A380404 Number of prime powers that do not exceed the primorial number A002110(n).

0, 1, 4, 16, 60, 377, 3323, 42518, 646580, 12285485, 300378113, 8028681592, 259488951722, 9414917934636, 362597756958862, 15397728568256861, 742238179325555125, 40068968503380861518, 2251262473065725514585, 139566579946046888545036

Offset: 0

Author

Michael De Vlieger, Jan 24 2025

Keywords

• nonn
• hard
• more

Examples

			Let P = A002110 and let s = A246655.
a(0) = 0 since P(0) = 1, and the smallest term in s is 2.
a(1) = 1 since P(1) = 2.
a(2) = 4 since P(2) = 6 and the terms in s that do not exceed 6 are {2, 3, 4, 5}.
a(3) = 16 since P(3) = 30; the numbers 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, and 29 are less than 30, etc.
		

Crossrefs

Cf. A000849, A002110, A182908, A246655, A380402.

Programs

Mathematica

Table[Sum[PrimePi[Floor[#^(1/k)]], {k, Floor@ Log2[#]}] &[Product[Prime[i], {i, n}]], {n, 0, 14}]

Formula

a(n) = Sum_{k = 1..floor(log_2(P(n)))} pi(floor(P(n)^(1/k))), where P(n) = A002110(n).

a(n) = A000849(n) + A380402(n).