

A258455


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


9



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
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OFFSET

1,1


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_{dk} d for k > 2 is a prime p, then p1 is a square.
(2) except for n = 2, a(n)  1 are squares.
(3) subsequence of A062459 (primes of form x^2 + mu(x)).
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)


LINKS



EXAMPLE

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.


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] (* JeanFrançois Alcover, May 31 2015 *)
Take[Sort[Select[Table[Times@@Divisors[n]+1, {n, 3000}], PrimeQ]], 40] (* Harvey P. Dale, Apr 18 2018 *)


PROG

(Magma) Set(Sort([&*(Divisors(n))+1: n in [1..1000000]  IsPrime(&*(Divisors(n))+1)]))
(PARI) list(lim)=my(v=List()); lim\=1; for(n=1, sqrtint(lim1), 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


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



