A380404 - OEIS

%I #14 May 25 2026 08:50:04

%S 0,1,4,16,60,377,3323,42518,646580,12285485,300378113,8028681592,

%T 259488951722,9414917934636,362597756958862,15397728568256861,

%U 742238179325555124,40068968503380861517,2251262473065725514584,139566579946046888545035

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

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

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

%e Let P = A002110 and let s = A246655.

%e a(0) = 0 since P(0) = 1, and the smallest term in s is 2.

%e a(1) = 1 since P(1) = 2.

%e a(2) = 4 since P(2) = 6 and the terms in s that do not exceed 6 are {2, 3, 4, 5}.

%e 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.

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

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

%K nonn,hard,more

%O 0,3

%A _Michael De Vlieger_, Jan 24 2025

%E a(16)-a(19) corrected by _Falk Hüffner_, May 25 2026