.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000720, A001221, A001222, A001358, A055932, A056239, A071625, A112798, A181819, A182850, A182857, A275870, A304465, A304660.

Programs

Maple

a:= n-> (l-> add(i*numtheory[pi](l[i]), i=1..nops(l)))(
             sort(map(i-> i[1]$i[2], ifactors(n)[2]))):
seq(a(n), n=1..100);  # Alois P. Heinz, May 20 2018

Mathematica

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[With[{y=primeMS[n]},Sum[y[[i]]*i,{i,Length[y]}]],{n,20}]

PARI

a(n) = {my(f = factor(n), s = 0, i = 0); for (k=1, #f~, for (kk = 1, f[k, 2], i++; s += i*primepi(f[k,1]););); s;} \\ Michel Marcus, May 19 2018

PARI

vf(n) = {my(f=factor(n), nb = bigomega(n), g = vector(nb), i = 0); for (k=1, #f~, for (kk = 1, f[k, 2], i++; g[i] = primepi(f[k,1]););); return(g);} \\ A112798
a(n) = {my(g = vf(n)); sum(k=1, #g, k*g[k]);} \\ Michel Marcus, May 19 2018

Formula

a(n) = A056239(A304660(n)).