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.

0, 0, 1, 2, 2, 3, 4, 6, 8, 11, 15, 21, 27, 36, 47, 61, 79, 104, 133, 170, 215, 272, 343, 433, 542, 678, 845, 1050, 1300, 1608, 1981, 2437, 2988, 3655, 4460, 5433, 6603, 8014, 9705, 11731, 14155, 17055, 20509, 24624, 29512, 35313, 42184, 50315, 59916, 71248, 84598

Offset: 1

Author

Felix Huber, Mar 23 2025

Keywords

• nonn

Comments

a(n) is the number of positive integers k for A008474(k) = n.

Examples

			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.
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.
		

Links

• Felix Huber, Table of n, a(n) for n = 1..5000
• Eric Weisstein's World of Mathematics, Fundamental Theorem of Arithmetic

Crossrefs

Cf. A008474, A219180, A377505, A377537.

Programs

Maple

# processes b and T from Alois P. Heinz (A219180).
b:= proc(n,i) option remember;
      `if`(n=0,[1],`if`(i<1,[],zip((x,y)->x+y,b(n,i-1),
       [0,`if`(ithprime(i)>n,[],b(n-ithprime(i),i-1))[]],0)))
    end:
T:= proc(n) local l;l:=b(n,NumberTheory:-pi(n));
       while nops(l)>0 and l[-1]=0 do l:=subsop(-1=NULL,l) od; l[]
    end:
A382330:=proc(n)
    local a,k,s,i,j,L;
    a:=0;k:=1;s:=0;
    while s+k<=n do
        s:=s+ithprime(k);k:=k+1
    od;
    for i to k-1 do
        for j to n-i do
            L:=[T(j)];
            if nops(L)>=i+1 then
                a:=a+L[i+1]*binomial(n-j-1,n-j-i);
            fi
        od
    od;
    return a
end proc;
seq(A382330(n),n=1..51);