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Prime numbers p such that primepi(p) + p is a square.
1

%I #23 Oct 27 2021 11:58:55

%S 11,37,443,571,1049,1307,1451,1523,2837,3593,5233,8539,9257,9439,

%T 10391,10987,17579,21881,23321,23909,25117,30557,30893,31231,42239,

%U 47123,64811,65789,83089,91631,92219,95747,97549,99971,101197,101807,110603,114487,120431

%N Prime numbers p such that primepi(p) + p is a square.

%C A064371(p) + A000040(A064371(p)) = A086968(p)^2.

%C p^2 is prime + its index A086968; p + p-th prime is a square A064371.

%C Equals the prime terms of A073945. - _Bill McEachen_, Oct 26 2021

%F a(n) = A086968(n)^2 - pi(a(n)).

%e 37 is a term because 37 is 12th prime and 37 + 12 = 49 = 7^2.

%p q:= n-> isprime(n) and issqr(n+numtheory[pi](n)):

%p select(q, [$0..150000])[]; # _Alois P. Heinz_, Oct 27 2021

%t Select[Prime@Range[10^4],IntegerQ@Sqrt[PrimePi@#+#]&] (* _Giorgos Kalogeropoulos_, Oct 26 2021 *)

%o (PARI) isok(n) = isprime(n) && issquare(n + primepi(n)); \\ _Michel Marcus_, Oct 05 2013

%Y Cf. A000040, A000720, A064371, A086968, A064370, A073945.

%K nonn

%O 1,1

%A _Zak Seidov_, Feb 26 2005

%E Definition corrected by _Michel Marcus_, Oct 05 2013