A258455 -id:A258455 - cp's OEIS frontend

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

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

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)]));

A258897 Divisorial primes p such that p-1 = Product_{d|k} d for some k < sqrt(p-1).

Original entry on oeis.org

331777, 8503057, 9834497, 59969537, 562448657, 916636177, 3208542737, 3782742017, 5006411537, 7676563457, 11574317057, 19565295377, 34188010001, 38167092497, 49632710657, 56712564737, 59553569297, 61505984017, 104086245377, 114733948177

Offset: 1

Jaroslav Krizek, Jun 20 2015

nonn

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 from A258455 such that p-1 = A007955(k) for some k < 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 second kind. Divisorial primes of the first kind are in A258896.
With number 3, complement of A258896 with respect to A258455.
With numbers 2 and 3, divisorial primes p that are not of the form 4*q^2 + 1 where q = prime.
See A259023 - numbers n such that Product_{d|n} d is a divisorial prime from this sequence.

			Prime p = 331777 is in sequence because p - 1 = 331776 = 576^2 is the product of divisors of 24 and 24 < 576.

Jaroslav Krizek and Chai Wah Wu, Table of n, a(n) for n = 1..500 [a(n) for n = 1..43 from Jaroslav Krizek].

Cf. A007955, A258455, A258896, A259023, A259199.

Set(Sort([&*(Divisors(n))+1: n in [1..1000] | &*(Divisors(n)) ne n^2 and IsSquare(&*(Divisors(n))) and IsPrime(&*(Divisors(n))+1)]));

[n: n in [A258455(n)] | not IsPrime(Floor(Sqrt(n-1)) div 2)];

A118370 Divisorial primes: Primes p such that p = 1 + Product_{d|n} d for some n (ordered by n).

Original entry on oeis.org

2, 3, 37, 101, 197, 331777, 677, 8503057, 9834497, 5477, 59969537, 8837, 17957, 21317, 562448657, 916636177, 42437, 3208542737, 3782742017, 5006411537, 7676563457, 98597, 106277, 11574317057, 19565295377, 416806419029812551937, 148997, 34188010001, 38167092497

Offset: 1

Rick L. Shepherd, Apr 25 2006

nonn

See A118369 for the corresponding n. These are primes in the sequence 1 + A007955. (The suggested name "divisorial prime" is obviously analogous to that of factorial primes (A088332) and primorial primes (A014545).).

			The prime 37 is a(3) as there exists a number, A118369(3)=6, such that 37 = 6*3*2*1 + 1, where {1,2,3,6} are all the positive divisors of 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A118369, A007955.
Cf. A014545, A088332.
Cf. A258455 (sorted).

Reap[For[n = 1, n <= 500, n++, p = Times @@ Divisors[n]; If[PrimeQ[p+1], Sow[p+1]]]][[2, 1]] (* Jean-François Alcover, Oct 07 2016 *)

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

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).

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.

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

cp's OEIS Frontend

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

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A258897 Divisorial primes p such that p-1 = Product_{d|k} d for some k < sqrt(p-1).

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A118370 Divisorial primes: Primes p such that p = 1 + Product_{d|n} d for some n (ordered by n).

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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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A259199 Divisorial primes ending with digit 1.

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