A294602 - OEIS

%I #24 Sep 08 2022 08:46:20

%S 0,0,1,1,1,1,1,2,2,1,1,2,2,2,2,2,2,3,3,4,4,3,3,4,4,3,3,3,3,4,4,5,5,4,

%T 4,4,4,4,4,4,4,5,5,6,6,5,5,6,6,6,6,6,6,7,7,7,7,6,6,7,7,7,7,7,7,7,7,8,

%U 8,8,8,9,9,9,9,9,9,9,9,10,10,9,9,10,10,9

%N a(n) = pi(n-1) - pi(floor(n/2)), where pi is A000720.

%C Number of primes in the interval (n/2, n).

%C Number of primes among the larger parts of the partitions of n into two distinct parts. For n=8, the partitions of 8 into two distinct parts are (7,1), (6,2), (5,3); 7 and 5 are prime so a(8) = 2. - _Wesley Ivan Hurt_, Apr 07 2018

%F a(n) = A056171(n) - A010051(n).

%F a(n) = Sum_{i=1..floor((n-1)/2)} A010051(n-i). - _Wesley Ivan Hurt_, Apr 07 2018

%e a(8) = 2 because there are 2 primes between 4 and 8: 5, 7.

%e a(19) = 3 because there are 3 primes between 9 and 19: 11, 13, 17.

%p A294602 := proc(n)

%p     numtheory[pi](n-1)-numtheory[pi](floor(n/2)) ;

%p end proc:

%p seq(A294602(n),n=1..120) ; # _R. J. Mathar_, Dec 17 2017

%t Array[PrimePi[# - 1] - PrimePi[Floor[#/2]] &, 86] (* _Michael De Vlieger_, Nov 03 2017 *)

%o (Magma) [0, 0] cat [#PrimesInInterval(Floor(n/2)+1, n-1): n in [3..86]];

%o (PARI) vector(86, n, primepi(n-1)-primepi(n\2))

%Y Cf. A000720, A001221, A010051, A056171.

%K nonn,easy

%O 1,8

%A _Arkadiusz Wesolowski_, Nov 03 2017