A086169 -id:A086169 - cp's OEIS frontend

A239309 a(n) is the smallest k such that prime(n) divides Sum_{i=1..k} A086169(i), or 0 if no such k exists, where A086169(i) is the sum of the first i twin prime pairs.

1, 0, 2, 5, 3, 37, 21, 29, 67, 71, 23, 11, 15, 7, 58, 12, 41, 8, 66, 25, 35, 370, 35, 17, 75, 159, 198, 30, 37, 153, 232, 333, 170, 507, 108, 279, 41, 61, 486, 9, 194, 211, 29, 73, 173, 575, 152, 214, 10, 147, 126, 672, 388, 77, 358, 1048, 528, 291, 322, 1491

Offset: 1

Michel Lagneau, Mar 15 2014

nonn

a(2) = 0. Proof

It is easy to see that A054735(1)= 8 ==2 (mod 3) and A054735(n)==0 mod 3 for n > 1 where A054735 is the sum of twin pairs. Hence A086169(n)==2 (mod 3) and the prime 3 is never a divisor of A086169(n).

			a(1)=1 because A086169(1)=(3+5)=8 and prime(1)= 2 divides 8;
a(2)=0 because prime(2)=3 is never a divisor of A086169(n);
a(3)=2 because A086169(2)=(3+5)+(5+7)=20 and prime(3)= 5 divides 20.

Michel Lagneau, Table of n, a(n) for n = 1..1000

Cf. A000040, A054735, A086169.

Transpose[With[{aprs=Thread[{Range[5000],Accumulate[Select[Table[Prime[n]+1,{n,45900}],PrimeQ[#+1]&]*2]}]},Flatten[Table[Select[aprs,Divisible[Last[#],Prime[m]]&,1],{m,1,60}],1]]][[1]]

A146536 Sum of the first 10^n twin primes.

Original entry on oeis.org

8, 908, 328184, 69004076, 11556327260, 1707243198956, 237064232862404, 31153163750203064, 3947120494191630260, 486665774050923191336, 58727077924563028184984

Offset: 0

Cino Hilliard, Oct 31 2008

The author's Gcc/Gmp program is in the links section. The page also has the PARI bisection algorithms which give a very good approximation for the n-th prime number and the n-th lower twin prime number. The first 5 terms can be computed from the PARI script although 11556327260 takes 4 hours on a 2.53ghz 2 gig ram p4.

			The 10^0-th twin prime pair is (3,5). This adds up to 8, the first entry in the sequence.
The first 10^1 twin prime pairs are (3,5),(5,7),(11,13) (17,19),(29,31),(41,43),(59,61),(71,73),(101,103),(107,109). This adds up to 908, the second entry in the table.

Cino Hilliard, Counting and summing primes [Dead link]
Kevin Ryde, C Code

Cf. A086169, A118552.

C
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// See links.
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a(n)={my(k=10^n,s=k,q); forprime(p=3, oo, if(p==q+2, s+=q; k--; if(!k, return(2*s))); q=p)} \\ Andrew Howroyd, Oct 22 2023

a(n) = A086169(10^n). - Andrew Howroyd, Oct 22 2023

a(10) corrected by Bill McEachen, Oct 16 2023

cp's OEIS Frontend

A239309 a(n) is the smallest k such that prime(n) divides Sum_{i=1..k} A086169(i), or 0 if no such k exists, where A086169(i) is the sum of the first i twin prime pairs.

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