A289829 Perfect squares of the form prime(k+1)^2 - prime(k)^2 + 1 where prime(k) is the k-th prime number.

25, 49, 121, 169, 289, 361, 841, 961, 1681, 1849, 2401, 2809, 3721, 5929, 6889, 7921, 8281, 10201, 11449, 11881, 14161, 14641, 17689, 24649, 26569, 32041, 38809, 41209, 43681, 44521, 61009, 63001, 69169, 76729, 80089, 85849, 89401, 94249, 96721, 97969, 108241

Offset

1,1

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Example

7^2 - 5^2 + 1 = 5^2, 17^2 - 13^2 + 1 = 11^2, 47^2 - 43^2 + 1 = 19^2, etc.

Mathematica

TakeWhile[#, # < 110000 &] &@ Union@ Select[Array[Prime[# + 1]^2 - Prime[#]^2 + 1 &, 10^4], IntegerQ@ Sqrt@ # &] (* Michael De Vlieger, Jul 13 2017 *)

Prog

(Python)
from __future__ import division
from sympy import divisors, isprime, prevprime, nextprime
A289829_list = []
for n in range(10**4):
    m = n**2-1
    for d in divisors(m):
        if d*d >= m:
            break
        r = m//d
        if not r % 2:
            r = r//2
            if not isprime(r):
                p, q = prevprime(r), nextprime(r)
                if m == (q-p)*(q+p):
                    A289829_list.append(n**2)
                    break # Chai Wah Wu, Jul 15 2017
(PARI) is(n) = if(!issquare(n), return(0), my(p=2); while(1, if(n==nextprime(p+1)^2-p^2+1, return(1)); p=nextprime(p+1); if(p > n, return(0)))) \\ Felix Fröhlich, Jul 15 2017

Crossrefs

Sequence in context: A106564 A308177 A104777 * A358060 A131706 A110015

Adjacent sequences: A289826 A289827 A289828 * A289830 A289831 A289832

Keyword

nonn

Author

Joseph Wheat, Jul 12 2017

Extensions

More terms from Alois P. Heinz, Jul 13 2017

Status

approved