A244572 -id:A244572 - cp's OEIS frontend

A244570 Smallest prime p >= prime(n) such that p*prime(n)+2 is prime.

3, 7, 11, 19, 17, 37, 23, 37, 43, 47, 47, 61, 53, 73, 109, 79, 89, 71, 109, 227, 113, 97, 139, 137, 127, 149, 127, 131, 283, 137, 139, 181, 173, 151, 167, 191, 173, 193, 193, 181, 239, 199, 233, 223, 239, 251, 239, 241, 233, 313, 241, 251, 271, 277, 373, 271

Offset: 2

Vladimir Shevelev, Jun 30 2014

nonn

See comment in A244572.

Jens Kruse Andersen, Table of n, a(n) for n = 2..10000
V. Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 [math.GM], 2009-2014. Sections 10,14.

Cf. A242719, A242720.

a[n_] := For[p = Prime[n], True, p = NextPrime[p], If[PrimeQ[p Prime[n] + 2], Return[p]]];
Table[a[n], {n, 2, 60}] (* Jean-François Alcover, Nov 18 2018 *)
sp[n_]:=Module[{p=n},While[CompositeQ[p*n+2],p=NextPrime[p]];p]; Table[ sp[k],{k,Prime[Range[2,60]]}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 19 2020 *)

s=[]; for(n=2, 100, p=q=prime(n); while(!isprime(p*q+2), p=nextprime(p+1)); s=concat(s, p)); s \\ Colin Barker, Jun 30 2014

More terms from Peter J. C. Moses, Jun 30 2014

A244571 Smallest prime p >= prime(n) such that p*prime(n)-2 is prime.

Original entry on oeis.org

3, 5, 7, 23, 13, 23, 19, 41, 29, 61, 37, 59, 43, 47, 71, 107, 61, 73, 71, 97, 109, 113, 89, 157, 113, 103, 107, 127, 167, 127, 131, 149, 139, 179, 163, 163, 181, 227, 173, 251, 199, 191, 223, 257, 229, 211, 223, 233, 271, 233, 239, 271, 281, 257, 443, 389, 373

Offset: 2

Vladimir Shevelev, Jun 30 2014

nonn

See comment in A244572.

Jens Kruse Andersen, Table of n, a(n) for n = 2..10000
V. Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 [math.GM], 2009-2014. Sections 10, 14.

Cf. A244570, A242719, A242720.

a[n_] := For[p = Prime[n], True, p = NextPrime[p], If[PrimeQ[p Prime[n] - 2], Return[p]]];
Table[a[n], {n, 2, 60}] (* Jean-François Alcover, Nov 18 2018 *)

s=[]; for(n=2, 100, p=q=prime(n); while(!isprime(p*q-2), p=nextprime(p+1)); s=concat(s, p)); s \\ Colin Barker, Jun 30 2014

More terms from Peter J. C. Moses, Jun 30 2014

A245457 a(n) = ((prime(n)-1)!+2) mod prime(n)# (cf. A002110).

Original entry on oeis.org

1, 4, 26, 92, 2102, 23102, 60062, 510512, 29099072, 3792578792, 84106011992, 2005604901302, 252305096583542, 11561510014033982, 52331045326680122, 31359378912013061912, 1792403716245452460152, 98060777857864844592572, 3401363059422802158514832

Offset: 1

Vladimir Shevelev, Jul 22 2014

nonn

Smallest positive residue modulo p_1*...*p_n (cf. A002110) of (p_n-1)!+2, where p_n=prime(n).

See comment in A245460.

Jens Kruse Andersen, Table of n, a(n) for n = 1..100
V. Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 (Sections 10,11,14-18)

Cf. A002110, A244570, A244571, A244572.

a(n) = ((prime(n)-1)!+2) % prod(i=1, n, prime(i)) \\ Jens Kruse Andersen, Jul 22 2014

More terms and simpler definition from Jens Kruse Andersen, Jul 22 2014

A245460 Max (A245457(n), A245458(n)).

Original entry on oeis.org

1, 4, 26, 122, 2102, 23102, 450452, 9189182, 193993802, 3792578792, 116454478142, 5415133233512, 252305096583542, 11561510014033982, 562558737261811292, 31359378912013061912, 1792403716245452460152, 98060777857864844592572, 4456958491657464897364262

Offset: 1

Vladimir Shevelev, Jul 22 2014

nonn

Knowing a(n) <= (prime(n))^4 would yield an infinity of twin primes (in fact it is sufficient if this inequality holds for an arbitrary infinite subsequence k = k_n). See the Shevelev link, Section 17, Corollary 6.

Of course, (p_n)^4/A002110(n) is very small, but remember that sequence k_n could have arbitrary fast growth, for example, as (A002110(n)/(p_n)^4)^n. - Vladimir Shevelev, Jul 24 2014

Jens Kruse Andersen, Table of n, a(n) for n = 1..100
V. Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 (Sections 10,11,14-18)

Cf. A245457, A245458, A244570, A244571, A244572.

f(n,k) = ((prime(n)-k)!+2) % prod(i=1, n, prime(i))
a(n) = max(f(n,1), f(n,2)) \\ Jens Kruse Andersen, Jul 22 2014

More terms from Jens Kruse Andersen, Jul 22 2014

A245458 a(n) = ((prime(n)-2)!+2) mod prime(n)# (cf. A002110).

Original entry on oeis.org

1, 3, 8, 122, 212, 6932, 450452, 9189182, 193993802, 2677114442, 116454478142, 5415133233512, 51945166943672, 1521251317636052, 562558737261811292, 1229779565176982822, 130356633908760178922, 19227603501542126390702, 4456958491657464897364262

Offset: 1

Vladimir Shevelev, Jul 22 2014

nonn

Smallest positive residue modulo p_1*...*p_n (cf. A002110) of (p_n-2)!+2, where p_n=prime(n).

See comment in A245460.

Jens Kruse Andersen, Table of n, a(n) for n = 1..100
V. Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 (Sections 10,11,14-18)

Cf. A002110, A245457, A244570, A244571, A244572.

a[n_] := Mod[(Prime[n]-2)! + 2, Product[Prime[i], {i, 1, n}]];
Array[a, 19] (* Jean-François Alcover, Dec 15 2018 *)

a(n) = ((prime(n)-2)!+2) % prod(i=1, n, prime(i)) \\ Jens Kruse Andersen, Jul 22 2014

More terms and simpler definition from Jens Kruse Andersen, Jul 22 2014

cp's OEIS Frontend

A244570 Smallest prime p >= prime(n) such that p*prime(n)+2 is prime.

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A244571 Smallest prime p >= prime(n) such that p*prime(n)-2 is prime.

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A245457 a(n) = ((prime(n)-1)!+2) mod prime(n)# (cf. A002110).

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PARI

Extensions

A245460 Max (A245457(n), A245458(n)).

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A245458 a(n) = ((prime(n)-2)!+2) mod prime(n)# (cf. A002110).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions