cp's OEIS Frontend

A080362 a(n) is the number of positive integers x such that the number of unitary-prime-divisors of x! equals n. Same as the number of positive integers x such that the number of primes in (x/2,x] equals n.

%I A080362 #8 Feb 16 2025 08:32:48
%S A080362 4,10,7,14,7,10,12,5,14,16,3,10,18,16,15,11,7,16,19,14,9,2,14,14,8,11,
%T A080362 18,19,24,10,14,16,20,10,11,3,6,13,18,21,9,31,37,10,15,6,2,6,21,12,7,
%U A080362 6,6,16,15,34,14,10,15,29,22,9,4,14,16,17,25,36,12,15,13,19,19,8,10,5,12
%N A080362 a(n) is the number of positive integers x such that the number of unitary-prime-divisors of x! equals n. Same as the number of positive integers x such that the number of primes in (x/2,x] equals n.
%H A080362 J. Sondow, <a href="https://mathworld.wolfram.com/RamanujanPrime.html">Ramanujan Prime in MathWorld</a> [From _Jonathan Sondow_, Aug 10 2008]
%F A080362 a(n)=Card{x; Pi[x]-Pi[x/2]=n}, where Pi()=A000720().
%e A080362 n=5,a(5)=7 because in 7 factorials 5 primes arise with exponent 1: in factorials of 31,32,33,37,41,46; e.g. in 37! these are {19,23,29,31,37}, or 10 numbers x, exist such ones that number of unitary prime divisors of x! equals 2, namely in factorials of {3,5,7,8,9,11,12,13,15,16}.
%Y A080362 Cf. A056171, A056169, A000720, A000142, A080359, A080360, A080361.
%Y A080362 Cf. A104272 Ramanujan primes. [From _Jonathan Sondow_, Aug 10 2008]
%K A080362 nonn
%O A080362 1,1
%A A080362 _Labos Elemer_, Feb 21 2003
%E A080362 Definition corrected by _Jonathan Sondow_, Aug 10 2008