A258897 -id:A258897 - 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

A258896 Divisorial primes p of the form p = 1 + k^2 where k^2 = Product_{d|k} d= A007955(k) for some k.

Original entry on oeis.org

2, 37, 101, 197, 677, 5477, 8837, 17957, 21317, 42437, 98597, 106277, 148997, 217157, 401957, 454277, 1196837, 1378277, 1674437, 1705637, 1833317, 1865957, 2390117, 2735717, 3118757, 3147077, 3587237, 3865157, 4104677, 4519877, 4726277, 5410277, 6728837, 7043717

Offset: 1

Jaroslav Krizek, Jun 20 2015

nonn

Sequence lists divisorial primes p from A258455 such that p-1 = A007955(sqrt(p-1)).
If 1 + Product_{d|k} d for some k > 1 is a prime p other than 3, then p-1 is a square and p is either of the form k^2 + 1 or h^2 + 1 where h>k. In this sequence are divisorial primes of the first kind. Divisorial primes of the second kind are in A258897.
With number 3, complement of A258897 with respect to A258455.
All terms > 2 are of the form 4*q^2 + 1 where q = prime (see A052292).
Subsequence of A002496 (primes of the form k^2 + 1), and the corresponding k are a subsequence of A007422. - Michel Marcus, Jul 09 2015

			Number 101 is in sequence because 100 is the product of divisors of 10; 101 - 1 = 100 = A007955(sqrt(101 - 1)).

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A007955, A052292, A258455, A258897, A259199.

[n: n in [1..10000000] | IsPrime(n) and n-1 eq (&*(Divisors(Floor(Sqrt(n-1)))))];

lista(nn) = {forprime(p=2, nn, if (issquare(pp=(p-1)) && (k=sqrtint(pp)) && (d=divisors(k)) && (1+prod(j=1, #d, d[j])==p), print1(p, ", ")););} \\ Michel Marcus, Jul 08 2015

For n>1; a(n) = 4*(A052291(n))^2 + 1 = A052292(n).

A259021 Numbers k such that k^2 = Product_{d|k} d (= A007955(k)) and simultaneously k^2 + 1 is a divisorial prime (A258455).

Original entry on oeis.org

1, 6, 10, 14, 26, 74, 94, 134, 146, 206, 314, 326, 386, 466, 634, 674, 1094, 1174, 1294, 1306, 1354, 1366, 1546, 1654, 1766, 1774, 1894, 1966, 2026, 2126, 2174, 2326, 2594, 2654, 2746, 2974, 2986, 3046, 3106, 3134, 3214, 3254, 3274, 3314, 3326, 3334, 3446

Offset: 1

Jaroslav Krizek, Sep 01 2015

nonn

First deviation from A259020 is at a(15).
With number 2 complement of A259023 with respect to A118369.
1 together with squarefree semiprimes (A006881) k such that k^2 + 1 is prime. Without the squarefree restriction there will be only one more term, 4. - Amiram Eldar, Sep 25 2022

			The number 10 is in sequence because 10^2 = 1*2*5*10 = 100 and simultaneously 101 is prime.

Danny Rorabaugh, Table of n, a(n) for n = 1..10000

Union of {1} and (intersection of A005574 and A006881).
Subsequence of A007422, A048943, A259020, A118369.
Cf. A007955, A007422, A048943, A052291, A118369, A258455, A259020, A259023, A258897.

[Floor(Sqrt(n-1)): n in [1..10000000] | IsPrime(n) and n-1 eq (&*(Divisors(Floor(Sqrt(n-1)))))];

Prepend[2*Select[Prime[Range[2, 300]], PrimeQ[4 #^2 + 1] &], 1] (* Amiram Eldar, Sep 25 2022 *)

a = [n for n in range(1,100000) if is_prime(n^2+1) and n^2==prod(list(divisors(n)))] # Danny Rorabaugh, Sep 21 2015

a(n) = 2*A052291(n) for n > 1. - Amiram Eldar, Sep 25 2022

A259023 Numbers n such that Product_{d|n} d = k^2 for some k > n and simultaneously number k^2 + 1 is a divisorial prime (A258455).

Original entry on oeis.org

24, 54, 56, 88, 154, 174, 238, 248, 266, 296, 328, 374, 378, 430, 442, 472, 488, 494, 498, 510, 568, 582, 584, 680, 710, 730, 742, 786, 856, 874, 894, 918, 962, 986, 1038, 1246, 1270, 1406, 1434, 1442, 1446, 1542, 1558, 1586, 1598

Offset: 1

Jaroslav Krizek, Sep 01 2015

nonn

Product_{d|n} d is the product of divisors of n (A007955).
If 1+ Product_{d|k} d for k > 2 is a prime p, then p-1 is a square.
With number 2 complement of A259021 with respect to A118369.
See A258897 - divisorial primes of the form 1 + Product_{d|a(n)} d.

			The number 24 is in sequence because A007955(24) = 331776 = 576^2 and simultaneously 331777 is prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A048943 (product of divisors of n is a square) and A118369 (numbers n such that Prod_{d|n} d + 1 is prime).

Cf. A007955, A258455, A259020, A259021, A258897.

[n: n in [1..2000] | &*(Divisors(n)) ne n^2 and IsSquare(&*(Divisors(n))) and IsPrime(&*(Divisors(n))+1)];

A007955(n)=if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2))
is(n)=my(t=A007955(n)); t>n^2 && issquare(t) && isprime(t+1) \\ Charles R Greathouse IV, Sep 01 2015

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.

A259020 Numbers k such that k^2 + 1 is a divisorial prime (A258455).

Original entry on oeis.org

1, 6, 10, 14, 26, 74, 94, 134, 146, 206, 314, 326, 386, 466, 576, 634, 674, 1094, 1174, 1294, 1306, 1354, 1366, 1546, 1654, 1766, 1774, 1894, 1966, 2026, 2126, 2174, 2326, 2594, 2654, 2746, 2916, 2974, 2986, 3046, 3106, 3134, 3136, 3214, 3254, 3274, 3314, 3326

Offset: 1

Jaroslav Krizek, Sep 01 2015

nonn

The divisorial primes are primes of the form p = 1 + Product_{d|k} d = 1 + A007955(k) for some k.

Supersequence of A259021. Subsequence of A005574. First deviation from A259021 is at a(15).

			The number 6 is in sequence because prime 37 = 6^2 + 1 is prime of the form p = 1 + Product_{d|k} d = 1 + A007955(k) for k = 6.

OEIS wiki, Divisorial primes

Cf. A005574, A007955, A048943, A118369, A258455, A258897, A259021, A259023.

Set(Sort([1] cat [Floor(Sqrt(&*(Divisors(n)))): n in [3..10000] | IsPrime(&*(Divisors(n))+1)]));

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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A258896 Divisorial primes p of the form p = 1 + k^2 where k^2 = Product_{d|k} d= A007955(k) for some k.

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A259021 Numbers k such that k^2 = Product_{d|k} d (= A007955(k)) and simultaneously k^2 + 1 is a divisorial prime (A258455).

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A259023 Numbers n such that Product_{d|n} d = k^2 for some k > n and simultaneously number k^2 + 1 is a divisorial prime (A258455).

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A259199 Divisorial primes ending with digit 1.

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A259020 Numbers k such that k^2 + 1 is a divisorial prime (A258455).

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