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%I A096462 #14 Sep 24 2018 16:53:14
%S A096462 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,
%T A096462 1,47,1,145,18,34,13,100,1,37,21,120,1,64,1,85,51,44,1,200,15,70,26,
%U A096462 101,1,125,18,165,30,56,1,153,1,59,69,363,20,101,1,135,35,94,1,274,1,73,70
%N A096462 Sum of index values of the prime factors (with multiplicity) of n.
%C A096462 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.
%H A096462 Robert Israel, <a href="/A096462/b096462.txt">Table of n, a(n) for n = 2..10000</a>
%F A096462 a(p)=1 where p is a prime.
%F A096462 If p is a prime, a(p^k) = k*((p^k-1)/(p-1) - (k-1)/2). - _Robert Israel_, Dec 01 2016
%e A096462 2 = 2_1, thus a(2)=1.
%e A096462 3 = 3_1, thus a(3)=1.
%e A096462 4 = 2_2 * 2_3, thus a(4)=5.
%e A096462 5 = 5_1, thus a(5)=1.
%e A096462 6 = 2_4 * 3_2, thus a(6)=6.
%e A096462 7 = 7_1, thus a(7)=1.
%e A096462 8 = 2_5 * 2_6 * 2_7, thus a(8) = 5 + 6 + 7 = 18, etc.
%p A096462 for n from 2 to 100 do
%p A096462   F:= ifactors(n)[2];
%p A096462   t:= 0;
%p A096462   for f in F do
%p A096462     if not assigned(R[f[1]]) then R[f[1]]:= 0 fi;
%p A096462     t:= t + R[f[1]]*f[2] + f[2]*(f[2]+1)/2;
%p A096462     R[f[1]]:= R[f[1]]+f[2];
%p A096462   od;
%p A096462   A[n]:= t;
%p A096462 od:
%p A096462 seq(A[i],i=2..100); # _Robert Israel_, Dec 01 2016
%t A096462 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}]
%K A096462 nonn,look
%O A096462 2,3
%A A096462 _Andrew S. Plewe_, Aug 10 2004
%E A096462 Edited and extended by _Robert G. Wilson v_, Aug 10 2004

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