A006862 -id:A006862 - cp's OEIS frontend

A005235 Fortunate numbers: least m > 1 such that m + prime(n)# is prime, where p# denotes the product of all primes <= p.

3, 5, 7, 13, 23, 17, 19, 23, 37, 61, 67, 61, 71, 47, 107, 59, 61, 109, 89, 103, 79, 151, 197, 101, 103, 233, 223, 127, 223, 191, 163, 229, 643, 239, 157, 167, 439, 239, 199, 191, 199, 383, 233, 751, 313, 773, 607, 313, 383, 293, 443, 331, 283, 277, 271, 401, 307, 331

Offset: 1

N. J. A. Sloane

Reo F. Fortune conjectured that a(n) is always prime.
You might be searching for Fortunate Primes, which is an alternative name for this sequence. It is not the official name yet, because it is possible, although unlikely, that not all the terms are primes. - N. J. A. Sloane, Sep 30 2020
The first 500 terms are primes. - Robert G. Wilson v. The first 2000 terms are primes. - Joerg Arndt, Apr 15 2013
The strong form of Cramér's conjecture implies that a(n) is a prime for n > 1618, as previously noted by Golomb. - Charles R Greathouse IV, Jul 05 2011
a(n) is the smallest m such that m > 1 and A002110(n) + m is prime. For every n, a(n) must be greater than prime(n+1) - 1. - Farideh Firoozbakht, Aug 20 2003
If a(n) < prime(n+1)^2 then a(n) is prime. According to Cramér's conjecture a(n) = O(prime(n)^2). - Thomas Ordowski, Apr 09 2013
Conjectures from Pierre CAMI, Sep 08 2017: (Start)
If all terms are prime, then lim_{N->oo} (Sum_{n=1..N} primepi(a(n))) / (Sum_{n=1..N} n) = 3/2, and primepi(a(n))/n < 6 for all n.
Limit_{N->oo} (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} prime(n)) = Pi/2.
a(n)/prime(n) < 8 for all n. (End)
Conjecture: Limit_{N->oo} (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} prime(n)) = 3/2. - Alain Rocchelli, Dec 24 2022
The name "Fortunate numbers" was coined by Golomb (1981) after the New Zealand social anthropologist Reo Franklin Fortune (1903 - 1979). According to Golomb, Fortune's conjecture first appeared in print in Martin Gardner's Mathematical Games column in 1980. - Amiram Eldar, Aug 25 2020

			a(4) = 13 because P_4# = 2*3*5*7 = 210, plus one is 211, the next prime is 223 and the difference between 210 and 223 is 13.

Martin Gardner, The Last Recreations, Chapter 12: Strong Laws of Small Primes, Springer-Verlag, 1997, pp. 191-205, especially pp. 194-195.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 1994, Section A2, p. 11.
Stephen P. Richards, A Number For Your Thoughts, 1982, p. 200.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 114-115.
David Wells, Prime Numbers: The Most Mysterious Figures In Math, Hoboken, New Jersey: John Wiley & Sons (2005), pp. 108-109.

Pierre CAMI, Table of n, a(n) for n = 1..3000 (first 2000 terms from T. D. Noe)
Ray Abrahams and Huon Wardle, Fortune's 'Last Theorem', Cambridge Anthropology, Vol. 23, No. 1 (2002), pp. 60-62.
Cyril Banderier, Conjecture checked for n < 1000 [It has been reported that the data given here contains several errors]
C. K. Caldwell, Fortunate number, The Prime Glossary.
Antonín Čejchan, Michal Křížek, and Lawrence Somer, On Remarkable Properties of Primes Near Factorials and Primorials, Journal of Integer Sequences, Vol. 25 (2022), Article 22.1.4.
Martin Gardner, Patterns in primes are a clue to the strong law of sma11 numbers, Mathematical Games, Scientific American, Vol. 243, No. 6 (December, 1980), pp. 18-28.
Solomon W. Golomb, The evidence for Fortune's conjecture, Mathematics Magazine, Vol. 54, No. 4 (1981), pp. 209-210.
Richard K. Guy, Letter to N. J. A. Sloane, 1987
Richard K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
Richard K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Bill McEachen, McEachen Conjecture
Romeo Meštrović, 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.
Eric Weisstein's World of Mathematics, Fortunate Prime
Robert G. Wilson v, Letter to N. J. A. Sloane with attachment, Jan 1992

Cf. A046066, A002110, A006862, A035345, A035346, A055211, A129912, A010051, A005408, A038771, A038711.

a005235 n = head [m | m <- [3, 5 ..], a010051'' (a002110 n + m) == 1]
-- Reinhard Zumkeller, Apr 02 2014

Primorial:= 2:
p:= 2:
A[1]:= 3:
for n from 2 to 100 do
  p:= nextprime(p);
  Primorial:= Primorial * p;
  A[n]:= nextprime(Primorial+p+1)-Primorial;
od:
seq(A[n],n=1..100); # Robert Israel, Dec 02 2015

NPrime[n_Integer] := Module[{k}, k = n + 1; While[! PrimeQ[k], k++]; k]; Fortunate[n_Integer] := Module[{p, q}, p = Product[Prime[i], {i, 1, n}] + 1; q = NPrime[p]; q - p + 1]; Table[Fortunate[n], {n, 60}]
r[n_] := (For[m = (Prime[n + 1] + 1)/2, ! PrimeQ[Product[Prime[k], {k, n}] + 2 m - 1], m++]; 2 m - 1); Table[r[n], {n, 60}]
FN[n_] := Times @@ Prime[Range[n]]; Table[NextPrime[FN[k] + 1] - FN[k], {k, 60}] (* Jayanta Basu, Apr 24 2013 *)
NextPrime[#]-#+1&/@(Rest[FoldList[Times,1,Prime[Range[60]]]]+1) (* Harvey P. Dale, Dec 15 2013 *)

a(n)=my(P=prod(k=1,n,prime(k)));nextprime(P+2)-P \\ Charles R Greathouse IV, Jul 15 2011; corrected by Jean-Marc Rebert, Jul 28 2015

from sympy import nextprime, primorial
def a(n): psharp = primorial(n); return nextprime(psharp+1) - psharp
print([a(n) for n in range(1, 59)]) # Michael S. Branicky, Jan 15 2022

def P(n): return prod(nth_prime(k) for k in range(1, n + 1))
it = (P(n) for n in range(1, 31))
print([next_prime(Pn + 2) - Pn for Pn in it]) # F. Chapoton, Apr 28 2020

If x(n) = 1 + Product_{i=1..n} prime(i), q(n) = least prime > x(n), then a(n) = q(n) - x(n) + 1.
a(n) = 1 + the difference between the n-th primorial plus one and the next prime.
a(n) = A035345(n) - A002110(n). - Jonathan Sondow, Dec 02 2015

A057588 Kummer numbers: -1 + product of first n consecutive primes.

Original entry on oeis.org

1, 5, 29, 209, 2309, 30029, 510509, 9699689, 223092869, 6469693229, 200560490129, 7420738134809, 304250263527209, 13082761331670029, 614889782588491409, 32589158477190044729, 1922760350154212639069, 117288381359406970983269, 7858321551080267055879089

Offset: 1

Mario Velucchi (mathchess(AT)velucchi.it), Oct 05 2000

a(n) is congruent to -1 modulo the first n primes. - Michael Engling, Mar 31 2011
Named after the German mathematician Ernst Eduard Kummer (1810-1893). - Amiram Eldar, Jun 19 2021
Subsequence of A048103. Proof: For all primes p, when i >= A000720(p), neither p itself nor p^p divides a(i), but neither does p^p divide a(i) when i < A000720(p), as p^p > -1 + A034386(p). - Antti Karttunen, Nov 17 2024

T. D. Noe, Table of n, a(n) for n = 1..100
E. E. Kummer, Neuer elementarer Beweis des Satzes, dass die Anzahl aller Primzahlen eine unendliche ist, Monatsber. Preuss. Akad. Wiss., Berlin 1878/9, pp. 777-778. [Collected Papers, II, pp. 669-670, Springer, Berlin-Heidelberg, 1975.] Cited in Mestrovic.
Romeo Meštrović, 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. - From _N. J. A. Sloane_, Jun 13 2012
Hisanori Mishima, Factorizations of many number sequences.
Robert G. Wilson v, Explicit factorizations.
Index entries for sequences related to primorial base

Cf. A002110, A006862, A057705 (primes), A000040, A000720.

Subsequence of A048103.

a057588 = (subtract 1) . product . (flip take a000040_list)
-- Reinhard Zumkeller, Mar 27 2013

seq(mul(ithprime(k), k=1..n) - 1, n=1..100); # Muniru A Asiru, Jan 19 2018

Table[Product[Prime[k], {k, 1, n}] - 1, {n, 1, 18}] (* Artur Jasinski, Jan 01 2007 *)
FoldList[Times,1,Prime[Range[20]]]-1  (* Harvey P. Dale, Apr 17 2011 *)
Table[ChineseRemainder[PadRight[{},n,-1],Prime[Range[n]]],{n,20}] (* Harvey P. Dale, Jul 01 2017 *)

a(n) = prod(k=1, n, prime(k)) - 1; \\ Michel Marcus, Oct 02 2015

from sympy import primorial
def A057588(n): return primorial(n)-1 # Chai Wah Wu, Feb 25 2023

a(n) = A002110(n) - 1. - Altug Alkan, Oct 02 2015

a(n) = A006862(n) - 2. - Antti Karttunen, Nov 17 2024

More terms from Larry Reeves (larryr(AT)acm.org), Oct 05 2000

A018239 Primorial primes: primes of the form 1 + product of first k primes, for some k.

Original entry on oeis.org

2, 3, 7, 31, 211, 2311, 200560490131

Offset: 1

Murray R. Bremner

Prime numbers that are the sum of two primorial numbers. - Juri-Stepan Gerasimov, Nov 08 2010

			From _M. F. Hasler_, Jun 23 2019: (Start)
a(1) = 2 = 1 + product of the first 0 primes (i.e., the empty product = 1).
a(2) = 3 = 1 + 2 = 1 + product of the first prime (= 2).
a(3) = 7 = 1 + 2*3 = 1 + product of the first two primes.
a(4) = 31 = 1 + 2*3*5 = 1 + product of the first three primes.
a(5) = 211 = 1 + 2*3*5*7 = 1 + product of the first four primes.
a(6) = 2311 = 1 + 2*3*5*7*11 = 1 + product of the first five primes.
Then the product of the first 6, 7, ..., 9 or 10 primes does not yield a primorial prime, the next one is:
a(7) = 200560490131 = 1 + 2*3*5*7*11*13*17*19*23*29*31 = 1 + product of the first eleven primes,
and so on. See A014545 = (0, 1, 2, 3, 4, 5, 11, 75, 171, 172, ...) for the k's that yield a term. (End)

F. Iacobescu, Smarandache Partition Type and Other Sequences, Bull. Pure Appl. Sciences, Vol. 16E, No. 2 (1997), pp. 237-240.

Michel Marcus, Table of n, a(n) for n = 1..10
M. Fleuren, Factors and primes of Smarandache sequences.
M. Fleuren, Smarandache Prime Product Sequence.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
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], 2012-2023. - From N. J. A. Sloane, Jun 13 2012
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.
R. Ondrejka, Primorial-Plus-One Primes

Primes in A006862 (primorials plus 1).
A005234 and A014545 (which are the main entries for this sequence) give more terms.
Cf. A002110.

Select[FoldList[Times, 1, Prime[Range[200]]] + 1, PrimeQ] (* Loreno Heer (helohe(AT)bluewin.ch), Jun 29 2004 *)

P=1;print1(2); forprime(p=2,1e6, if(isprime(1+P*=p), print1(", "P+1))) \\ Charles R Greathouse IV, Apr 28 2015

a(n) = 1 + A002110(A014545(n)), where A002110(k) is the product of the first k primes. - M. F. Hasler, Jun 23 2019

Edited by N. J. A. Sloane at the suggestion of Vladeta Jovovic, Jun 18 2007

Name edited by M. F. Hasler, Jun 23 2019

A369054 Number of representations of n as a sum (pq + pr + q*r) with three odd primes p <= q <= r.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jan 20 2024

nonn

Number of solutions to n = x', where x' is the arithmetic derivative of x (A003415), and x is a product of three odd primes (not all necessarily distinct, A046316).

See the conjecture in A369055.

			a(27) = 1 as 27 can be expressed in exactly one way in the form (p*q + p*r + q*r), with p, q, r all being 3 in this case, as 27 = (3*3 + 3*3 + 3*3).
a(311) = 5 as 311 = (3*5 + 3*37 + 5*37) = (3*7 + 3*29 + 7*29) = (3*13 + 3*17 + 13*17) = (5*7 + 5*23 + 7*23) = (7*11 + 7*13 + 11*13). Expressed in the terms of arithmetic derivatives, of the A099302(311) = 8 antiderivatives of 311 [366, 430, 494, 555, 609, 663, 805, 1001], only the last five are products of three odd primes: 555 = 3*5*37, 609 = 3*7*29, 663 = 3*13*17, 805 = 5*7*23, 1001 = 7 * 11 * 13.

Antti Karttunen, Table of n, a(n) for n = 0..50000
Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.

Cf. A369055 [quadrisection, a(4n-1)], and its trisections A369460 [= a((12*n)-9)], A369461 [= a((12*n)-5)], A369462 [= a((12*n)-1)].
Cf. A002620, A003415, A046316, A099302, A369056, A369058, A369252.
Cf. A369251 (positions of terms > 0), A369464 (positions of 0's).
Cf. A369063 (positions of records), A369064 (values of records).
Cf. A369241 [= a(2^n - 1)], A369242 [= a(n!-1)], A369245 [= a(A006862(n))], A369247 [= a(3*A057588(n))].

\\ Use this for building up a list up to a certain n. We iterate over weakly increasing triplets of odd primes:
A369054list(up_to) = { my(v = [3,3,3], ip = #v, d, u = vector(up_to)); while(1, d = ((v[1]*v[2]) + (v[1]*v[3]) + (v[2]*v[3])); if(d > up_to, ip--, ip = #v; u[d]++); if(!ip, return(u)); v[ip] = nextprime(1+v[ip]); for(i=1+ip,#v,v[i]=v[i-1])); };
v369054 = A369054list(100001);
A369054(n) = if(!n,n,v369054[n]);

\\ Use this for computing the value of arbitrary n. We iterate over weakly increasing pairs of odd primes:
A369054(n) = if(3!=(n%4),0, my(v = [3,3], ip = #v, r, c=0); while(1, r = (n-(v[1]*v[2])) / (v[1]+v[2]); if(r < v[2], ip--, ip = #v; if(1==denominator(r) && isprime(r),c++)); if(!ip, return(c)); v[ip] = nextprime(1+v[ip]); for(i=1+ip,#v,v[i]=v[i-1])));

a(n) = Sum_{i=1..A002620(n)} A369058(i)*[A003415(i)==n], where [ ] is the Iverson bracket.

For n >= 2, a(n) <= A099302(n).

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

A002585 Largest prime factor of 1 + (product of first n primes).

Original entry on oeis.org

3, 7, 31, 211, 2311, 509, 277, 27953, 703763, 34231, 200560490131, 676421, 11072701, 78339888213593, 13808181181, 18564761860301, 19026377261, 525956867082542470777, 143581524529603, 2892214489673, 16156160491570418147806951, 96888414202798247, 1004988035964897329167431269

Offset: 1

N. J. A. Sloane

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

The products of the first primes are called primorial numbers. - Franklin T. Adams-Watters, Jun 12 2014

M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors).
M. Kraitchik, Introduction à la Théorie des Nombres. Gauthier-Villars, Paris, 1952, p. 2.
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).

Table of n, a(n) for n = 1..98
A. Borning, Some results for k!+-1 and 2.3.5...p+-1, Math. Comp., 26 (1972), 567-570.
M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors). [Annotated scanned copy]
S. Kravitz and D. E. Penney, An extension of Trigg's table, Math. Mag., 48 (1975), 92-96.
S. Kravitz and D. E. Penney, An extension of Trigg's table, Math. Mag., 48 (1975), 92-96. [Annotated scanned copy; also letter from N. J. A. Sloane to John Selfridge]
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. - From _N. J. A. Sloane_, Jun 13 2012
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
John Selfridge, Marvin Wunderlich, Robert Morris, N. J. A. Sloane, Correspondence, 1975
Eric Weisstein's World of Mathematics, Euclid Number.
R. G. Wilson v, Explicit factorizations

Cf. A002110, A002584, A006530, A006862, A051342.

FactorInteger[#][[-1,1]]&/@Rest[FoldList[Times,1,Prime[Range[30]]]+1] (* Harvey P. Dale, Apr 10 2012 *)

a(n)=my(f=factor(prod(i=1,n,prime(i))+1)[,1]); f[#f] \\ Charles R Greathouse IV, Feb 07 2017

a(n) = A006530(A006862(n)). - Amiram Eldar, Feb 13 2020

More terms from Labos Elemer, May 02 2000
More terms from Robert G. Wilson v, Mar 24 2001
Terms up to a(81) in b-file added by Sean A. Irvine, Apr 19 2014
Terms a(82)-a(87) in b-file added by Amiram Eldar, Feb 13 2020
Terms a(88)-a(98) in b-file added by Max Alekseyev, Aug 26 2021

A054988 Number of prime divisors of 1 + (product of first n primes), with multiplicity.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 2, 2, 3, 1, 3, 3, 2, 3, 4, 4, 2, 2, 4, 2, 3, 2, 4, 3, 2, 4, 4, 3, 3, 5, 3, 6, 2, 3, 2, 5, 4, 4, 2, 6, 3, 4, 3, 5, 6, 7, 2, 6, 3, 5, 3, 4, 2, 6, 5, 4, 5, 3, 5, 5, 5, 3, 3, 5, 5, 6, 3, 4, 4, 7, 5, 3, 4, 1, 2, 5, 5, 5, 4, 5, 3, 5, 4, 6, 5, 8

Offset: 1

Arne Ring (arne.ring(AT)epost.de), May 30 2000

Prime divisors are counted with multiplicity. - Harvey P. Dale, Oct 23 2020

It is an open question as to whether omega(p#+1) = bigomega(p#+1) = a(n); that is, as to whether the Euclid numbers are squarefree. Any square dividing p#+1 must exceed 2.5*10^15 (see Vardi, p. 87). - Sean A. Irvine, Oct 21 2023

			a(6)=2 because 2*3*5*7*11*13+1 = 30031 = 59 * 509.

Ilan Vardi, Computational Recreations in Mathematica, Addison-Wesley, 1991.

Max Alekseyev, Table of n, a(n) for n = 1..98
Hisanori Mishima, Factorizations of many number sequences
R. G. Wilson v, Explicit factorizations

Cf. A001222, A002585, A006862, A014545, A054989, A054990, A054991, A054992.

A054988 := proc(n)
    numtheory[bigomega](1+mul(ithprime(i),i=1..n)) ;
end proc:
seq(A054988(n),n=1..20) ; # R. J. Mathar, Mar 09 2022

a[q_] := Module[{x, n}, x=FactorInteger[Product[Table[Prime[i], {i, q}][[j]], {j, q}]+1]; n=Length[x]; Sum[Table[x[[i]][[2]], {i, n}][[j]], {j, n}]]
PrimeOmega[#+1]&/@FoldList[Times,Prime[Range[90]]] (* Harvey P. Dale, Oct 23 2020 *)

a(n) = bigomega(1+prod(k=1, n, prime(k))); \\ Michel Marcus, Mar 07 2022

a(n) = Omega(1 + Product_{k=1..n} prime(k)). - Wesley Ivan Hurt, Mar 06 2022
a(n) = A001222(A006862(n)). - Michel Marcus, Mar 07 2022
a(n) = 1 iff n is in A014545. - Bernard Schott, Mar 07 2022

More terms from Robert G. Wilson v, Mar 24 2001
a(44)-a(81) from Charles R Greathouse IV, May 07 2011
a(82)-a(87) from Amiram Eldar, Oct 03 2019

A104365 a(n) = A104350(n) + 1.

Original entry on oeis.org

2, 3, 7, 13, 61, 181, 1261, 2521, 7561, 37801, 415801, 1247401, 16216201, 113513401, 567567001, 1135134001, 19297278001, 57891834001, 1099944846001, 5499724230001, 38498069610001, 423478765710001, 9740011611330001

Offset: 1

Reinhard Zumkeller, Mar 06 2005

nonn

Reinhard Zumkeller, Products of largest prime factors of numbers <= n.

Cf. A006530, A006862, A038507, A104366, A104367, A104368, A104369, A104370, A104371, A104357.

FoldList[Times, Array[FactorInteger[#][[-1, 1]] &, 30]] + 1 (* Amiram Eldar, Apr 08 2024 *)

a(n) = (a(n-1) - 1) * A006530(n) + 1 for n>1, a(1) = 0;

A113165 Numbers that divide primorial numbers plus one (p#+1).

Original entry on oeis.org

2, 3, 7, 19, 31, 59, 61, 73, 97, 131, 139, 149, 167, 173, 181, 211, 223, 271, 277, 307, 313, 317, 331, 347, 463, 467, 509, 571, 601, 673, 809, 827, 877, 881, 953, 983, 997, 1031, 1033, 1039, 1051, 1063, 1069, 1109, 1259, 1279, 1283, 1291, 1297, 1361, 1381

Offset: 1

Franklin T. Adams-Watters, Jan 05 2006

The smallest composite member of the sequence is 1843 (19 * 97), which divides 17#+1 (19 * 97 * 277). Based on Euclid's proof that there are infinitely many primes.

			59 is in the sequence because 13#+1 = 30031 = 59 * 509.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..1000

Cf. A002110 (primorials), A018239 (primorial primes), A000945 (Euclid-Mullin sequence), A006862 (primorials plus one).

n=0;for(i=2,1e5,p=Mod(1,i);forprime(q=2,factor(i)[1,1],if(p==-1,print(n++," ",i);break());p*=q)) \\ Jeppe Stig Nielsen, Mar 25 2017

A058233 Primes p such that p#+1 is divisible by the next prime after p.

Original entry on oeis.org

2, 17, 1459, 2999

Offset: 1

Carlos Rivera, Dec 01 2000

No additional terms through the 100000th prime. - Harvey P. Dale, Mar 12 2014

a(5) > prime(1400000) = 22182343. - Robert Price, Apr 02 2018

			2*3*5*7*11*13*17+1 is divisible by 19.

Carlos Rivera, Puzzle 117: Certain p#+1 values, The Prime Puzzles and Problems Connection.

Cf. A006862, A066735, A341804.

primorial[n_] := Product[ Prime[k], {k, 1, PrimePi[n]}]; Select[ Prime[ Range[1000]], Divisible[ primorial[#] + 1, NextPrime[#]] &] (* Jean-François Alcover, Aug 19 2013 *)
Module[{prs=Prime[Range[500]]},Transpose[Select[Thread[{Rest[ FoldList[ Times, 1,prs]], prs}], Divisible[ First[#]+1, NextPrime[Last[#]]]&]][[2]]] (* Harvey P. Dale, Mar 12 2014 *)

from sympy import nextprime
A058233_list, p, q, r = [], 2, 3, 2
for _ in range(10**3):
    if (r+1) % q == 0:
        A058233_list.append(p)
    r *= q
    p, q = q, nextprime(q) # Chai Wah Wu, Sep 27 2021

cp's OEIS Frontend

A005235 Fortunate numbers: least m > 1 such that m + prime(n)# is prime, where p# denotes the product of all primes <= p.

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A057588 Kummer numbers: -1 + product of first n consecutive primes.

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A018239 Primorial primes: primes of the form 1 + product of first k primes, for some k.

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A369054 Number of representations of n as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

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

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A002585 Largest prime factor of 1 + (product of first n primes).

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A369054 Number of representations of n as a sum (pq + pr + q*r) with three odd primes p <= q <= r.