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A382330 a(n) is the number of positive integers k for which Sum_{i=1..j} (p_i+e_i) = n, where p_1^e_1*...*p_j^e_j is the prime factorization of k.

%I A382330 #11 Mar 29 2025 18:38:18
%S A382330 0,0,1,2,2,3,4,6,8,11,15,21,27,36,47,61,79,104,133,170,215,272,343,
%T A382330 433,542,678,845,1050,1300,1608,1981,2437,2988,3655,4460,5433,6603,
%U A382330 8014,9705,11731,14155,17055,20509,24624,29512,35313,42184,50315,59916,71248,84598
%N A382330 a(n) is the number of positive integers k for which Sum_{i=1..j} (p_i+e_i) = n, where p_1^e_1*...*p_j^e_j is the prime factorization of k.
%C A382330 a(n) is the number of positive integers k for A008474(k) = n.
%H A382330 Felix Huber, <a href="/A382330/b382330.txt">Table of n, a(n) for n = 1..5000</a>
%H A382330 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FundamentalTheoremofArithmetic.html">Fundamental Theorem of Arithmetic</a>
%e A382330 The a(7) = 4 positive integers k are 32 = 2^5, 81 = 3^4, 25 = 5^2, 6 = 2^1*3^1 because 2 + 5 = 3 + 4 = 5 + 2 = 2 + 1 + 3 + 1 = 7 and there is no further positive integer with that property.
%e A382330 The a(11) = 15 positive integers k are 512 = 2^9, 6561 = 3^8, 15625 = 5^6, 2401 = 7^4, 96 = 2^5*3^1, 144 = 2^4*3^2, 216 = 2^3*3^3, 324 = 2^2*3^4, 486 = 2^1*3^5, 40 = 2^3*5^1, 100 = 2^2*5^2, 250 = 2^1*5^3, 14 = 2^1*7^1, 45 = 3^2*5^1, 75 = 3^1*5^2 because 2 + 9 = 3 + 8 = 5 + 6 = 7 + 4 = 2 + 5 + 3 + 1 = 2 + 4 + 3 + 2 = 2 + 3 + 3 + 3 = 2 + 2 + 3 + 4 = 2 + 1 + 3 + 5 = 2 + 3 + 5 + 1 = 2 + 2 + 5 + 2 = 2 + 1 + 5 + 3 = 2 + 1 + 7 + 1 = 3 + 2 + 5 + 1 = 3 + 1 + 5 + 2 = 11 and there is no further positive integer with that property.
%p A382330 # processes b and T from Alois P. Heinz (A219180).
%p A382330 b:= proc(n,i) option remember;
%p A382330       `if`(n=0,[1],`if`(i<1,[],zip((x,y)->x+y,b(n,i-1),
%p A382330        [0,`if`(ithprime(i)>n,[],b(n-ithprime(i),i-1))[]],0)))
%p A382330     end:
%p A382330 T:= proc(n) local l;l:=b(n,NumberTheory:-pi(n));
%p A382330        while nops(l)>0 and l[-1]=0 do l:=subsop(-1=NULL,l) od; l[]
%p A382330     end:
%p A382330 A382330:=proc(n)
%p A382330     local a,k,s,i,j,L;
%p A382330     a:=0;k:=1;s:=0;
%p A382330     while s+k<=n do
%p A382330         s:=s+ithprime(k);k:=k+1
%p A382330     od;
%p A382330     for i to k-1 do
%p A382330         for j to n-i do
%p A382330             L:=[T(j)];
%p A382330             if nops(L)>=i+1 then
%p A382330                 a:=a+L[i+1]*binomial(n-j-1,n-j-i);
%p A382330             fi
%p A382330         od
%p A382330     od;
%p A382330     return a
%p A382330 end proc;
%p A382330 seq(A382330(n),n=1..51);
%Y A382330 Cf. A008474, A219180, A377505, A377537.
%K A382330 nonn
%O A382330 1,4
%A A382330 _Felix Huber_, Mar 23 2025