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A258897 Divisorial primes p such that p-1 = Product_{d|k} d for some k < sqrt(p-1). 7

%I #20 Dec 13 2023 08:46:28

%S 331777,8503057,9834497,59969537,562448657,916636177,3208542737,

%T 3782742017,5006411537,7676563457,11574317057,19565295377,34188010001,

%U 38167092497,49632710657,56712564737,59553569297,61505984017,104086245377,114733948177

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

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

%C Sequence lists divisorial primes p from A258455 such that p-1 = A007955(k) for some k < sqrt(p-1).

%C 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.

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

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

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

%H Jaroslav Krizek and Chai Wah Wu, <a href="/A258897/b258897.txt">Table of n, a(n) for n = 1..500</a> [a(n) for n = 1..43 from Jaroslav Krizek].

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

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

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

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

%K nonn

%O 1,1

%A _Jaroslav Krizek_, Jun 20 2015

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