A065316 -id:A065316 - cp's OEIS frontend

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

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.

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.

A309671 Primes prime(m) such that G = prime(m-1)# - prime(m) is prime.

Original entry on oeis.org

7, 11, 13, 17, 23, 83, 89, 97, 151, 373, 433, 857, 4013, 8821, 12959

Offset: 1

Mohamed Sami Gattoufi, Aug 11 2019

G = prime(n-1)# - prime(n) where G is a prime is a special case of A090188 where (k=1).

			7 is a term because 23 = 2*3*5 - 7 is prime.

Cf. A002110, A065314, A060882, A096649, A090188, A065316.

primo(p) = my(ip=primepi(p)); factorback(primes(ip)); \\ A002110
isok(p) = isprime(p) && isprime(primo(precprime(p-1)) - p);

cp's OEIS Frontend

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

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A309671 Primes prime(m) such that G = prime(m-1)# - prime(m) is prime.

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