A273529 Primes p of the form x^2 + y^2 such that p+1 is the sum of the two nonzero squares in exactly 2 ways.

337, 409, 449, 577, 1009, 1129, 1489, 1801, 2377, 2521, 2609, 2689, 2833, 3041, 3169, 3329, 3361, 3433, 3529, 3889, 4049, 4513, 4657, 5209, 5569, 5689, 5857, 5881, 5953, 6529, 6553, 6569, 7177, 7297, 8009, 8089, 8209, 8329, 8641, 8737, 8761, 9433, 9697, 9769, 10169, 10321

Offset: 1

Author

Altug Alkan, May 24 2016

Keywords

• nonn

Comments

Number of prime divisors (counted with multiplicity) of p+1 is 3, 3, 5, 3, 3, 3, 3, 3, 3, 3, 5, 3, 3, 5, 3, 5, 3, 3, 3, 3, 7, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 3, 3, 3, 3, 5, 3, 7, 3, 3, 5, 3, 3, 3, 5, 5, 3, 3, 3, 3, 3, ...

In this sequence, 20249 is the first p such that p+1 has even number of prime divisors (counted with multiplicity).

Examples

			The prime 409 is a term because 409 = 3^2 + 20^2 and 410 = 7^2 + 19^2 = 11^2 + 17^2.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002313, A025285.

Programs

PARI

is(n, k)=nb = 0; lim = sqrtint(n); for (x=1, lim, if ((n-x^2 >= x^2) && issquare(n-x^2), nb++); ); nb == k;
isok(n) = isprime(n) && is(n, 1) && is(n+1, 2);

PARI

is(n)=if(n%8!=1 || !isprime(n), return(0)); my(f=factor((n+1)/2), t=1); for(i=1, #f~, if(f[i, 1]%4==1, t*=f[i, 2]+1, if(f[i, 2]%2, return(0)))); t==3 || t==4 \\ Charles R Greathouse IV, May 24 2016

Formula

a(n) mod 8 = 1.