A258897 Divisorial primes p such that p-1 = Product_{d|k} d for some k < sqrt(p-1).

331777, 8503057, 9834497, 59969537, 562448657, 916636177, 3208542737, 3782742017, 5006411537, 7676563457, 11574317057, 19565295377, 34188010001, 38167092497, 49632710657, 56712564737, 59553569297, 61505984017, 104086245377, 114733948177

Offset: 1

Author

Jaroslav Krizek, Jun 20 2015

Keywords

• nonn

Comments

A divisorial prime is a prime p of the form p = 1 + Product_{d|k} d for some k (see A007955 and A258455).

Sequence lists divisorial primes p from A258455 such that p-1 = A007955(k) for some k < 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 second kind. Divisorial primes of the first kind are in A258896.

With number 3, complement of A258896 with respect to A258455.

With numbers 2 and 3, divisorial primes p that are not of the form 4*q^2 + 1 where q = prime.

See A259023 - numbers n such that Product_{d|n} d is a divisorial prime from this sequence.

Examples

			Prime p = 331777 is in sequence because p - 1 = 331776 = 576^2 is the product of divisors of 24 and 24 < 576.
		

Links

• Jaroslav Krizek and Chai Wah Wu, Table of n, a(n) for n = 1..500 [a(n) for n = 1..43 from Jaroslav Krizek].

Crossrefs

Cf. A007955, A258455, A258896, A259023, A259199.

Programs

Magma

Set(Sort([&*(Divisors(n))+1: n in [1..1000] | &*(Divisors(n)) ne n^2 and IsSquare(&*(Divisors(n))) and IsPrime(&*(Divisors(n))+1)]));

Magma

[n: n in [A258455(n)] | not IsPrime(Floor(Sqrt(n-1)) div 2)];