A002585 -id:A002585 - cp's OEIS frontend

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

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

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

Original entry on oeis.org

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

A002584 Largest prime factor of product of first n primes - 1, or 1 if no such prime exists.

Original entry on oeis.org

1, 5, 29, 19, 2309, 30029, 8369, 929, 46027, 81894851, 876817, 38669, 304250263527209, 92608862041, 59799107, 1143707681, 69664915493, 1146665184811, 17975352936245519, 2140320249725509

Offset: 1

N. J. A. Sloane

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).

Sean A. Irvine and Amiram Eldar, Table of n, a(n) for n = 1..99 (terms 1..91 from Sean A. Irvine)
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]
Hisanori Mishima, Factorizations of many number sequences
John Selfridge, Marvin Wunderlich, Robert Morris, N. J. A. Sloane, Correspondence, 1975
R. G. Wilson v, Explicit factorizations.

Cf. A002110, A002585, A006530, A057588.

Prepend[Table[ Max[Transpose[FactorInteger[(Times @@ Prime[Range[i]]) - 1]][[1]]], {i, 2, 20}], 1]
FactorInteger[#][[-1,1]]&/@Rest[FoldList[Times,1,Prime[Range[20]]]-1] (* Harvey P. Dale, Feb 27 2013 *)

a(n)=if(n>1, my(f=factor(prod(i=1,f,prime(i)))[,1]); f[#f], 1) \\ Charles R Greathouse IV, Feb 08 2017

a(n) = A006530(A057588(n)). - Amiram Eldar, Feb 13 2020

More terms from J. L. Selfridge

Further terms from Labos Elemer, Oct 25 2000

A104367 Greatest prime factor of A104365(n) = A104350(n) + 1.

Original entry on oeis.org

2, 3, 7, 13, 61, 181, 97, 2521, 7561, 367, 415801, 1247401, 97103, 594311, 2689891, 269, 415147, 1434493, 1099944846001, 13421, 938977307561, 1687166397251, 6121943187511, 13027211250107, 146100174169950001, 1389833, 10603380543703, 2129284819, 1156675078903494150001, 132597517693, 47172675889, 11159737, 20350106034371

Offset: 1

Reinhard Zumkeller, Mar 06 2005

nonn

Max Alekseyev, Table of n, a(n) for n = 1..158 (terms 1..76 from Amiram Eldar, terms 142..151 from Tyler Busby)
Reinhard Zumkeller, Products of largest prime factors of numbers <= n

Cf. A006530, A002583, A002585, A104350, A104359, A104365, A104366, A104368, A104369, A104370, A104371, A104372.

a[n_] := FactorInteger[1 + Product[FactorInteger[k][[-1, 1]], {k, 1, n}]][[-1, 1]]; Array[a, 76] (* Amiram Eldar, Feb 12 2020 *)

a(n) = A006530(A104365(n)).

Corrected by T. D. Noe, Nov 15 2006

A057713 Smallest prime divisor of Kummer numbers ( = primorials - 1), or 1 if no such prime exists.

Original entry on oeis.org

1, 5, 29, 11, 2309, 30029, 61, 53, 37, 79, 228737, 229, 304250263527209, 141269, 191, 87337, 27600124633, 1193, 163, 260681003321, 313, 163, 139, 23768741896345550770650537601358309, 66683, 2990092035859, 15649, 17515703, 719, 295201, 15098753, 10172884549, 20962699238647, 4871, 673, 311, 1409, 1291, 331, 1450184819, 23497, 711427, 521, 673, 519577, 1372062943, 56543, 811, 182309, 53077, 641, 349, 389

Offset: 1

Labos Elemer, Oct 25 2000

nonn

			6th term in the sequence corresponds to 7th primorial = 510510 and 510509 = 61 * 8369, so a(7) = 61.

Amiram Eldar, Table of n, a(n) for n = 1..99
M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors). [Annotated scanned copy]
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. - _N. J. A. Sloane_, Jun 13 2012
Hisanori Mishima, Factorizations of many number sequences
R. G. Wilson v, Explicit factorizations

Cf. A002110, A002584, A002585, A006862, A020639, A057588.

Map[If[PrimeQ@ #, #, FactorInteger[#][[1, 1]]] &, FoldList[#1 #2 &, Prime@ Range@ 36] - 1] (* Michael De Vlieger, Feb 18 2017 *)

a(n) = A020639(A057588(n)). - Amiram Eldar, Feb 13 2020

More terms from Klaus Brockhaus, Larry Reeves (larryr(AT)acm.org) and Robert G. Wilson v, Apr 02 2001

A065314 Smallest prime divisor of (n-th primorial - (n+1)-st prime).

Original entry on oeis.org

23, 199, 2297, 30013, 41, 9699667, 2819, 53, 21701, 79, 163, 181, 61, 1619, 14669, 307, 103, 306091, 907, 3217644767340672907899084554047, 267064515689275851355624017992701, 23768741896345550770650537601358213

Offset: 3

Labos Elemer, Oct 29 2001

nonn

			For n=3, 3rd primorial=30, prime(4)=7, difference=23, so a(3)=23.

Tyler Busby, Table of n, a(n) for n = 3..104 (terms 3..40 from Michel Marcus, terms 41..80 from Sean A. Irvine)
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], 2012-2023. - _N. J. A. Sloane_, Jun 13 2012

Cf. A002110, A000040, A006530, A006530, A057713, A002584, A002585, A051342.

Cf. A065315, A065316, A065317.

Map[FactorInteger[Times @@ #1 - #2][[1, 1, 1]] & @@ Reverse@ TakeDrop[#, -1] &, Drop[#, 3] &@ FoldList[Flatten@ Append[{#1}, #2] &, Prime@ Range@ 25]] (* Michael De Vlieger, Jul 16 2017 *)

a(n) = vecmin(factor(prod(k=1, n, prime(k)) - prime(n+1))[,1]); \\ Michel Marcus, Jul 16 2017

a(n) = A020639( A002110(n) - A000040(n+1) ).

A065315 Smallest prime divisor of n-th primorial + (n+1)-st prime.

Original entry on oeis.org

5, 11, 37, 13, 23, 30047, 510529, 9699713, 127, 107, 433, 1093, 375569, 13082761331670077, 941879, 32589158477190044789, 1922760350154212639131, 4129, 92388407, 5879, 40729680599249024150621323549, 1783, 4903, 10279098043, 191, 131, 109, 163, 337, 20261, 673327, 6599, 181

Offset: 1

Labos Elemer, Oct 29 2001

nonn

			For n=3, 3rd primorial=30, prime(4)=7, sum=37, so a(3)=37.

Tyler Busby, Table of n, a(n) for n = 1..84 (terms 1..79 from Sean A. Irvine)
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, 2012-2023. - From N. J. A. Sloane, Jun 13 2012

Cf. A002110, A000040, A006530, A057713, A002584, A002585, A051342, A060881.

A065316 Largest prime divisor of n-th primorial - (n+1)-st prime.

Original entry on oeis.org

23, 199, 2297, 30013, 12451, 9699667, 79139, 122069683, 9241993, 77184383, 211941187, 72280449346243, 73629553, 142226610221, 131076443530861861, 382046844818915214929, 1348764323657, 1822793973448088839487, 217379667530071, 3217644767340672907899084554047, 267064515689275851355624017992701

Offset: 3

Labos Elemer, Oct 29 2001

nonn

			For n=3, 3rd primorial=30, 4th prime=7, difference=23, so a(3)=23.

Tyler Busby, Table of n, a(n) for n = 3..89 (terms 3..80 from Sean A. Irvine)
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], 2012-2023. - From _N. J. A. Sloane_, Jun 13 2012

Cf. A002110, A000040, A006530, A006530, A057713, A002584, A002585, A051342.

A066735 Primes p dividing 1 + the product of the primes less than p.

Original entry on oeis.org

2, 3, 19, 1471, 3001

Offset: 1

Joseph L. Pe, Jan 15 2002

No further terms up to prime(216816) = 2999999. Is the sequence finite? - Klaus Brockhaus, Jan 17 2002
From Lévai Gábor (gablevai(AT)vipmail.hu), Nov 23 2004: (Start)
Let p(1)=2, p(2)=3, p(3)=5, ... denote the primes and let E(n) = 1 + p(1) * p(2) * ... * p(n). For k >= 1, list the primes p such that p(n+k) | E(n). For k=1 we get this sequence, for k=2 we get A100465.
For k >= 3 the known results are as follows: if k = 3: no solutions for p < 80000000; if k = 4: 463, 2908123 and no others for p < 80000000; if k = 5: 61, 73 and no others for p < 80000000; if k = 6: 21687203 and no others for p < 80000000; if k = 7: 149, 43951591 and no others for p < 80000000; if k = 8: 31, 131 and no others for p < 80000000; if k = 9: 58691999 and no others for p < 80000000. (End)
No further terms up to 80000000. - Lévai Gábor (gablevai(AT)vipmail.hu), Nov 23 2004
a(6) > 179424673 = prime(10^7). - Giovanni Resta, Apr 13 2017
a(6) > 914799232 > prime(46727379). - Max Z. Scialabba, Feb 26 2024

			1 + Product of the primes < 19 = 1 + 2*3*5*7*11*13*17 = 510511 = 19*26869; so 19 is a term of the sequence.

Hisanori Mishima, Factorization results for #Pn (Primorial) + 1

Cf. A002110, A002585, A100465, A081618.

p = 2; Do[q = Prime[n]; If[ IntegerQ[(p + 1)/q], Print[q]]; p = p*q, {n, 2, 86120} ]

a066735(m) =local(k,p); k=1; forprime(p=2,m, if((k+1)%p==0,print1(p,",")); k=k*p)

A065317 Largest prime divisor of the sum of the n-th primorial and the (n+1)-st prime.

Original entry on oeis.org

5, 11, 37, 17, 101, 30047, 510529, 9699713, 1427, 76829, 789077, 659863, 810104837, 13082761331670077, 652833094897, 32589158477190044789, 1922760350154212639131, 28406001782370300553, 770555057, 94904036422299534098897, 40729680599249024150621323549

Offset: 1

Labos Elemer, Oct 29 2001

nonn

			For n = 4, 4th primorial = 210, prime(5) = 11, sum = 210 + 11 = 13 * 17, a(4) = 17.

Tyler Busby, Table of n, a(n) for n = 1..84 (terms 1..68 from Daniel Suteu, terms 69..79 from Sean A. Irvine)
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], 2012-2023. - From _N. J. A. Sloane_, Jun 13 2012

Cf. A002110, A000040, A060881, A006530, A057713, A002584, A002585, A051342.

cp's OEIS Frontend

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

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

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A002584 Largest prime factor of product of first n primes - 1, or 1 if no such prime exists.

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A104367 Greatest prime factor of A104365(n) = A104350(n) + 1.

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A057713 Smallest prime divisor of Kummer numbers ( = primorials - 1), or 1 if no such prime exists.

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A065314 Smallest prime divisor of (n-th primorial - (n+1)-st prime).

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A065315 Smallest prime divisor of n-th primorial + (n+1)-st prime.

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