A057704 -id:A057704 - cp's OEIS frontend

A002109 Hyperfactorials: Product_{k = 1..n} k^k.

1, 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, 55696437941726556979200000, 21577941222941856209168026828800000, 215779412229418562091680268288000000000000000, 61564384586635053951550731889313964883968000000000000000

Offset: 0

N. J. A. Sloane

A054374 gives the discriminants of the Hermite polynomials in the conventional (physicists') normalization, and A002109 (this sequence) gives the discriminants of the Hermite polynomials in the (in my opinion more natural) probabilists' normalization. See refs Wikipedia and Szego, eq. (6.71.7). - Alan Sokal, Mar 02 2012
a(n) = (-1)^n/det(M_n) where M_n is the n X n matrix m(i,j) = (-1)^i/i^j. - Benoit Cloitre, May 28 2002
a(n) = determinant of the n X n matrix M(n) where m(i,j) = B(n,i,j) and B(n,i,x) denote the Bernstein polynomial: B(n,i,x) = binomial(n,i)*(1-x)^(n-i)*x^i. - Benoit Cloitre, Feb 02 2003
Partial products of A000312. - Reinhard Zumkeller, Jul 07 2012
Number of trailing zeros (A246839) increases every 5 terms since the exponent of the factor 5 increases every 5 terms and the exponent of the factor 2 increases every 2 terms. - Chai Wah Wu, Sep 03 2014
Also the number of minimum distinguishing labelings in the n-triangular honeycomb rook graph. - Eric W. Weisstein, Jul 14 2017
Also shows up in a term in the solution to the generalized version of Raabe's integral. - Jibran Iqbal Shah, Apr 24 2021

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 135-145.
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 50.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 477.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
G. Szego, Orthogonal Polynomials, American Mathematical Society, 1981 edition, 432 Pages.

N. J. A. Sloane, Table of n, a(n) for n = 0..36
Christian Aebi and Grant Cairns, Generalizations of Wilson's Theorem for Double-, Hyper-, Sub-and Superfactorials, The American Mathematical Monthly 122.5 (2015): 433-443.
Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics 36(2), 2007, pp. 251-257. MR2312537. Zbl 1133.11012.
blackpenredpen, What is a Hyperfactorial? Youtube video (2018).
CreativeMathProblems, A beautiful integral | Raabe's integral, Youtube Video (2021).
Steven R. Finch, Glaisher-Kinkelin Constant (gives asymptotic expressions for A002109, A000178) [Broken link]
Steven R. Finch, Glaisher-Kinkelin Constant (gives asymptotic expressions for A002109, A000178) [From the Wayback machine]
Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
A. M. Ibrahim, Extension of factorial concept to negative numbers, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, 2, 30-42.
Jeffrey C. Lagarias and Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, arXiv:1409.4145 [math.NT], 2014.
Jean-Christophe Pain, Series representations for the logarithm of the Glaisher-Kinkelin constant, arXiv:2304.07629 [math.NT], 2023.
Jean-Christophe Pain, Bounds on the p-adic valuation of the factorial, hyperfactorial and superfactorial, arXiv:2408.00353 [math.NT], 2024. See p. 5.
Vignesh Raman, The Generalized Superfactorial, Hyperfactorial and Primorial functions, arXiv:2012.00882 [math.NT], 2020.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl., 332 (2007), 292-314; see Section 5.
László Tóth, Weighted gcd-sum functions, J. Integer Sequences, 14 (2011), Article 11.7.7.
Eric Weisstein's World of Mathematics, Hyperfactorial.
Eric Weisstein's World of Mathematics, K-Function.
Wikipedia, Hermite polynomials.
Index entries for sequences related to factorial numbers.
Index to divisibility sequences.

Cf. A000178, A000142, A000312, A001358, A002981, A002982, A100015, A005234, A014545, A018239, A006794, A057704, A057705, A054374.
Cf. A074962 [Glaisher-Kinkelin constant, also gives an asymptotic approximation for the hyperfactorials].
Cf. A001923, A051675, A240993, A255321, A255323, A255344.
Cf. A246839 (trailing 0's).
Cf. A347345, A347352.
Cf. A261175 (number of digits).

a002109 n = a002109_list !! n
a002109_list = scanl1 (*) a000312_list  -- Reinhard Zumkeller, Jul 07 2012

f := proc(n) local k; mul(k^k,k=1..n); end;
A002109 := n -> exp(Zeta(1,-1,n+1)-Zeta(1,-1));
seq(simplify(A002109(n)),n=0..11); # Peter Luschny, Jun 23 2012

Table[Hyperfactorial[n], {n, 0, 11}] (* Zerinvary Lajos, Jul 10 2009 *)
Hyperfactorial[Range[0, 11]] (* Eric W. Weisstein, Jul 14 2017 *)
Join[{1},FoldList[Times,#^#&/@Range[15]]] (* Harvey P. Dale, Nov 02 2023 *)

a(n)=prod(k=2,n,k^k) \\ Charles R Greathouse IV, Jan 12 2012

a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/prod(j=1,k+1,(1+j^j*x+x*O(x^n)) )), n) \\ Paul D. Hanna, Oct 02 2013

A002109 = [1]
for n in range(1, 10):
    A002109.append(A002109[-1]*n**n) # Chai Wah Wu, Sep 03 2014

a = lambda n: prod(falling_factorial(n,k) for k in (1..n))
[a(n) for n in (0..10)]  # Peter Luschny, Nov 29 2015

a(n)*A000178(n-1) = (n!)^n = A036740(n) for n >= 1.
Determinant of n X n matrix m(i, j) = binomial(i*j, i). - Benoit Cloitre, Aug 27 2003
a(n) = exp(zeta'(-1, n + 1) - zeta'(-1)) where zeta(s, z) is the Hurwitz zeta function. - Peter Luschny, Jun 23 2012
G.f.: 1 = Sum_{n>=0} a(n)*x^n / Product_{k=1..n+1} (1 + k^k*x). - Paul D. Hanna, Oct 02 2013
a(n) = A240993(n) / A000142(n+1). - Reinhard Zumkeller, Aug 31 2014
a(n) ~ A * n^(n*(n+1)/2 + 1/12) / exp(n^2/4), where A = 1.2824271291006226368753425... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Feb 20 2015
a(n) = Product_{k=1..n} ff(n,k) where ff denotes the falling factorial. - Peter Luschny, Nov 29 2015
log a(n) = (1/2) n^2 log n - (1/4) n^2 + (1/2) n log n + (1/12) log n + log(A) + o(1), where log(A) = A225746 is the logarithm of Glaisher's constant. - Charles R Greathouse IV, Mar 27 2020
From Amiram Eldar, Apr 30 2023: (Start)
Sum_{n>=1} 1/a(n) = A347345.
Sum_{n>=1} (-1)^(n+1)/a(n) = A347352. (End)
From Andrea Pinos, Apr 04 2024: (Start)
a(n) = e^(Integral_{x=1..n+1} (x - 1/2 - log(sqrt(2*Pi)) + (n+1-x)*Psi(x)) dx), where Psi(x) is the digamma function.
a(n) = e^(Integral_{x=1..n} (x + 1/2 - log(sqrt(2*Pi)) + log(Gamma(x+1))) dx). (End)

A014545 Numbers k such that the k-th Euclid number A006862(k) = 1 + (Product of first k primes) is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, 1391, 1613, 2122, 2647, 2673, 4413, 13494, 31260, 33237, 304723, 365071, 436504, 498865, 637491

Offset: 1

Eric W. Weisstein, Murray R. Bremner

			a(1) = 0 because the (empty) product of 0 primes is 1, plus 1 yields the prime 2.
prime(4413) = 42209 and Primorial(4413) + 1 = 42209# + 1 is a 18241-digit prime.
prime(13494) = 145823 and Primorial(13494) + 1 = 145823# + 1 is a 63142-digit prime.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 211, p. 61, Ellipses, Paris 2008.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 114.

C. K. Caldwell, Prime Pages: Database Search
C. K. Caldwell, Primorial Primes.
H. Ibstedt, A Few Smarandache Sequences, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, 170-183.
Benny Lim, Prime Numbers Generated From Highly Composite Numbers, Parabola (2018) Vol. 54, Issue 3.
Eric Weisstein's World of Mathematics, Euclid Number
Eric Weisstein's World of Mathematics, Primorial Prime
Eric Weisstein's World of Mathematics, Integer Sequence Primes

Cf. A005234 (values of p such that 1 + product of primes <= p is prime).
Cf. A018239 (primorial plus 1 primes).
Cf. A002110, A006862, A057704.

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, Nov 04 2015

Flatten[Position[Rest[FoldList[Times,1,Prime[Range[180]]]]+1,?PrimeQ]] (* _Harvey P. Dale, May 04 2012 *) (* this program generates the first 9 positive terms of the sequence; changing the Range constant to 33237 will generate all 23 terms above, but it will take a long time to do so *)

is(n)=ispseudoprime(prod(i=1,n,prime(i))+1) \\ Charles R Greathouse IV, Mar 21 2013

P=1; n=0; forprime(p=1, 10^5, if(ispseudoprime(P+1), print1(n", ")); n=n+1; P*=p;) \\ Hans Loeblich, May 10 2019

a(n+1) = A000720(A005234(n)). - M. F. Hasler, May 31 2018

More terms from Labos Elemer
a(21) from Arlin Anderson (starship1(AT)gmail.com), Oct 20 2000
a(22)-a(23) from Eric W. Weisstein, Mar 13 2004 (based on information in A057704)
Offset and first term changed by Altug Alkan, Nov 27 2015
a(24) from Jeppe Stig Nielsen, Aug 08 2024
a(25) from Jeppe Stig Nielsen, Sep 01 2024
a(26) from Jeppe Stig Nielsen, Sep 24 2024
a(27) from Jeppe Stig Nielsen, Nov 10 2024
a(28) from Jeppe Stig Nielsen, Aug 21 2025

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

Original entry on oeis.org

3, 5, 11, 13, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, 4583, 6569, 13033, 15877, 843301, 1098133, 3267113, 4778027, 6354977, 6533299

Offset: 1

N. J. A. Sloane

Or, p such that primorial(p) - 1 is prime.

Conjecture: if p# - 1 is a prime number, then the previous prime is greater than p# - exp(1)*p. - Arkadiusz Wesolowski, Jun 19 2016

H. Dubner, Factorial and primorial primes, J. Rec. Math., 19 (No. 3, 1987), 197-203.
R. K. Guy, Unsolved Problems in Number Theory, Section A2.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 4-5.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 111.

C. K. Caldwell, Prime Pages: Database Search.
C. K. Caldwell, Primorial Primes.
C. K. Caldwell, On the primality of n! +- 1 and 2*3*5*...*p +- 1, Math. Comput. 64, 889-890, 1995.
C. K. Caldwell and Y. Gallot, On the primality of n!+-1 and 2*3*5*...*p+-1, Math. Comp., 71 (2001), 441-448.
H. Dubner, Factorial and primorial primes, J. Rec. Math., 19 (No. 3, 1987), 197-203. (Annotated scanned copy)
Des MacHale, Infinitely many proofs that there are infinitely many primes, Math. Gazette, 97 (No. 540, 2013), 495-498.
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - _N. J. A. Sloane_, Jun 13 2012
PrimeGrid, Primorial Prime Search - Primes by User.
Eric Weisstein's World of Mathematics, Primorial Prime.

Cf. A057704 (Primorial - 1 prime indices: integers n such that the n-th primorial minus 1 is prime).

Cf. A057705, A002110, A005234, A014545, A018239.

primorial[p_] := Product[Prime[k], {k, 1, PrimePi[p]}]; Select[Prime[Range[1900]], PrimeQ[primorial[#] - 1] &] (* Jean-François Alcover, Mar 16 2011 *)
Transpose[With[{pr=Prime[Range[2000]]},Select[Thread[{Rest[FoldList[ Times,1,pr]], pr}], PrimeQ[ First[#]-1]&]]][[2]] (* Harvey P. Dale, Jun 21 2011 *)
With[{p = Prime[Range[200]]}, p[[Flatten[Position[Rest[FoldList[Times, 1, p]] - 1, ?PrimeQ]]]]] (* _Eric W. Weisstein, Nov 03 2015 *)

is(n)=isprime(n) && ispseudoprime(prod(i=1,primepi(n),prime(i))-1) \\ Charles R Greathouse IV, Apr 29 2015

from sympy import nextprime, isprime
A006794_list, p, q = [], 2, 2
while p < 10**5:
    if isprime(q-1):
        A006794_list.append(p)
    p = nextprime(p)
    q *= p # Chai Wah Wu, Apr 03 2021

a(n) = A000040(A057704(n)).

a(n) = prime(A057704(n)).

Stated incorrectly in CRC Standard Mathematical Tables and Formulae, 30th ed., 1996, p. 101; corrected in 2nd printing.
Corrected by Arlin Anderson (starship1(AT)gmail.com), who reports that he and Don Robinson have checked this sequence through about 63000 digits without finding another term (Jul 04 2000).
a(19)-a(20) from Eric W. Weisstein, Dec 08 2015 (Mark Rodenkirch confirms based on saved log files that all p < 700000 have been tested)
a(21) from Jeppe Stig Nielsen, Oct 19 2021
a(22)-a(24) from Jeppe Stig Nielsen, Dec 16 2024

A103514 a(n) is the smallest m such that primorial(n)/2 - 2^m is prime.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 25, 2, 1, 6, 6, 19, 1, 13, 3, 3, 11, 29, 2, 1, 6, 3, 4, 2, 6, 4, 15, 6, 4, 20, 4, 1, 7, 16, 4, 7, 22, 3, 12, 13, 9, 35, 2, 3, 3, 52, 35, 3, 32, 15, 13, 10, 53, 56, 9, 16, 36, 5, 8, 5, 22, 3, 14, 2, 64, 37, 8, 22, 42, 11, 22, 22, 12, 11, 26, 1, 54, 187, 20, 9

Offset: 2

Lei Zhou, Feb 15 2005

nonn

			P(2)/2-2^0=2 is prime, so a(2)=0;
P(10)/2-2^3=3234846607 is Prime, so a(10)=3.

R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2018.

Cf. A002110, A005234, A014545, A018239, A006794, A057704, A057705, A103153, A067026, A067027, A139439, A139440, A139441, A139442, A139443, A139444, A139445, A139446, A139447, A139448, A139449, A139450, A139451, A139452, A139453, A139454, A139455, A139456, A139457, A103514.

nmax = 2^8192; npd = 1; n = 2; npd = npd*Prime[n]; While[npd < nmax, tn = 1; tt = 2; cp = npd - tt; While[(cp > 1) && (! (PrimeQ[cp])), tn = tn + 1; tt = tt*2; cp = npd - tt]; If[cp < 2, Print["*"], Print[tn]]; n = n + 1; npd = npd*Prime[n]]
(* Second program: *)
k = 1; a = {}; Do[k = k*Prime[n]; m = 1; While[ ! PrimeQ[k - 2^m], m++ ]; Print[m]; AppendTo[a, m], {n, 2, 200}]; a (* Artur Jasinski, Apr 21 2008 *)

a(n)=my(t=prod(i=2,n,prime(i)),m); while(!isprime(t-2^m),m++); m \\ Charles R Greathouse IV, Apr 28 2015

Edited by N. J. A. Sloane, May 16 2008 at the suggestion of R. J. Mathar

A005234 Primes p such that 1 + product of primes up to p is prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547, 4787, 11549, 13649, 18523, 23801, 24029, 42209, 145823, 366439, 392113, 4328927, 5256037, 6369619, 7351117, 9562633

Offset: 1

N. J. A. Sloane

Conjecture: if p# + 1 is a prime number, then the next prime is less than p# + exp(1)*p. - Arkadiusz Wesolowski, Feb 20 2013

Conjecture: if p# + 1 is a prime, then the next prime is less than p# + p^2. - Thomas Ordowski, Apr 07 2013

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 134.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 211, p. 61, Ellipses, Paris 2008.
H. Dubner, A new primorial prime, J. Rec. Math., 21 (No. 4, 1989), 276.
R. K. Guy, Unsolved Problems in Number Theory, Section A2.
F. Le Lionnais, Les Nombres Remarquables, Paris, Hermann, 1983, p. 109, 1983.
Paulo Ribenboim, The New Book of Prime Number Records, p. 13.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 4-5.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 112.

C. K. Caldwell, Prime Pages: Database Search.
C. K. Caldwell, Primorial Primes.
C. K. Caldwell and Y. Gallot, On the primality of n!+-1 and 2*3*5*...*p+-1, Math. Comp., 71 (2001), 441-448.
H. Dubner, Factorial and primorial primes, J. Rec. Math., 19 (No. 3, 1987), 197-203. (Annotated scanned copy)
H. Dubner, A new primorial prime, J. Rec. Math., 21.4 (1989), 276. (Annotated scanned copy)
H. Dubner and N. J. A. Sloane, Correspondence, 1991.
Des MacHale, Infinitely many proofs that there are infinitely many primes, Math. Gazette, 97 (No. 540, 2013), 495-498.
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv:1202.3670 [math.HO], 2012.
R. Ondrejka, The Top Ten: a Catalogue of Primal Configurations.
Eric Weisstein's World of Mathematics, Euclid Number.
Eric Weisstein's World of Mathematics, Primorial Prime.

Cf. A006862 (Euclid numbers).
Cf. A014545 (Primorial plus 1 prime indices: n such that 1 + (Product of first n primes) is prime).
Cf. A018239 (Primorial plus 1 primes).
Cf. A002110, A006794, A057704.

[p:p in PrimesUpTo(3000)|IsPrime(&*PrimesUpTo(p)+1)]; // Marius A. Burtea, Mar 25 2019

N:= 5000: # to get all terms <= N
Primes:= select(isprime, [$2..N]):
P:= 1: count:= 0:
for n from 1 to nops(Primes) do
   P:= P*Primes[n];
   if isprime(P+1) then
     count:= count+1; A[count]:= Primes[n]
   fi
od:
seq(A[i],i=1..count); # Robert Israel, Nov 03 2015

(* This program is not convenient for large values of p *) p = pp = 1; Reap[While[p < 5000, p = NextPrime[p]; pp = pp*p; If[PrimeQ[1 + pp], Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Dec 31 2012 *)
With[{p = Prime[Range[200]]}, p[[Flatten[Position[Rest[FoldList[Times, 1, p]] + 1, ?PrimeQ]]]]] (* _Eric W. Weisstein, Nov 03 2015 *)

is(n)=isprime(n) && ispseudoprime(prod(i=1,primepi(n),prime(i))+1) \\ Charles R Greathouse IV, Feb 20 2013

is(n)=isprime(n) && ispseudoprime(factorback(primes([2,n]))+1) \\ M. F. Hasler, May 31 2018

a(n) = A000040(A014545(n+1)). - M. F. Hasler, May 31 2018

42209 sent in by Chris Nash (chrisnash(AT)cwix.com).
145823 discovered and sent in by Arlin Anderson (starship1(AT)gmail.com) and Don Robinson (donald.robinson(AT)itt.com), Jun 01 2000
366439, 392113 from Eric W. Weisstein, Mar 13 2004 (based on information in A014545)
a(23) from Jeppe Stig Nielsen, Aug 08 2024
a(24) from Jeppe Stig Nielsen, Sep 01 2024
a(25) from Jeppe Stig Nielsen, Sep 24 2024
a(26) from Jeppe Stig Nielsen, Nov 10 2024
a(27) from Jeppe Stig Nielsen, Aug 21 2025

A057705 Primorial primes: primes p such that p+1 is a primorial number (A002110).

Original entry on oeis.org

5, 29, 2309, 30029, 304250263527209, 23768741896345550770650537601358309, 19361386640700823163471425054312320082662897612571563761906962414215012369856637179096947335243680669607531475629148240284399976569

Offset: 1

Labos Elemer, Oct 24 2000

Chris K. Caldwell's The Top Twenty, Primorial.
Romeo Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - _N. J. A. Sloane_, Jun 13 2012

See A006794 and A057704 (the main entries for this sequence) for more terms.
Cf. A014545, A018239.
Cf. A002110, A010051.
Subsequence of A057588.

a057705 n = a057705_list !! (n-1)
a057705_list = filter ((== 1) . a010051) a057588_list
-- Reinhard Zumkeller, Mar 27 2013

Select[FoldList[Times, 1, Prime[Range[70]]], PrimeQ[# - 1] &] - 1 (* Harvey P. Dale, Jan 27 2014 *)

a(n) = A002110(A057704(n)) - 1.

A060270 Distance of n-th primorial from previous prime.

Original entry on oeis.org

1, 1, 11, 1, 1, 29, 23, 43, 41, 73, 59, 1, 89, 67, 73, 107, 89, 101, 127, 97, 83, 89, 1, 251, 131, 113, 151, 263, 251, 223, 179, 389, 281, 151, 197, 173, 239, 233, 191, 223, 223, 293, 593, 293, 457, 227, 311, 373, 257, 307, 313, 607, 347, 317, 307, 677, 467, 317

Offset: 2

Labos Elemer, Mar 23 2001

nonn

			Before 7th primorial 510481 is the largest prime. Its distance from 510510 is a(7)=29.

Robert G. Wilson v, Table of n, a(n) for n = 2..1000

Cf. A002110, A038710, A007014, A058044, A038711, A055211.

[seq(product(ithprime(j),j=1..n)-prevprime(product(ithprime(j),j=1..n)), n=2..50)];

Map[# - NextPrime[#, -1] &, Rest@ FoldList[Times, Prime@ Range[59]]] (* Michael De Vlieger, Aug 10 2023 *)

a(n) = my(P=vecprod(primes(n))); P-precprime(P-1); \\ Michel Marcus, Aug 11 2023

a(n)=1 for n=2, 3, 5, 6, 13, 24, 66, 68, 167, ... (A057704); a(n)=A055211(n) otherwise. - Jeppe Stig Nielsen, Oct 31 2003

More terms from Jeppe Stig Nielsen, Oct 31 2003

A128420 Numbers n such that p(n)# + p(n+1)# -1 is prime, where p(n)# is the product of first n primes (A002110).

Original entry on oeis.org

0, 1, 3, 6, 8, 10, 12, 37, 72, 92, 142, 295, 1529, 1625, 1914, 2276, 4423

Offset: 1

Pierre CAMI, Mar 02 2007

nonn

Cf. A002110, A057704, A128421.

Corrected by Emeric Deutsch, Mar 06 2007

Edited by Ray Chandler, Mar 13 2007

A057706 Smaller of twin primes whose average is a primorial number.

Original entry on oeis.org

5, 29, 2309

Offset: 1

Labos Elemer, Oct 24 2000

According to Caldwell, the next term, if it exists, has more than 100000 digits. - T. D. Noe, May 08 2012

			(5+7)/2 = 6 = 2*3, (29+31)/2 = 30 = 2*3*5, (2309+2311)/2 = 2310 = 2*3*5*7*11.

Chris K. Caldwell, Primorial prime.
Rudolf Ondrejka, The Top Ten: a Catalogue of Primal Configurations, 2001, tables 20A, 20B.
Eric Weisstein's World of Mathematics, Primorial Prime.
Index entries for sequences related to primorial numbers.

Cf. A000040 (primes), A002110 (primorials, p#).
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).

Select[FoldList[Times, Prime@ Range@ 40], AllTrue[# + {-1, 1}, PrimeQ] &] - 1 (* Michael De Vlieger, Jul 15 2017 *)

from sympy import isprime, prime, primerange
def auptoprimorial(limit):
  phash, alst = 1, []
  for p in primerange(1, prime(limit)+1):
    phash *= p
    if isprime(phash-1) and isprime(phash+1): alst.append(phash-1)
  return alst
print(auptoprimorial(5)) # Michael S. Branicky, May 29 2021

Offset corrected by Arkadiusz Wesolowski, May 08 2012

A103515 Primes of the form primorial P(k)*2^n-1 with minimal n, n>=0, k>=2.

Original entry on oeis.org

5, 29, 419, 2309, 30029, 1021019, 19399379, 892371479, 51757545839, 821495767572479, 14841476269619, 304250263527209, 54873078184468933509119, 2459559130353965639, 521426535635040715679, 15751252788463309939261439

Offset: 1

Lei Zhou, Feb 15 2005

nonn

Conjecture: sequence is defined for all k>=2

			P(2)*2^0-1=3*2-1=5 is prime, so a(2)=5;
P(4)*2^1-1=7*5*3*2*2-1=419 is prime, so a(4)=419;

Cf. A002110, A005234, A014545, A018239, A006794, A057704, A057705, A103153, A103154.

nmax = 2^2048; npd = 2; n = 2; npd = npd*Prime[n]; While[npd < nmax, tt = 1; cp = npd*tt - 1; While[ ! (PrimeQ[cp]), tt = tt*2; cp = npd*tt - 1]; Print[cp]; n = n + 1; npd = npd*Prime[n]]

cp's OEIS Frontend

A002109 Hyperfactorials: Product_{k = 1..n} k^k.

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A014545 Numbers k such that the k-th Euclid number A006862(k) = 1 + (Product of first k primes) is prime.

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

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A103514 a(n) is the smallest m such that primorial(n)/2 - 2^m is prime.

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A005234 Primes p such that 1 + product of primes up to p is prime.

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A057705 Primorial primes: primes p such that p+1 is a primorial number (A002110).

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