A051342 -id:A051342 - cp's OEIS frontend

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

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

A104366 Smallest prime factor of A104365(n) = A104350(n) + 1.

Original entry on oeis.org

2, 3, 7, 13, 61, 181, 13, 2521, 7561, 103, 415801, 1247401, 167, 191, 211, 127, 23, 40357, 1099944846001, 349, 41, 251, 37, 2243, 146100174169950001, 103, 53, 1217, 1156675078903494150001, 47, 2939, 251, 857, 41, 547, 13127, 47, 48563, 281, 1336484560722851, 479, 373, 2179, 577670972464621571, 17491, 1399, 97, 22893547

Offset: 1

Reinhard Zumkeller, Mar 06 2005

nonn

a(n) = A020639(A104365(n)).

Tyler Busby, Table of n, a(n) for n = 1..170 (terms 1..100 from Chai Wah Wu, terms 101..165 from Max Alekseyev)
Reinhard Zumkeller, Products of largest prime factors of numbers <= n

Cf. A104367, A104358, A051301, A051342, A104358, A104365, A104367, A104368, A104369, A104370, A104371, A104372.

gpf(n) = if (n==1, 1, vecmax(factor(n)[,1])); \\ A006530
spf(n) = if (n==1, 1, vecmin(factor(n)[,1])); \\ A020639
a(n) = spf(prod(i=2, n, gpf(i))+1); \\ Michel Marcus, Feb 21 2023

Corrected by D. S. McNeil, Dec 10 2010

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.

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.

A051454 a(n) is the smallest prime factor of 1 + lcm(1..k) where k is the n-th prime power A000961(n).

Original entry on oeis.org

2, 3, 7, 13, 61, 421, 29, 2521, 19, 89, 71, 1693, 232792561, 6659, 26771144401, 331, 101, 72201776446801, 1801, 173, 54941, 89, 442720643463713815201, 593, 5171, 239, 1222615931, 103, 7265496855919, 6562349363, 4447, 147099357127, 1931

Offset: 1

Labos Elemer

nonn

			1 + lcm(1..8) = 29^2, so its smallest prime divisor is 29; it occurs as the 7th term in the sequence because 8 is the 7th prime power: A000961(7) = 8.

Sean A. Irvine, Table of n, a(n) for n = 1..82

Cf. A000961, A003418, A049536, A049537, A051301, A051342, A051452.

a:=[]; lcm:=1; for k in [1..83] do if (k eq 1) or IsPrimePower(k) then lcm:=Lcm(lcm,k); a:=a cat [Factorization(1+lcm)[1][1]]; end if; end for; a; // Jon E. Schoenfield, May 28 2018

Join[{2},With[{ppwr=Select[Range[200],PrimePowerQ]},Table[FactorInteger[LCM@@Take[ ppwr,n]+ 1][[1,1]],{n,40}]]] (* Harvey P. Dale, May 28 2024 *)

a(n) = {my(nb = 1, lc = 1, k = 2); while (nb != n, if (isprimepower(k), nb++; lc = lcm(lc, k)); k++;); vecmin(factor(lc +1)[,1]);} \\ Michel Marcus, May 29 2018

from math import prod
from sympy import primepi, integer_nthroot, integer_log, primerange, primefactors
def A051454(n):
    def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return min(primefactors(1+prod(p**integer_log(m, p)[0] for p in primerange(m+1)))) # Chai Wah Wu, Aug 15 2024

A068488 m for which p(m) is the least prime dividing #p(n) + 1, i.e., primorial n-th prime augmented by 1 (A005234).

Original entry on oeis.org

2, 4, 11, 47, 344, 17, 8, 69, 66, 67, 8028643011, 42, 18, 39, 162, 21, 59, 48, 2311331257, 179, 369, 2477, 23289, 32, 172011, 75668, 342, 35, 28757, 356411, 243, 297, 152

Offset: 1

Lekraj Beedassy, Mar 11 2002

Since #P34 + 1 has two rather large factors, we need the number of primes less than or equal to 678279959005528882498681487.

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, Factorization results #Pn (Primorial) + 1

Cf. A068489.

Do[ Print[ PrimePi[ FactorInteger[ Product[ Prime[k], {k, 1, n}] + 1] [[1, 1]]]], {n, 1, 20} ]

a(n) = PrimePi(A051342).

Edited and extended by Robert G. Wilson v, Mar 12 2002

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A104366 Smallest prime factor of A104365(n) = A104350(n) + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A051454 a(n) is the smallest prime factor of 1 + lcm(1..k) where k is the n-th prime power A000961(n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica