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%I #39 Sep 08 2022 08:46:12
%S 2,3,37,101,197,677,5477,8837,17957,21317,42437,98597,106277,148997,
%T 217157,331777,401957,454277,1196837,1378277,1674437,1705637,1833317,
%U 1865957,2390117,2735717,3118757,3147077,3587237,3865157,4104677,4519877,4726277,5410277,6728837
%N Divisorial primes: primes p of the form p = 1 + Product_{d|k} d for some k.
%C Primes p of the form p = A007955(k) + 1 for some k.
%C This sequence is a sorted version of A118370.
%C Corresponding values of k are in A118369.
%C Conjectures:
%C (1) if 1+ Product_{d|k} d for k > 2 is a prime p, then p-1 is a square.
%C (2) except for n = 2, a(n) - 1 are squares.
%C (3) subsequence of A062459 (primes of form x^2 + mu(x)).
%C From _Robert Israel_, Jun 08 2015: (Start)
%C The first n > 4 for which a(n) does not end in 7 is a(918) = 34188010001.
%C Statements (1) and (2) are true.
%C 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.
%C 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)
%C 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
%C 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
%H Giovanni Resta, <a href="/A258455/b258455.txt">Table of n, a(n) for n = 1..10000</a>
%e 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.
%p N:= 10^8: # to get all terms <= N
%p K:= floor(sqrt(N)):
%p 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
%t 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 *)
%t Take[Sort[Select[Table[Times@@Divisors[n]+1,{n,3000}],PrimeQ]],40] (* _Harvey P. Dale_, Apr 18 2018 *)
%o (Magma) Set(Sort([&*(Divisors(n))+1: n in [1..1000000] | IsPrime(&*(Divisors(n))+1)]))
%o (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
%Y Cf. A007955, A048943, A118369, A118370, A174895.
%K nonn
%O 1,1
%A _Jaroslav Krizek_, May 30 2015
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