A245061 Prime numbers p such that p - primepi(p) is a square, where primepi is the prime counting function.

2, 3, 37, 541, 647, 881, 1151, 1301, 1627, 2377, 3271, 5179, 5641, 10501, 11597, 11821, 18503, 20543, 23339, 31259, 35461, 38669, 39499, 42901, 43331, 44201, 45523, 51973, 53407, 67213, 67757, 70489, 72169, 77291, 98893, 99551, 128291, 139721, 145207, 150011

Offset: 1

Author

Chai Wah Wu, Jul 10 2014

Keywords

• nonn

Examples

			37 is in the sequence because primepi(37) = 12, and 37 - 12 = 5^2.
541 is in the sequence because primepi(541) = 100, and 541 - 100 = 21^2.
547 is not in the sequence because primepi(547) = 101, and 547 - 101 = 446, which is not a perfect square.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A104269, A113410, A064370.

Programs

Maple

with(numtheory): A245061:=n->`if`(type(sqrt(n-pi(n)),integer) and type(n,prime), n, NULL): seq(A245061(n), n=2..10^5); # Wesley Ivan Hurt, Jul 10 2014

Mathematica

Select[Prime[Range[200]], IntegerQ[Sqrt[# - PrimePi[#]]] &] (* Alonso del Arte, Jul 11 2014 *)

PARI

select(p->issquare(p-primepi(p)), primes(15000)) \\ Michel Marcus, Jul 11 2014

Python

import sympy,gmpy2
[sympy.prime(n) for n in range(1,10**6) if gmpy2.is_square(sympy.prime(n)-n)] # Chai Wah Wu, Jul 11 2014

Formula

a(n) = prime(A064370(n+1)). - Michel Marcus, Jul 11 2014

Extensions

More terms from Michel Marcus, Jul 11 2014