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A258897
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Divisorial primes p such that p-1 = Product_{d|k} d for some k < sqrt(p-1).
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7
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331777, 8503057, 9834497, 59969537, 562448657, 916636177, 3208542737, 3782742017, 5006411537, 7676563457, 11574317057, 19565295377, 34188010001, 38167092497, 49632710657, 56712564737, 59553569297, 61505984017, 104086245377, 114733948177
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OFFSET
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1,1
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COMMENTS
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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 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.
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LINKS
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EXAMPLE
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Prime p = 331777 is in sequence because p - 1 = 331776 = 576^2 is the product of divisors of 24 and 24 < 576.
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PROG
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(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)]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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