A014545 -id:A014545 - cp's OEIS frontend

A051342 Smallest prime factor of 1 + (product of first n primes).

3, 7, 31, 211, 2311, 59, 19, 347, 317, 331, 200560490131, 181, 61, 167, 953, 73, 277, 223, 54730729297, 1063, 2521, 22093, 265739, 131, 2336993, 960703, 2297, 149, 334507, 5122427, 1543, 1951, 881, 678279959005528882498681487, 87549524399, 23269086799180847

Offset: 1

Labos Elemer

nonn

Based on Euclid's proof that there are infinitely many primes.

Amiram Eldar, Table of n, a(n) for n = 1..87
M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors). [Annotated scanned copy]
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
R. G. Wilson v, Explicit factorizations

Cf. A014545, A002585.

a:= proc(n)
local N, F, i;
  N:= 1 + mul(ithprime(i),i=1..n);
  F:= select(type,map(t->t[1],ifactors(N,easy)[2]),integer);
if nops(F) >= 1 then return min(F) fi;
  min(map(t->t[1],ifactors(N)[2]))
end proc:
seq(a(n),n=1..40); # Robert Israel, Oct 19 2014

Map[FactorInteger,
   Table[Product[Prime[n], {n, 1, m}] + 1, {m, 1, 36}]][[All,
1]][[All, 1]] (* Geoffrey Critzer, Oct 19 2014 *)
FactorInteger[#][[1,1]]&/@(FoldList[Times,Prime[Range[40]]]+1) (* Harvey P. Dale, Oct 08 2021 *)

a(n) = factor(1 + prod(i=1, n, prime(i)))[1, 1]; \\ Michel Marcus, Dec 10 2013

a(n) = A020639(1+A002110(n)).

One more term from Michel Marcus, Dec 10 2013

A096177 Primes p such that primorial(p)/2 + 2 is prime.

Original entry on oeis.org

2, 3, 5, 7, 13, 29, 31, 37, 47, 59, 109, 223, 307, 389, 457, 1117, 1151, 2273, 9137, 10753, 15727, 25219, 26459, 29251, 30259, 52901, 194471

Offset: 1

Hugo Pfoertner, Jun 27 2004

a(27) > 172000. - Robert Price, May 10 2019

Some of the results were computed using the PrimeFormGW (PFGW) primality-testing program. - Hugo Pfoertner, Nov 16 2019

			a(3)=7 because primorial(7)/2 + 2 = A070826(4) + 2 = 2*3*5*7/2 + 2 = 107 is prime.

Cf. A070826, A096178 primes of the form primorial(p)/2+2, A096547 primes p such that primorial(p)/2-2 is prime, A067024 smallest p+2 that has n distinct prime factors, A014545 primorial primes, A087398.

k = 1; Do[If[PrimeQ[n], k = k*n; If[PrimeQ[k/2 + 2], Print[n]]], {n, 2, 100000}] (* Ryan Propper, Jul 03 2005 *)

P=1/2;forprime(p=2,1e4,if(isprime((P*=p)+2), print1(p", "))) \\ Charles R Greathouse IV, Mar 14 2011

7 additional terms, corresponding to probable primes, from Ryan Propper, Jul 03 2005
Edited by T. D. Noe, Oct 30 2008
a(26) from Robert Price, May 10 2019
a(27) from Tyler Busby, Mar 17 2024

A128421 Numbers k such that p(k)# + p(k+1)# + 1 is prime, where p(k)# is the product of first k primes (A002110).

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 20, 56, 101, 108, 141, 202, 265, 364, 401, 1035, 1588, 3062, 4191, 4579, 10373

Offset: 1

Pierre CAMI, Mar 02 2007

Cf. A002110, A014545, A128420.

Module[{nn=450,prmrl},prmrl=Partition[FoldList[Times,Prime[Range[nn]]],2,1];Position[prmrl,?(PrimeQ[#[[1]]+#[[2]]+ 1]&),1,Heads-> False]]//Flatten (* The program generates the first 15 terms of the sequence. *) (* _Harvey P. Dale, Sep 26 2024 *)

pd(n)=prod(i=1,n,prime(i)) \\ A002110
for(k=1,10^4,a=pd(k)+pd(k+1)+1;if(isprime(a),print1(k,", "))) \\ Alexandru Petrescu,Jun 17 2022

Corrected by Emeric Deutsch, Mar 06 2007
Edited by Ray Chandler, Mar 13 2007
a(21) from Michael S. Branicky, Oct 01 2024

A370129 Triangle read by rows: T(n,k) = A003415(A002110(n)+A002110(k)), 0 <= k <= n; arithmetic derivatives of the sums of two primorial numbers.

Original entry on oeis.org

1, 1, 4, 1, 12, 16, 1, 80, 60, 92, 1, 216, 540, 608, 704, 1, 3740, 3100, 4548, 6324, 8164, 568, 60080, 40060, 56292, 116208, 61768, 110752, 33975, 1021040, 1041768, 794468, 2415104, 1091004, 1357128, 1942844, 28300, 9789116, 29099520, 19722884, 18576860, 35347200, 35779644, 26575580, 37935056, 704080, 335024060

Offset: 0

Antti Karttunen, Feb 29 2024

Apart from those positions (A014545) at the left edge where a(n) = 1, a(n) <= A087112(1+n) only at n=2, 4 and 5, i.e., never after the third row.

			Triangle begins as:
      1;
      1,       4;
      1,      12,       16;
      1,      80,       60,       92;
      1,     216,      540,      608,      704;
      1,    3740,     3100,     4548,     6324,     8164;
    568,   60080,    40060,    56292,   116208,    61768,   110752;
  33975, 1021040,  1041768,   794468,  2415104,  1091004,  1357128,  1942844;
  28300, 9789116, 29099520, 19722884, 18576860, 35347200, 35779644, 26575580, 37935056;

Cf. A002110, A003415, A024451, A370121, A370135, A370136.
Cf. A014545 (positions of 1's at the left edge), A087112.
Cf. also A024451 (arithmetic derivatives of primorials).

A002110(n) = prod(i=1,n,prime(i));
A370121(n) = { my(c = (sqrtint(8*n + 1) - 1) \ 2); (A002110(c) + A002110(n - binomial(c + 1, 2))); };
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A370129(n) = A003415(A370121(n));

a(n) = A003415(A370121(n)).

For n, k >= 1, T(n,k) = A002110(k)*A370136(n,k) + A024451(k)*A370135(n,k).

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

A096547 Primes p such that primorial(p)/2 - 2 is prime.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 23, 31, 41, 53, 71, 103, 167, 431, 563, 673, 727, 829, 1801, 2699, 4481, 6121, 7283, 9413, 10321, 12491, 17807, 30307, 31891, 71917, 172517

Offset: 1

Hugo Pfoertner, Jun 27 2004

Some of the results were computed using the PrimeFormGW (PFGW) primality-testing program. - Hugo Pfoertner, Nov 14 2019

a(32) > 180000. - Tyler Busby, Mar 29 2024

			Prime 7 is a term because primorial(7)/2 - 2 = A034386(7)/2 - 2 = 2*3*5*7/2 - 2 = 103 is prime.

Cf. A070826, A096177 primes p such that primorial(p)/2+2 is prime, A096178 primes of the form primorial(p)/2+2, A014545 primorial primes, A087398.

Cf. A034386.

b:= proc(n) b(n):= `if`(n=0, 1, `if`(isprime(n), n, 1)*b(n-1)) end:
q:= p-> isprime(p) and isprime(b(p)/2-2):
select(q, [$1..500])[];

k = 1; Do[k *= Prime[n]; If[PrimeQ[k - 2], Print[Prime[n]]], {n, 2, 3276}] (* Ryan Propper, Oct 25 2005 *)
Prime[#]&/@Flatten[Position[FoldList[Times,Prime[Range[1000]]]/2-2,?PrimeQ]] (* _Harvey P. Dale, Jun 09 2023 *)

5 more terms from Ryan Propper, Oct 25 2005

a(29)-a(31) from Tyler Busby, Mar 16 2024

A136351 Primorial numbers p# such that p# + 1 is a prime.

Original entry on oeis.org

1, 2, 6, 30, 210, 2310, 200560490130

Offset: 1

Enoch Haga, Dec 25 2007

This sequence is different from A121069 and A002110.

a(8) = A002110(75) has 154 digits and is too long to be listed. - R. J. Mathar, Jul 23 2008

			a(6)=2310 is followed by prime 2311 whereas 30030 is not followed by a prime.

James C. McMahon, Table of n, a(n) for n = 1..10 (a(n) for n = 2..10 by Miles Englezou).

Cf. A002110, A014545, A018239, A121069, A136349, A136350, A136352.

Select[FoldList[Times, 1, Prime[Range[18]]],PrimeQ[#+1]&] (* James C. McMahon, May 08 2025 *)

S=[];for(n=0, 80, k=vecprod(primes(n)); if(isprime(k+1), S=concat(S,k))); S \\ Miles Englezou, Oct 28 2024

{A002110(j): A002110(j)+1 in A000040}. - R. J. Mathar, Jul 23 2008
a(n) = A002110(A014545(n)). - Michel Marcus, Apr 05 2021
a(n) = A018239(n) - 1. - James C. McMahon, May 08 2025

Changed a(1) from 4 to 2 and edited by R. J. Mathar, Jul 23 2008

a(1)=1 inserted by James C. McMahon, May 08 2025

A087398 Primes of the form primorial(P(k))/2-2.

Original entry on oeis.org

13, 103, 1153, 15013, 255253, 4849843, 111546433, 100280245063, 152125131763603, 16294579238595022363, 278970415063349480483707693, 11992411764462614086353260819346129198103, 481473710367991963528473107950567214598209565303106537707981745633

Offset: 1

Cino Hilliard, Oct 21 2003

nonn

Twinmorial numbers are the partial products of twin primes. Sum of reciprocals = 0.08756985926348207565388288916..

The next term (a(14)) has 174 digits. - Harvey P. Dale, Mar 30 2013

Charles R Greathouse IV, Table of n, a(n) for n = 1..19

Cf. A096177 primes k such that primorial(k)/2+2 is prime, A096178 primes of the form primorial(k)/2+2, A096547 Primes k such that primorial(k)/2-2 is prime, A067024 smallest p+2 that has n distinct prime factors, A014545 primorial primes.

Select[#/2-2&/@Rest[FoldList[Times,1,Prime[Range[100]]]],PrimeQ] (* Harvey P. Dale, Mar 30 2013 *)

twimorial(n) = { s=0; p=3; forprime(x=5,n, if(isprime(x-2),c1++); p=p*x; if(isprime(p-2), print1(p-2","); c2++; s+=1.0/(p-2); ) ); print(); print(s) }

v=[];pr=1; forprime(p=3,2357,pr*=p; if(ispseudoprime(pr-2),v=concat(v,pr-2))) \\ Charles R Greathouse IV, Feb 14 2011

Twins 3*5 = 15 = p+2. p=13.

Description corrected by Hugo Pfoertner, Jun 25 2004

One more term (a(13)) added by Harvey P. Dale, Mar 30 2013

A096178 Primes of the form primorial(p)/2+2.

Original entry on oeis.org

3, 5, 17, 107, 15017, 3234846617, 100280245067, 3710369067407, 307444891294245707, 961380175077106319537, 139867498408927468089138080936033904837498617

Offset: 1

Hugo Pfoertner, Jun 27 2004

nonn

Primes of the form A070826(n)+2.

			a(4) = 107 because 107 is a prime of the form primorial(7)/2 + 2 = A070826(4) + 2 = 2*3*5*7/2 + 2.

Amiram Eldar, Table of n, a(n) for n = 1..18

Cf. A070826, A096177 (primorial(p)/2+2 is prime), A096547 (primorial(p)/2-2 is prime), A067024 (smallest p+2 that has n distinct prime factors), A014545 (primorial primes), A087398.

for(n=1,30,p=prod(k=1,n,prime(k))/2+2;if(ispseudoprime(p),print1(p,", "))) \\ Hugo Pfoertner, Dec 26 2019

a(n) = A070826(A096177(n)) + 2. - Amiram Eldar, Dec 26 2019

a(1) inserted by Amiram Eldar, Dec 26 2019

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,", ")))

cp's OEIS Frontend

A051342 Smallest prime factor of 1 + (product of first n primes).

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A096177 Primes p such that primorial(p)/2 + 2 is prime.

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A128421 Numbers k such that p(k)# + p(k+1)# + 1 is prime, where p(k)# is the product of first k primes (A002110).

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A370129 Triangle read by rows: T(n,k) = A003415(A002110(n)+A002110(k)), 0 <= k <= n; arithmetic derivatives of the sums of two primorial numbers.

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

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A096547 Primes p such that primorial(p)/2 - 2 is prime.

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A136351 Primorial numbers p# such that p# + 1 is a prime.

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