A390784 a(n) is the number of positive integers k for which Sum_{i=1..r} p_i = Sum_{i=1..r} e_i = n, where p_1^e_1*...*p_r^e_r is the canonical prime factorization of k.

0, 1, 1, 0, 5, 0, 7, 7, 8, 45, 1, 66, 13, 91, 105, 135, 561, 306, 172, 380, 1540, 462, 3543, 575, 4624, 3250, 6175, 21231, 10963, 24650, 13951, 10447, 21856, 93907, 26741, 119735, 38851, 165871, 120251, 366418, 721969, 450344, 186019, 571427, 1442694, 982215, 3005089

Offset

1,5

Comments

a(n) is the number of positive integers k for which A008472(k) = A001222(k) = n.

Links

Felix Huber, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{i=1..A024936(n)} T(n,i)*binomial(n-1,i-1), where T(n,i) is the number of partitions of n into i distinct prime parts.

Example

The a(5) = 5 positive integers are 2^4*3^1 = 48, 2^3*3^2 = 72, 2^2*3^3 = 108, 2^1*3^4 = 162, 5^5 = 3125.

Maple

# Processes b and T (slightly adapted) by 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, numtheory[pi](n));
   while nops(l)>0 and l[-1]=0 do
      l:=subsop(-1=NULL, l)
   od;
   `if`(nops(l)>0, subsop(1=NULL, l), l)
end:
A390784:=proc(n)
   local i, l;
   l:=T(n);
   add(l[i]*binomial(n-1, i-1), i=1..nops(l))
end:
seq(A390784(n), n=1..47);

Crossrefs

Cf. A001222, A008472, A024936, A097805, A219180, A377505, A377537, A382330, A390375.

Sequence in context: A112254 A180660 A221711 * A200400 A190147 A346190

Adjacent sequences: A390781 A390782 A390783 * A390785 A390786 A390787

Keyword

nonn

Author

Felix Huber, Nov 20 2025

Status

approved