cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

331777, 8503057, 9834497, 59969537, 562448657, 916636177, 3208542737, 3782742017, 5006411537, 7676563457, 11574317057, 19565295377, 34188010001, 38167092497, 49632710657, 56712564737, 59553569297, 61505984017, 104086245377, 114733948177
Offset: 1

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Author

Jaroslav Krizek, Jun 20 2015

Keywords

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.
		

Crossrefs

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)];

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

Original entry on oeis.org

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

Views

Author

Jaroslav Krizek, Jun 20 2015

Keywords

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)).
		

Crossrefs

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).
Showing 1-2 of 2 results.