A057706 -id:A057706 - cp's OEIS frontend

A057704 Primorial - 1 prime indices: integers m such that the m-th primorial minus 1 is prime.

2, 3, 5, 6, 13, 24, 66, 68, 167, 287, 310, 352, 564, 590, 620, 849, 1552, 1849, 67132, 85586, 234725, 334023, 435582, 446895

Offset: 1

Labos Elemer, Oct 24 2000

There are two versions of "primorial": this is using the definition in A002110. - Robert Israel, Dec 30 2014

As of 28 February 2012, the largest known primorial prime is A002110(85586) - 1 with 476311 digits, found by the PrimeGrid project (see link). - Dmitry Kamenetsky, Aug 11 2015

			The 6th primorial is A002110(6) = 2*3*5*7*11*13 = 30030, and 30030 - 1 = 30029 is a prime, so 6 is in the sequence.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 114.

Chris K. Caldwell, Prime Pages: Database Search
Chris K. Caldwell, The top 20: primorial primes
Sílvia Casacuberta, On the divisibility of binomial coefficients, arXiv:1906.07652 [math.NT], 2019. Mentions this sequence.
Benny Lim, Prime Numbers Generated From Highly Composite Numbers, Parabola (2018) Vol. 54, Issue 3.
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Eric Weisstein's World of Mathematics, Primorial Prime
Wikipedia, Primorial prime

Cf. A006794 (Primorial -1 primes: Primes p such that -1 + product of primes up to p is prime).

Cf. A057705, A014545, A005234, A002110, A057706.

P:= 1:
p:= 1:
count:= 0:
for n from 1 to 1000 do
  p:= nextprime(p);
  P:= P*p;
  if isprime(P-1) then
    count:= count+1;
    A[count]:= n;
  fi
od:
seq(A[i],i=1..count); # Robert Israel, Dec 25 2014

a057704[n_] :=
Flatten@Position[
Rest[FoldList[Times, 1, Prime[Range[n]]]] - 1, Integer?PrimeQ]; a057704[500] (* _Michael De Vlieger, Dec 25 2014 *)

lista(nn) = {s = 1; for(k=1, nn, s *= prime(k); if(ispseudoprime(s - 1), print1(k, ", ")); ); } \\ Altug Alkan, Dec 08 2015

is(n) = ispseudoprime(prod(k=1, n, prime(k)) - 1); \\ Altug Alkan, Dec 08 2015

a(n) = A000720(A006794(n)).

a(n) = primepi(A006794(n)).

Corrected by Holzer Werner, Nov 28 2002
a(19)-a(20) from Eric W. Weisstein, Dec 08 2015 (Mark Rodenkirch confirms based on saved log files that all p < 700,000 have been tested)
a(21) from Jeppe Stig Nielsen, Oct 19 2021
a(22)-a(24) from Jeppe Stig Nielsen, Dec 16 2024

A060256 Smallest multiple a(n) of n-th primorial q(n) such that a(n)q(n)-1 and a(n)q(n)+1 are a pair of twin primes.

Original entry on oeis.org

2, 1, 1, 2, 1, 6, 8, 11, 4, 16, 22, 4, 74, 24, 37, 28, 14, 11, 242, 11, 91, 20, 83, 91, 35, 80, 48, 47, 226, 2, 12, 203, 30, 38, 356, 54, 266, 108, 305, 227, 173, 1185, 738, 13, 382, 277, 455, 433, 173, 1303, 926, 1162, 164, 298, 69, 121, 702, 1670, 36, 570, 170, 204

Offset: 1

Labos Elemer, Mar 22 2001

nonn

			30030*j-1 or 30030*j+1 are not both primes for j=1,2,3,4,5. But for j=6 {180179,180181} are twin primes. So a(6)=6.

Pierre CAMI, Table of n, a(n) for n = 1..500

Cf. A001359, A002110, A060229-A060232, A060255, A057706.

smp[n_]:=Module[{k=1},While[!PrimeQ[k*n+1]||!PrimeQ[k*n-1],k++];k]; Table[ smp[n],{n,FoldList[Times,Prime[Range[70]]]}] (* Harvey P. Dale, Oct 27 2016 *)

a(n)=p=vecprod(primes(n));for(k=1,+oo,ispseudoprime(k*p+1)&&ispseudoprime(k*p-1)&&return(k)) \\ Jeppe Stig Nielsen, Nov 09 2024

Corrected and extended by Ray Chandler, Apr 03 2009

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

A057822 Smaller of pair of twin primes whose average is lcm(1,...,m) for some m.

Original entry on oeis.org

5, 11, 59, 419, 232792559, 442720643463713815199

Offset: 1

Labos Elemer, Nov 08 2000

Known values of m such that lcm(1,...,m) is a twin prime mean value are as follows: {3, 4, 5, 6, 7, 19, 20, 21, 22, 47, 48}.
No more such primes occurs below m < 2000.
No more such primes occurs below m < 30000. - Amiram Eldar, Aug 18 2024

			419 and 421 are twin primes, (419 + 421)/2 = 420 = lcm(1,2,3,4,5,6,7).

Intersection of A057824 and {A049536(n)-2}.

Cf. A003418, A051451, A057706.

Select[FoldList[LCM, Select[Range[50], PrimePowerQ]] - 1, And @@ PrimeQ[# + {0, 2}] &] (* Amiram Eldar, Aug 18 2024 *)

lista(nn=50) = {for (i=1, nn, if (isprimepower(i), if (isprime(p=lcm([2..i])-1) && isprime(p+2), print1(p, ", "));););} \\ Michel Marcus, Aug 25 2019

A333058 0, 1, or 2 primes at primorial(n) +- 1.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Frank Ellermann, Mar 06 2020

nonn

a(n) = 0 marks a prime gap size of at least 2*prime(n+1)-1, e.g., primorial(8) +- prime(9) = {9699667,9699713} are primes, gap 2*23-1.
Mathworld reports that it is not known if there are an infinite number of prime Euclid numbers.
The tables in Ondrejka's collection contain no further primorial twin primes after {2309,2311} = primorial(13) +- 1 up to primorial(15877) +- 1 with 6845 digits.

			a(2) = a(3) = a(5) = 2: 2*3 +-1 = {5,7}, 6*5 +-1 = {29,31} and 210*11 +-1 = {2309,2311} are twin primes.
a(1) = a(4) = a(6) = 1: 1, 30*7 - 1 = 209 and 2310*13 + 1 = 30031 are not primes.
a(7) = 0: 510509 = 61 * 8369 and 510511 = 19 * 26869 are not primes.

H. Dubner, A new primorial prime, J. Rec. Math., 21 (No. 4, 1989), 276.

Chris K. Caldwell, the top 20: Primorial, 2012.
H. Dubner & N. J. A. Sloane, Correspondence, 1991, on A005234.
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 30029.
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 9699667.
Rudolf Ondrejka, The Top Ten: a Catalogue of Primal Configurations, 2001, tables 20, 20A, 20B.
Eric Weisstein's World of Mathematics, Primorial Prime.
Eric Weisstein's World of Mathematics, Euclid Number.

Cf. A096831, A002110 (primorials, p#), A057706.
Cf. A006862 (Euclid, p#+1), A005234 (prime p#+1), A014545 (index prime p#+1).
Cf. A057588 (Kummer, p#-1), A006794 (prime p#-1), A057704 (index prime p#-1).
Cf. A010051, A088411 (where a(n) is positive), A088257.

p:= proc(n) option remember; `if`(n<1, 1, ithprime(n)*p(n-1)) end:
a:= n-> add(`if`(isprime(p(n)+i), 1, 0), i=[-1, 1]):
seq(a(n), n=0..120);  # Alois P. Heinz, Mar 18 2020

primorial[n_] := primorial[n] = Times @@ Prime[Range[n]];
a[n_] := Boole@PrimeQ[primorial[n] - 1] + Boole@PrimeQ[primorial[n] + 1];
a /@ Range[0, 105] (* Jean-François Alcover, Nov 30 2020 *)

S = ''                     ;  Q = 1
do N = 1 to 27
   Q = Q * PRIME( N )
   T = ISPRIME( Q - 1 ) + ISPRIME( Q + 1 )
   S = S || ',' T
end N
S = substr( S, 3 )
say S                      ;  return S

a(n) = [ isprime(primorial(n) - 1) ] + [ isprime(primorial(n) + 1) ].

a(n) = Sum_{i in {-1,1}} A010051(primorial(n) + i).

cp's OEIS Frontend

A057704 Primorial - 1 prime indices: integers m such that the m-th primorial minus 1 is prime.

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A060256 Smallest multiple a(n) of n-th primorial q(n) such that a(n)*q(n)-1 and a(n)*q(n)+1 are a pair of 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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A057822 Smaller of pair of twin primes whose average is lcm(1,...,m) for some m.

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PARI

A333058 0, 1, or 2 primes at primorial(n) +- 1.

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A060256 Smallest multiple a(n) of n-th primorial q(n) such that a(n)q(n)-1 and a(n)q(n)+1 are a pair of 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.