A244571 -id:A244571 - cp's OEIS frontend

A244572 a(n) = max(A244570(n), A244571(n)).

3, 7, 11, 23, 17, 37, 23, 41, 43, 61, 47, 61, 53, 73, 109, 107, 89, 73, 109, 227, 113, 113, 139, 157, 127, 149, 127, 131, 283, 137, 139, 181, 173, 179, 167, 191, 181, 227, 193, 251, 239, 199, 233, 257, 239, 251, 239, 241, 271, 313, 241, 271, 281, 277, 443, 389

Offset: 2

Vladimir Shevelev, Jun 30 2014

nonn

a(n) < (prime(n))^3 yields an infinity of twin primes (it is sufficient, if this inequality holds for an arbitrary infinite subsequence n = n_k). For a proof, see the Shevelev link (Remark 8).
The author apparently claims to have proved the infinitude of twin primes. No alleged proof has been accepted by the mathematical community. - Jens Kruse Andersen, Jul 13 2014
In the statistical part of my link (Section 14), using the Chinese Remainder and Tolev's theorems, I reduced the supposition of the finiteness of twin primes to an arbitrarily long coin-flipping experiment in which only "heads" appear. There I gave only a "demonstration" of the infinity of twin primes. In the analytical part (Sections 15-18) I proved unconditionally till now only Theorem 13. - Vladimir Shevelev, Jul 22 2014

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], 2010-2014. (Sections 10,14-18). [Note this article has been changed many times.]

Cf. A244570, A244571, A242519, A242520.

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

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

A244572 a(n) = max(A244570(n), A244571(n)).

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

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

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PARI

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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