A258455 Divisorial primes: primes p of the form p = 1 + Product_{d|k} d for some k.

2, 3, 37, 101, 197, 677, 5477, 8837, 17957, 21317, 42437, 98597, 106277, 148997, 217157, 331777, 401957, 454277, 1196837, 1378277, 1674437, 1705637, 1833317, 1865957, 2390117, 2735717, 3118757, 3147077, 3587237, 3865157, 4104677, 4519877, 4726277, 5410277, 6728837

Offset: 1

Author

Jaroslav Krizek, May 30 2015

Keywords

• nonn

Comments

Primes p of the form p = A007955(k) + 1 for some k.

This sequence is a sorted version of A118370.

Corresponding values of k are in A118369.

Conjectures:

(1) if 1+ Product_{d|k} d for k > 2 is a prime p, then p-1 is a square.

(2) except for n = 2, a(n) - 1 are squares.

(3) subsequence of A062459 (primes of form x^2 + mu(x)).

From Robert Israel, Jun 08 2015: (Start)

The first n > 4 for which a(n) does not end in 7 is a(918) = 34188010001.

Statements (1) and (2) are true.

Note that if k = p_1^(a_1) ... p_m^(a_m) is the prime factorization of k, then A007955(k) = p_1^(a_1*M/2) ... p_m^(a_m*M/2) where M = (a_1+1)*...*(a_m+1). Now if M has any odd factor r > 1, A007955(k) = x^r for some x > 1 and then p = A007955(k)+1 is divisible by x+1. So for p to be prime, M must be a power of 2.

Now if A007955(k) is not a square, we need M/2 to be odd, so M = 2. That can only happen if m=1 and a_1=1. For p to be odd we need k to be even, so this means p_1 = 1, and then k=2. (End)

Union of prime 3 (where A007955(3-1) is not a square), A258896 (primes p such that p-1 = A007955(sqrt(p-1))) and A258897 (primes p such that p-1 = A007955(k) for some k < sqrt(p-1)). - Jaroslav Krizek, Jun 14 2015

Contrary to the above, this is not a subsequence of A062459: 24^4+1 = 331777 is in this sequence but not A062459. - Charles R Greathouse IV, Sep 22 2015

Examples

			The prime 37 is in sequence because there is n = 6 with divisors 1, 2, 3, 6 such that 6*3*2*1 + 1 = 37.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007955, A048943, A118369, A118370, A174895.

Programs

Magma

Set(Sort([&*(Divisors(n))+1: n in [1..1000000] | IsPrime(&*(Divisors(n))+1)]));

Maple

N:= 10^8: # to get all terms <= N
K:= floor(sqrt(N)):
sort(convert(select(t -> t <= N and isprime(t),{2,seq(convert(numtheory:-divisors(k),`*`)+1,k=2..K,2)}),list)); # Robert Israel, Jun 08 2015

Mathematica

terms = 35; n0 = 1000; Clear[f]; f[nmax_] := f[nmax] = Reap[For[n = 1, n <= nmax, n++, If[PrimeQ[p = Times @@ Divisors[n] + 1], Sow[p]]]][[2, 1]] // Sort // Take[#, terms]&; f[n0]; f[nmax = 2*n0]; While[f[nmax] != f[nmax/2], Print[nmax]; nmax = 2*nmax]; f[nmax] (* Jean-François Alcover, May 31 2015 *)
Take[Sort[Select[Table[Times@@Divisors[n]+1,{n,3000}],PrimeQ]],40] (* Harvey P. Dale, Apr 18 2018 *)

PARI

list(lim)=my(v=List()); lim\=1; for(n=1,sqrtint(lim-1), my(d=divisors(n), t=prod(i=2,#d,d[i])+1); if(t<=lim && isprime(t), listput(v, t))); Set(v) \\ Charles R Greathouse IV, Jun 08 2015