A258896 Divisorial primes p of the form p = 1 + k^2 where k^2 = Product_{d|k} d= A007955(k) for some k.

2, 37, 101, 197, 677, 5477, 8837, 17957, 21317, 42437, 98597, 106277, 148997, 217157, 401957, 454277, 1196837, 1378277, 1674437, 1705637, 1833317, 1865957, 2390117, 2735717, 3118757, 3147077, 3587237, 3865157, 4104677, 4519877, 4726277, 5410277, 6728837, 7043717

Offset: 1

Author

Jaroslav Krizek, Jun 20 2015

Keywords

• nonn

Comments

Sequence lists divisorial primes p from A258455 such that p-1 = A007955(sqrt(p-1)).

If 1 + Product_{d|k} d for some k > 1 is a prime p other than 3, then p-1 is a square and p is either of the form k^2 + 1 or h^2 + 1 where h>k. In this sequence are divisorial primes of the first kind. Divisorial primes of the second kind are in A258897.

With number 3, complement of A258897 with respect to A258455.

All terms > 2 are of the form 4*q^2 + 1 where q = prime (see A052292).

Subsequence of A002496 (primes of the form k^2 + 1), and the corresponding k are a subsequence of A007422. - Michel Marcus, Jul 09 2015

Examples

			Number 101 is in sequence because 100 is the product of divisors of 10; 101 - 1 = 100 = A007955(sqrt(101 - 1)).
		

Links

• Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A007955, A052292, A258455, A258897, A259199.

Programs

Magma

[n: n in [1..10000000] | IsPrime(n) and n-1 eq (&*(Divisors(Floor(Sqrt(n-1)))))];

PARI

lista(nn) = {forprime(p=2, nn, if (issquare(pp=(p-1)) && (k=sqrtint(pp)) && (d=divisors(k)) && (1+prod(j=1, #d, d[j])==p), print1(p, ", ")););} \\ Michel Marcus, Jul 08 2015

Formula

For n>1; a(n) = 4*(A052291(n))^2 + 1 = A052292(n).