A096462 Sum of index values of the prime factors (with multiplicity) of n.

1, 1, 5, 1, 6, 1, 18, 7, 10, 1, 24, 1, 13, 9, 54, 1, 31, 1, 39, 12, 21, 1, 73, 11, 25, 36, 53, 1, 47, 1, 145, 18, 34, 13, 100, 1, 37, 21, 120, 1, 64, 1, 85, 51, 44, 1, 200, 15, 70, 26, 101, 1, 125, 18, 165, 30, 56, 1, 153, 1, 59, 69, 363, 20, 101, 1, 135, 35, 94, 1, 274, 1, 73, 70

Offset

2,3

Comments

Let P be equal to the set of prime factors of the positive integers, counted with multiplicity. Order the members of this set into subsets such that each prime has its own set with an index value assigned to each instance of the prime. Therefore P = {{2_1, 2_2,..2_i}, {3_1, 3_2,..3_j}, . . {p_1, p_2,..p_x}}. In generating the sequence, each indexed instance of a prime can only be used once.

Links

Robert Israel, Table of n, a(n) for n = 2..10000

Formula

a(p)=1 where p is a prime.

If p is a prime, a(p^k) = k*((p^k-1)/(p-1) - (k-1)/2). - Robert Israel, Dec 01 2016

Example

2 = 2_1, thus a(2)=1.
3 = 3_1, thus a(3)=1.
4 = 2_2 * 2_3, thus a(4)=5.
5 = 5_1, thus a(5)=1.
6 = 2_4 * 3_2, thus a(6)=6.
7 = 7_1, thus a(7)=1.
8 = 2_5 * 2_6 * 2_7, thus a(8) = 5 + 6 + 7 = 18, etc.

Maple

for n from 2 to 100 do
  F:= ifactors(n)[2];
  t:= 0;
  for f in F do
    if not assigned(R[f[1]]) then R[f[1]]:= 0 fi;
    t:= t + R[f[1]]*f[2] + f[2]*(f[2]+1)/2;
    R[f[1]]:= R[f[1]]+f[2];
  od;
  A[n]:= t;
od:
seq(A[i], i=2..100); # Robert Israel, Dec 01 2016

Mathematica

PrimeFactors[n_Integer] := Flatten[ Table[ # [[1]], {1}] & /@ FactorInteger[n]]; f[n_, p_] := Block[{t = 0, q = p}, While[s = Floor[n/q]; t = t + s; s > 0, q *= p]; t]; g[n_] := Block[{s = 0, pf = PrimeFactors[n], k = 1}, l = Length[pf]; While[k <= l, s = s + Sum[i, {i, f[n - 1, pf[[k]]] + 1, f[n, pf[[k]]]}]; k++ ]; s]; Table[g[n], {n, 2, 75}]

Crossrefs

Sequence in context: A326142 A028284 A359169 * A348507 A066948 A064265

Adjacent sequences: A096459 A096460 A096461 * A096463 A096464 A096465

Keyword

nonn,look

Author

Andrew S. Plewe, Aug 10 2004

Extensions

Edited and extended by Robert G. Wilson v, Aug 10 2004

Status

approved