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

1,4

Comments

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

Links

Felix Huber, Table of n, a(n) for n = 1..5000

Eric Weisstein's World of Mathematics, Fundamental Theorem of Arithmetic

Example

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.

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);

Crossrefs

Cf. A008474, A219180, A377505, A377537.

Sequence in context: A098492 A217314 A173508 * A103632 A214076 A067859

Adjacent sequences: A382327 A382328 A382329 * A382331 A382332 A382333

Keyword

nonn

Author

Felix Huber, Mar 23 2025

Status

approved