A158866 -id:A158866 - cp's OEIS frontend

A158870 Sums of the form (twin primes + 1) which are also an upper twin prime.

13, 61, 1321, 1621, 4261, 5101, 6661, 6781, 11701, 12541, 21061, 66361, 83221, 88261, 107101, 110881, 114661, 127681, 130201, 140761, 141961, 144541, 148201, 149521, 157561, 161341, 163861, 175081, 186481, 204601, 230941, 249541, 267961

Offset: 1

Cino Hilliard, Mar 28 2009

nonn

If the sum is a member of a twin prime pair, it always is the upper member, shown in A158866.
Moreover, except the first term, these numbers are of the form 10k+1. [We prove this by exhausting the possibilities when calculating the upper, summing and inspecting the lower of the sum. Here are the possible outcomes.
p1(k), p2(k)  p2(m) = p1(k)+p2(k)+1
------------  ---------------------------------
10k+1 10k+3   20k+4+1 not prime
10k+3 10k+5   p2(k) not prime
10k+5 10k+7   p1(k) not prime
10k+7 10k+9   20k+16+1 upper => p1(m) not prime
10k+9 10k+11  20k+20+1 = 10(2k+2)+1
So the only form that was not eliminated, is 10k+1. 13 defies this scheme because 10k+5 is prime for k=0, Q.E.D.]

			The 30th lower twin prime is 659. 659+661+1 = 1321, prime and 1319 is too.
Then 1319 is the lower member of the twin prime pair (1319,1321). So 1321 is in the sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A158866.

With[{tws=Total/@Select[Partition[Prime[Range[25000]],2,1],#[[2]]-#[[1]] == 2&]+1},Select[tws,And@@PrimeQ[#+{0,-2}]&]] (* Harvey P. Dale, Apr 30 2014 *)

gp > g(n)=for(x=1,n,y=2*twinl(x)+3;if(isprime(y)&&isprime(y-2), print1(y",")))

{A054735(k)+1: A054735(k)+1 = A006512(j), any j,k} - R. J. Mathar, Apr 06 2009

Edited by R. J. Mathar, Apr 06 2009

A195104 Smallest prime p such that p*t(n) +- 1 is a twin prime pair, where t(n)=A014574(n) is the n-th twin prime average.

Original entry on oeis.org

3, 2, 5, 11, 2, 11, 3, 29, 101, 199, 29, 7, 13, 11, 29, 71, 13, 3, 71, 101, 29, 43, 79, 5, 11, 11, 5, 29, 61, 2, 2, 11, 19, 11, 29, 5, 11, 7, 41, 19, 181, 19, 59, 5, 11, 7, 29, 11, 41, 179, 41, 13, 61, 181, 19, 241, 139, 331, 271, 3, 59, 5, 41, 89, 19, 2, 5, 131, 59, 5, 5, 509, 2

Offset: 1

Juri-Stepan Gerasimov, Dec 16 2011

nonn

a(n) = 2 for n in A158866. - Robert Israel, Mar 07 2021

			a(4)=11 because 11*A014574(4) +- 1 = 11*18 +- 1 = 198 +- 1 is a twin prime pair, A014574(4) +- 1 = 18 +- 1 is also a twin prime pair, and 11 is prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A014574, A158866.

A195104 := proc(n)
    k := A014574(n) ;
    for i from 1 do
        p := ithprime(i) ;
        if isprime(p*k-1) and isprime(p*k+1) then
            return p;
        end if;
    end do:
end proc:
seq(A195104(n),n=1..60) ; # R. J. Mathar, Dec 16 2011

a(64) corrected by Robert Israel, Mar 07 2021

cp's OEIS Frontend

A158870 Sums of the form (twin primes + 1) which are also an upper twin prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions