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

Author

Joseph Wheat, Jul 12 2017

Keywords

• nonn

Examples

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

Links

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

Programs

Mathematica

TakeWhile[#, # < 110000 &] &@ Union@ Select[Array[Prime[# + 1]^2 - Prime[#]^2 + 1 &, 10^4], IntegerQ@ Sqrt@ # &] (* Michael De Vlieger, Jul 13 2017 *)
Take[Select[#[[2]]-#[[1]]+1&/@Partition[Prime[Range[3000]]^2,2,1],IntegerQ[Sqrt[#]]&]//Union,50] (* Harvey P. Dale, Jan 19 2025 *)

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

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

Extensions

More terms from Alois P. Heinz, Jul 13 2017