A060256 -id:A060256 - cp's OEIS frontend

A117847 Numbers k such that A060256(k)*prime(k)# - 1 is a Sophie Germain prime, where prime(k)# is the k-th primorial.

1, 2, 3, 4, 8, 9, 10, 15, 24, 35, 37, 79, 340

Offset: 1

Pierre CAMI, May 01 2006

This sequence gives the firsts of twin primes (A060256(n)*prime(n)# - 1, A060256(n)*prime(n)# + 1) which are also Sophie Germain primes.

a(14) > 367. - Amiram Eldar, Sep 11 2021

			16*(29#)-1 is the first of twin primes, 16 = A060256(10), 2*(16*(29#)-1)+1 is prime so 16*(29#)-1 is a Sophie Germain prime.

Cf. A002110, A005384, A060256.

q[n_] := Module[{p = Product[Prime[i], {i,1,n}], k=1}, While[!PrimeQ[k*p-1] || !PrimeQ[k*p+1], k++]; PrimeQ[2*k*p - 1]]; Select[Range[100], q] (* Amiram Eldar, Sep 11 2021 *)

a(1)-a(6) inserted by Amiram Eldar, Sep 11 2021

A063983 Least k such that k*2^n +/- 1 are twin primes.

Original entry on oeis.org

4, 2, 1, 9, 12, 6, 3, 9, 57, 30, 15, 99, 165, 90, 45, 24, 12, 6, 3, 69, 132, 66, 33, 486, 243, 324, 162, 81, 90, 45, 345, 681, 585, 375, 267, 426, 213, 429, 288, 144, 72, 36, 18, 9, 147, 810, 405, 354, 177, 1854, 927, 1125, 1197, 666, 333, 519, 1032, 516, 258, 129, 72

Offset: 0

Robert G. Wilson v, Sep 06 2001

nonn

Excluding the first three terms, all remaining terms have digital root 3, 6, or 9. - J. W. Helkenberg, Jul 24 2013

			a(3) = 9 because 9*2^3 = 72 and 71 and 73 are twin primes.
a(6) = 3 because 3*2^6 = 192 and {191, 193} are twin primes.
a(71) = 630 because 630*2^71 = 1487545442103938242314240 and {1487545442103938242314239, 1487545442103938242314241} are twin primes.

Richard Crandall and Carl Pomerance, 'Prime Numbers: A Computational Perspective,' Springer-Verlag, NY, 2001, page 12.

Pierre CAMI, Table of n, a(n) for n = 0..2300

Cf. A040040, A045753, A002822, A124065, A124518-A124522.
Cf. A071256, A060210, A060256. For records see A125848, A125019.
Cf. A076806 (requires odd k).

Table[Do[s=(2^j)*k; If[PrimeQ[s-1]&&PrimeQ[s+1],Print[{j,k}]], {k,1,2*j^2}],{j,0,100}]; (* outprint of a[j]=k *)
Do[ k = 1; While[ ! PrimeQ[ k*2^n + 1 ] || ! PrimeQ[ k*2^n - 1 ], k++ ]; Print[ k ], {n, 0, 50} ]
f[n_] := Block[{k = 1},While[Nand @@ PrimeQ[{-1, 1} + 2^n*k], k++ ];k];Table[f[n], {n, 0, 60}] (* Ray Chandler, Jan 09 2009 *)

More terms from Labos Elemer, May 24 2002

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar

A071407 Least k such that kprime(n) + 1 and kprime(n) - 1 are twin primes.

Original entry on oeis.org

2, 2, 6, 6, 18, 24, 6, 12, 6, 12, 42, 54, 30, 24, 6, 120, 18, 258, 24, 18, 84, 132, 54, 48, 114, 42, 6, 6, 48, 24, 144, 30, 6, 12, 12, 78, 24, 36, 30, 54, 132, 18, 90, 36, 66, 18, 42, 30, 120, 30, 36, 42, 18, 18, 54, 84, 60, 12, 210, 12, 6, 60, 150, 102, 6, 210, 30, 24, 6

Offset: 1

Labos Elemer, May 24 2002

Note that 6 divides a(n) for n > 2. - T. D. Noe, Jan 07 2013

			n=4: prime(4)=7, a(4)=6 because 6*prime(4)=42 and {41,43} are primes.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A071256, A071404-A071406, A060256, A060210, A090530, A294731.
Cf. A071558 (k at every integer).
Cf. A220141, A220142 (record values).

a071407 n = head [k | k <- [2,4..], let x = k * a000040 n,
                      a010051' (x - 1) == 1, a010051' (x + 1) == 1]
-- Reinhard Zumkeller, Feb 14 2013

Table[fl=1; Do[s=(Prime[j])*k; If[PrimeQ[s-1]&&PrimeQ[s+1]&&Equal[fl, 1], Print[{j, k}]; fl=0], {k, 1, 2*j^2}], {j, 0, 100}]

From Amiram Eldar, Aug 25 2025: (Start)
a(n) = A090530(n) / prime(n).
a(n) = 6 * A294731(n) for n >= 3. (End)

A060255 Smaller of twin primes {p, p+2} whose average p+1 = kq is the least multiple of the n-th primorial number q such that kq-1 and k*q+1 are twin primes.

Original entry on oeis.org

3, 5, 29, 419, 2309, 180179, 4084079, 106696589, 892371479, 103515091679, 4412330782859, 29682952539239, 22514519501013539, 313986271960080719, 22750921955774182169, 912496437361321252439, 26918644902158976946979, 1290172194953476680815969, 1901713815361424627522739779

Offset: 1

Labos Elemer, Mar 22 2001

nonn

a(349) has 1001 digits. - Michael S. Branicky, Apr 19 2025

			a(13) = -1 + (2*3*5*7*...*41)*k(13) = 304250263527210*74 and {22514519501013539, 22514519501013542} are the corresponding primes; k(13)=74 is the smallest suitable multiplier. Twin primes obtained from primorial numbers with k=1 multiplier seem to be much rarer (see A057706).
For j=1,2,3,4,5,6, a(j)=A001359(1), A059960(1), A060229(1), A060230(1), A060231(1), A060232(1) respectively.

Michael S. Branicky, Table of n, a(n) for n = 1..348

Cf. A001359, A002110, A006794, A014545, A057706, A059960, A060229, A060230, A060231, A060232.

a(n) = {my(q = prod(k=1, n, prime(k))); for(k=1, oo, if (isprime(q*k-1) && isprime(q*k+1), return(q*k-1)););} \\ Michel Marcus, Jul 10 2018

from itertools import count
from sympy import primorial, isprime
def a(n):
    p = primorial(n)
    return next(m-1 for m in count(p, p) if isprime(m-1) and isprime(m+1))
print([a(n) for n in range(1, 20)]) # Michael S. Branicky, Apr 18 2025

a(n) = p = k(n)*q(n)-1, where q(n)=A002110(n) and k(n)=A060256(n) is the smallest integer whose multiplication by the n-th primorial yields p+1.

a(2)=5 corrected by Ray Chandler, Apr 03 2009

a(18) and beyond from Michael S. Branicky, Apr 18 2025

A071406 a(n) is the smallest multiplier of n! such that -1+a(n)n! and 1+a(n)n! are both primes.

Original entry on oeis.org

4, 2, 1, 3, 2, 17, 7, 6, 3, 14, 29, 30, 48, 27, 9, 24, 12, 97, 78, 47, 71, 80, 55, 13, 57, 20, 81, 259, 108, 163, 81, 118, 63, 215, 173, 513, 420, 561, 537, 1162, 158, 33, 122, 286, 459, 391, 305, 288, 114, 307, 15, 680, 355, 365, 338, 70, 23

Offset: 1

Labos Elemer, May 24 2002

nonn

			n=7: a(7)=7, 7!=5040, 7.7!=35280 and {35279,35281} are primes.

Pierre CAMI, Table of n, a(n) for n = 1..300

Cf. A063983, A071256, A060256.

Table[fl=1; Do[s=(j!)*k; If[PrimeQ[s-1]&&PrimeQ[s+1]&&Equal[fl, 1], Print[{j, k}]; fl=0], {k, 1, 2*j^2}], {j, 0, 100}]
smnf[n_]:=Module[{k=1,f=n!},While[!PrimeQ[k*f+1]||!PrimeQ[k*f-1],k++]; k]; Array[smnf,60] (* Harvey P. Dale, May 24 2016 *)

A329916 Smallest k such that 6kA057130(n)-1 and 6kA057130(n)+1 are twin primes.

Original entry on oeis.org

1, 2, 3, 23, 11, 18, 77, 46, 84, 76, 22, 30, 3, 107, 26, 198, 136, 23, 236, 284, 167, 269, 381, 405, 379, 374, 620, 481, 606, 505, 163, 1414, 348, 639, 1696, 1429, 850, 2050, 740, 117, 362, 35, 3961, 72, 1307, 1816, 9410, 5705, 972, 368, 5083, 4387, 3296, 6039

Offset: 1

Pierre CAMI, Nov 24 2019

nonn

A057130 gives the product of prime numbers (-1 mod 6) in the order of occurrence.

			A057130(1)=5, 6*1*5-1=29, and 29 and 31 are twin primes, so a(1)=1.
A057130(2)=55, 6*2*55-1=659, and 659 and 661 are twin primes, so a(2)=2.

Cf. A001359, A006512, A007528, A057130, A060256, A173937, A329736.

lista(nn) = {my(pp = 1); forprime (p = 1, nn, if (Mod(p, 6) == -1, pp *= p; my(k=1); while (!isprime(6*k*pp-1) || !isprime(6*k*pp+1), k++); print1(k, ", ");););} \\ Michel Marcus, Nov 25 2019

A382785 a(n) is the least multiple of the n-th primorial such that both a(n)-1 and a(n)+1 are prime and the prime factors of a(n) do not exceed prime(n).

Original entry on oeis.org

4, 6, 30, 420, 2310, 180180, 4084080, 106696590, 892371480, 103515091680, 4412330782860, 29682952539240, 22514519501013540, 313986271960080720, 22750921955774182170, 912496437361321252440, 26918644902158976946980, 1290172194953476680815970, 1901713815361424627522739780

Offset: 1

Rory Pulvino, Apr 04 2025

nonn

a(n) is the smallest multiple k of the n-th primorial, prime(n)#, such that both k-1 and k+1 are prime and the prime factors of m = k/prime(n)# do not exceed prime(n).
From Michael S. Branicky, Apr 19 2025: (Start)
a(n) first differs from A060255(n) + 1 at n = 29.
a(349) has 1001 digits. (End)

			For a(2), (2*3)*1 = 6 and the first twin primes are 5, 7.
For a(3), (2*3*5)*1 = 30 and the first twin primes are 29, 31.
For a(4), (2*3*5*7)*2 = 420, the first twin primes are 419, 421 and 2 <= prime(4).
For a(5), (2*3*5*7*11)*1 = 2310 and the first twin primes are 3209, 3211.
For a(6), (2*3*5*7*11*13)*2*3 = 180180. the first twin primes are 180179, 180181 and 2, 3 <= prime(6).

Michael S. Branicky, Table of n, a(n) for n = 1..348

Supersequence of A088256.

Cf. A000040, A002110, A060255, A060256, A088257.

a[n_] := Module[{P,k},P=Product[Prime[i],{i, 1, n}];k = 1; While[!(PrimeQ[k*P-1] && PrimeQ[k*P+1]), k++];k*P] (* James C. McMahon, May 09 2025 *)

isok(k, p) = if (k>1, vecmax(factor(k)[,1])<=p, 1);
a(n) = my(P=vecprod(primes(n)), k=1, p=prime(n)); while(!(isok(k, p) && ispseudoprime(k*P-1) && ispseudoprime(k*P+1)), k++); k*P; \\ Michel Marcus, Apr 27 2025

from itertools import count
from sympy import factorint, isprime, prime, primorial
def a(n):
    pn, prn = prime(n), primorial(n)
    return next(k for m in count(1) if max(factorint(m), default=1)<=pn and isprime((k:=m*prn)-1) and isprime(k+1))
print([a(n) for n in range(1, 20)]) # Michael S. Branicky, Apr 18 2025

Data corrected by Michael S. Branicky, Apr 18 2025

A096937 Least k such that kP(n)#/2 - 2 and kP(n)#/2 + 2 are both primes, where P(i)= i-th prime, P(i)# = i-th primorial.

Original entry on oeis.org

5, 3, 1, 1, 3, 1, 3, 41, 27, 3, 1, 171, 97, 19, 35, 13, 217, 57, 79, 133, 41, 219, 85, 43, 477, 205, 35, 455, 635, 275, 2081, 33, 513, 671, 427, 177, 997, 2671, 601, 123, 525, 1139, 411, 479, 363, 1311, 4685, 109, 159, 3367, 2761, 257, 161, 137, 49, 393, 3553, 1807

Offset: 1

Pierre CAMI, Aug 18 2004

nonn

			1*2*3*5*7/2 - 2 = 103, 1*2*3*5*7/2 + 2 = 107, 103 and 107 are both primes, so for n=4, k=1.

Cf. A060256.

Primorial[n_] := Product[Prime[i], {i, 1, n}]; f[n_] := Block[{p = Primorial[n]/2, k = 1}, While[ !PrimeQ[k*p - 2] || !PrimeQ[k*p + 2], k++ ]; k]; Table[ f[n], {n, 50}] (* Robert G. Wilson v, Aug 19 2004 *)

More terms from Robert G. Wilson v, Aug 19 2004

A100804 Smallest prime P such that nP# -1 and nP# +1 are twin primes, where P#=primorial P, or 0 if no such prime exists.

Original entry on oeis.org

3, 2, 2, 11, 3, 2, 3, 5, 2, 3, 7, 3, 7, 5, 2, 7, 3, 3, 5, 5, 2, 5, 3, 11, 3

Offset: 1

Pierre CAMI, Jan 04 2005

No solutions found yet for n = {26, 39, 46, 59, 63, 68, 76, 81, 82, 84, 89} through prime(1700) = 14519. - Ray Chandler, Jan 23 2005

The sequence continues: a(26)=?, 5, 7, 7, 2, 19, 3, 3, 5, 5, 2, 19, 3, a(39)=?, 3, 5, 7, 5, 5, 3, a(46)=?, 3, 11, 17, 7, 2, 3, 43, 2, 7, 37, 7, 3, a(59)=?, 151, 31, 13, a(63)=?. - Robert G. Wilson v, Jan 12 2005

			For n=4:
4*2=8 8-1=7 prime but 8+1=9=3*3.
4*2*3=24 24-1=23 prime but 24+1=25=5*5.
4*2*3*5=120 120-1=119=7*17.
4*2*3*5*7=840 840-1=839 prime but 840+1=841=29*29.
4*2*3*5*7*11=9240 9240-1=9239 prime 9240+1=9241 prime so for n=4 P=11.

Cf. A060256.

Primorial[n_] := Product[Prime[i], {i, n}]; f[n_] := Block[{k = 1}, While[p = n*Primorial[k]; !PrimeQ[p - 1]\ || ! PrimeQ[p + 1], k++ ]; Prime[k]]; Table[ f[n], {n, 25}] (* Robert G. Wilson v, Jan 12 2005 *)

cp's OEIS Frontend

A117847 Numbers k such that A060256(k)*prime(k)# - 1 is a Sophie Germain prime, where prime(k)# is the k-th primorial.

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A063983 Least k such that k*2^n +/- 1 are twin primes.

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A071407 Least k such that k*prime(n) + 1 and k*prime(n) - 1 are twin primes.

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A060255 Smaller of twin primes {p, p+2} whose average p+1 = k*q is the least multiple of the n-th primorial number q such that k*q-1 and k*q+1 are twin primes.

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A071406 a(n) is the smallest multiplier of n! such that -1+a(n)*n! and 1+a(n)*n! are both primes.

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A329916 Smallest k such that 6*k*A057130(n)-1 and 6*k*A057130(n)+1 are twin primes.

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PARI

A382785 a(n) is the least multiple of the n-th primorial such that both a(n)-1 and a(n)+1 are prime and the prime factors of a(n) do not exceed prime(n).

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Extensions

A096937 Least k such that k*P(n)#/2 - 2 and k*P(n)#/2 + 2 are both primes, where P(i)= i-th prime, P(i)# = i-th primorial.

A071407 Least k such that kprime(n) + 1 and kprime(n) - 1 are twin primes.

A060255 Smaller of twin primes {p, p+2} whose average p+1 = kq is the least multiple of the n-th primorial number q such that kq-1 and k*q+1 are twin primes.

A071406 a(n) is the smallest multiplier of n! such that -1+a(n)n! and 1+a(n)n! are both primes.

A329916 Smallest k such that 6kA057130(n)-1 and 6kA057130(n)+1 are twin primes.

A096937 Least k such that kP(n)#/2 - 2 and kP(n)#/2 + 2 are both primes, where P(i)= i-th prime, P(i)# = i-th primorial.

A100804 Smallest prime P such that nP# -1 and nP# +1 are twin primes, where P#=primorial P, or 0 if no such prime exists.