A174895 -id:A174895 - cp's OEIS frontend

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

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

A174897 a(n) = characteristic function of numbers k such that A007955(m) = k has solution for some m, where A007955(m) = product of divisors of m.

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0

Offset: 1

Jaroslav Krizek, Apr 01 2010

nonn

a(n) = characteristic function of numbers from A174895(n).

a(n) = 1 if A007955(m) = n for any m, else 0.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for characteristic functions

Cf. A007955, A174895, A174898.

Block[{nn = 105, t}, t = ConstantArray[0, nn]; ReplacePart[t, Map[# -> 1 &, TakeWhile[Sort@ Array[Times @@ Divisors@ # &, nn], # <= 105 &]]]] (* Michael De Vlieger, Oct 20 2017 *)

up_to = 65537;
v174897 = vector(up_to);
A007955(n) = if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2)); \\ This function from Charles R Greathouse IV, Feb 11 2011
for(k=1, up_to, t=A007955(k); if(t<=up_to, v174897[t] = 1));
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
write_to_bfile(1,v174897,"b174897_upto65537.txt");
\\ Antti Karttunen, Oct 20 2017

a(n) = 1 - A174898(n).

Name edited and more terms added by Antti Karttunen, Oct 20 2017

A259199 Divisorial primes ending with digit 1.

Original entry on oeis.org

101, 34188010001, 254116810001, 283982410001, 2601446410001, 13308633610001, 39691260010001, 52361143210001, 58873394410001, 88828740010001, 155274028810001, 451651754410001, 1004693469610001, 1236570192010001, 2100654722410001, 2886794695210001, 3353811326410001

Offset: 1

Jaroslav Krizek, Jun 20 2015

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 of the form h*10^m + 1 (h, m are positive integers).
Sequence of numbers sqrt(a(n) - 1): 10, 184900, 504100, 532900, 1612900, 3648100, 6300100, 7236100, 7672900, ...
Sequence of numbers k such that 1 + Product_{d|k} d is a divisorial prime ending with digit 1: 10, 430, 510, 680, 710, 730, ...
Intersection of A030430 and A258455. - Michel Marcus, Sep 14 2015

			Prime 34188010001 is in sequence because 34188010000 is the product of divisors of 430.
1 + the product of divisors of 3000 = 43046721000000000000000000000000000000000000000000000001 is also a term of this sequence.

Chai Wah Wu, Table of n, a(n) for n = 1..145

Cf. A007955, A030430, A174895, A258455, A258896, A258897.

Set(Sort([&*(Divisors(n))+1: n in [1..10000] | IsSquare(&*(Divisors(n))) and IsPrime(&*(Divisors(n))+1) and (&*(Divisors(n))) mod 10 eq 0]))

Subsequence of A258455.

A174896 a(n) = numbers k in increasing order such that A007955(m) = k has no solution for any m, where A007955(m) = product of divisors of m.

Original entry on oeis.org

4, 6, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 28, 30, 32, 33, 34, 35, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95

Offset: 1

Jaroslav Krizek, Apr 01 2010

nonn

Complement of A174895(n). A174897(a(n)) = 0, A174898(a(n)) = 1.

More terms from Michel Marcus, Sep 18 2013

cp's OEIS Frontend

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

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A174897 a(n) = characteristic function of numbers k such that A007955(m) = k has solution for some m, where A007955(m) = product of divisors of m.

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A259199 Divisorial primes ending with digit 1.

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Examples

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Programs

Magma

Formula

A174896 a(n) = numbers k in increasing order such that A007955(m) = k has no solution for any m, where A007955(m) = product of divisors of m.

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