A057713 -id:A057713 - cp's OEIS frontend

A104358 Smallest prime factor of A104357(n) = A104350(n) - 1.

1, 5, 11, 59, 179, 1259, 11, 7559, 37799, 415799, 71, 227, 5981, 9067, 1135133999, 11717, 61, 79, 5499724229999, 97, 1543, 31, 29220034833989999, 8937119, 181, 401, 124759, 443851, 31, 2141, 3082663, 8191, 37230797, 1697, 1408101540804746673385499999, 10613, 73, 59, 107, 79, 617, 163, 173

Offset: 2

Reinhard Zumkeller, Mar 06 2005

nonn

a(n) = A020639(A104357(n)).

Tyler Busby, Table of n, a(n) for n = 2..181 (terms 2..120 from Chai Wah Wu, terms 121..160 from Max Alekseyev)
Reinhard Zumkeller, Products of largest prime factors of numbers <= n

Cf. A104357, A104359, A104360, A104361, A104362, A104363, A104364, A104366, A054415, A057713.

Typo in data corrected by Gionata Neri, Oct 20 2017

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.

A068489 m for which prime(m) is the least prime dividing #prime(n) - 1, i.e., one less than primorial n-th prime (A057588).

Original entry on oeis.org

3, 10, 5, 343, 3248, 18, 16, 12, 22, 20324, 50, 9414916809095, 13120, 43, 8481, 1200361259, 196, 38, 10326732314, 65, 38, 34

Offset: 2

Lekraj Beedassy, Mar 11 2002

Since #P13 - 1 is a prime, see A006794, we need the number of primes less than or equal to #P13 - 1. The sequence continues, for n=14 to 23: 13120, 43, 8481, 1200361259, 196, 38, 10326732314, 65, 38, 34.

a(24) = pi(23768741896345550770650537601358309). - Donovan Johnson, Dec 08 2009

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], 2012-2023. - From N. J. A. Sloane, Jun 13 2012
Hisanori Mishima, Factorization results #Pn (Primorial) - 1.

Cf. A000720, A006794, A057588, A057713, A068488.

Do[ Print[ PrimePi[ FactorInteger[ Product[ Prime[k], {k, 1, n}] - 1] [[1, 1]]]], {n, 2, 22} ]

a(n) = A000720(A057713(n)).

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

a(13) from Donovan Johnson, Dec 08 2009

cp's OEIS Frontend

A104358 Smallest prime factor of A104357(n) = A104350(n) - 1.

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

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

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A068489 m for which prime(m) is the least prime dividing #prime(n) - 1, i.e., one less than primorial n-th prime (A057588).

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