A118370 -id:A118370 - cp's OEIS frontend

A118369 Numbers k such that 1 + Product_{d|k} d is prime.

1, 2, 6, 10, 14, 24, 26, 54, 56, 74, 88, 94, 134, 146, 154, 174, 206, 238, 248, 266, 296, 314, 326, 328, 374, 378, 386, 430, 442, 466, 472, 488, 494, 498, 510, 568, 582, 584, 634, 674, 680, 710, 730, 742, 786, 856, 874, 894, 918, 962, 986, 1038, 1094, 1174

Offset: 1

Rick L. Shepherd, Apr 25 2006

nonn

See A118370 for the corresponding primes, 'divisorial primes'.

			The (positive) divisors of 6 are 1,2,3,6. As 6*3*2*1 + 1 = 37 is prime, 6 is a term.

Paolo P. Lava, Table of n, a(n) for n = 1..10000

Cf. A118370, A007955.

f[n_]:=PrimeQ[Times@@Divisors[n]+1]; lst={};Do[If[f[n],AppendTo[lst,n]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 14 2009 *)

for(n=1,2500, s=1; fordiv(n,d,s=s*d); if(isprime(s+1), print1(n,", ")))

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

Original entry on oeis.org

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

Jaroslav Krizek, May 30 2015

nonn

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

			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.

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

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

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

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

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 *)

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

A258456 Product of divisors of n is not a square.

Original entry on oeis.org

2, 3, 4, 5, 7, 9, 11, 12, 13, 17, 18, 19, 20, 23, 25, 28, 29, 31, 32, 36, 37, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 59, 61, 63, 64, 67, 68, 71, 73, 75, 76, 79, 80, 83, 89, 92, 97, 98, 99, 100, 101, 103, 107, 109, 112, 113, 116, 117, 121, 124, 127, 131, 137

Offset: 1

Jaroslav Krizek, May 30 2015

Numbers n such that A007955(n) is not a square.
Complement of A048943.
2 is only number n from this sequence such that 1 + Product_{d|n} d is a prime.
If 1 + Product_{d|n} d for n > 2 is a prime p, then Product_{d|n} d is a square (see A258455).
m is a term if and only if m is not a fourth power and the number of divisors of m is not a multiple of 4. - Chai Wah Wu, Mar 09 2016

			9 is in sequence because product of divisors of 9 = 1*3*9 = 27 is not square.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A007955, A118369, A118370, A258455.

[n: n in [1..200] | not IsSquare(&*(Divisors(n)))];

Select[Range@ 137, ! IntegerQ@ Sqrt[Times @@ Divisors@ #] &] (* Michael De Vlieger, Jun 02 2015 *)

for(n=1,100,d=divisors(n);p=prod(i=1,#d,d[i]);if(!issquare(p),print1(n,", "))) \\ Derek Orr, Jun 12 2015

from gmpy2 import iroot
from sympy import divisor_count
A258456_list = [i for i in range(1,10**3) if not iroot(i,4)[1] and divisor_count(i) % 4] # Chai Wah Wu, Mar 10 2016

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A118369 Numbers k such that 1 + Product_{d|k} d is prime.

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

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A258456 Product of divisors of n is not a square.

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