A118371 -id:A118371 - cp's OEIS frontend

A173520 Partial sums of A118371.

2, 5, 10, 17, 30, 49, 72, 103, 140, 183, 230, 283, 344, 423, 506, 607, 714, 823, 936, 1067, 1206, 1363, 1530, 1729, 1940, 2191, 2460, 2741, 3024, 3317, 3624, 3937, 4274, 4657, 5058, 5479, 5910, 6349, 6798, 7255, 7746, 8255, 8776, 9299, 9868, 10469

Offset: 1

Jonathan Vos Post, Feb 20 2010

Partial sums of fastest growing sequence of primes satisfying Goldbach's conjecture. The subsequence of primes in this partial sum begins: 2, 5, 17, 103, 283, 607, 823, 2741, 4657, 5479, 13177, 16369. The subsequence of primes in this partial sum which are also in the underlying sequence begins: 2, 5, 283, 5479.

			a(56) = 2 + 3 + 5 + 7 + 13 + 19 + 23 + 31 + 37 + 43 + 47 + 53 + 61 + 79 + 83 + 101 + 107 + 109 + 113 + 131 + 139 + 157 + 167 + 199 + 211 + 251 + 269 + 281 + 283 + 293 + 307 + 313 + 337 + 383 + 401 + 421 + 431 + 439 + 449 + 457 + 491 + 509 + 521 + 523 + 569 + 601 + 643 + 673 + 691 + 701 + 769 + 773 + 811 + 839 + 863 + 881.

Cf. A118371.

a(n) = SUM[i=1..n] A118371(i).

One of the 3937 replaced by 3624 - R. J. Mathar, Mar 07 2010

A187072 Prime numbers chosen such that the even numbers that are the sum of two consecutive terms occur only once and occur as early as possible.

Original entry on oeis.org

3, 3, 5, 5, 7, 7, 11, 5, 17, 3, 23, 5, 19, 11, 23, 13, 19, 19, 23, 17, 29, 19, 31, 13, 41, 11, 47, 13, 43, 19, 47, 17, 53, 19, 59, 17, 67, 7, 61, 19, 67, 23, 59, 29, 67, 31, 61, 41, 53, 47, 59, 53, 61, 43, 67, 41, 79, 37, 89, 29, 101, 23, 109, 13, 127, 7, 131, 5, 137, 7, 139, 11, 137, 17, 139, 13, 149, 11, 157, 7, 151, 19, 109, 67, 107, 59, 113, 67

Offset: 1

Reinhard Zumkeller, Mar 03 2011

The even numbers a(n) + a(n+1) are in sequence A187085.

The terms for even n grow rapidly; for odd n they grow slowly. It appears that primes occur at a consistent frequency: in the first 1000000 terms, primes 3 to 23 occur about 4.7%, 4.9%, 3.4%, 2.9%, 2.6%, 2.0%, 1.8%, and 1.4% of the time. - T. D. Noe, Mar 04 2011

			Primes: 3 3 5  5  7  7  11  5  17  3  23  5  19  11  23  13  19  19  23
Evens:   6 8 10 12 14 18  16 22  20 26  28 24  30  34  36  32  38  42

T. D. Noe, Table of n, a(n) for n = 1..5000
T. D. Noe, Plot of frequency (%) of primes in the first 10^9 terms
Index entries for sequences related to Goldbach conjecture

Cf. A065091, A118371, A187085, A187098.

import Data.Set (Set, empty, member, insert)
a187072 n = a187072_list !! (n-1)
a187072_list = goldbach 0 a065091_list empty where
  goldbach :: Integer -> [Integer] -> Set Integer -> [Integer]
  goldbach q (p:ps) gbEven
      | qp `member` gbEven = goldbach q ps gbEven
      | otherwise          = p : goldbach p a065091_list (insert qp gbEven)
      where qp = q + p
-- performance bug fixed: Reinhard Zumkeller, Mar 06 2011

lastE=10; eList=Range[6,lastE,2]; evens[k_] := If[k<=Length[eList], eList[[k]], lastE+=2; AppendTo[eList,lastE]; lastE]; Join[{lastP=3}, Table[k=1; While[p=evens[k]-lastP; p<0 || !PrimeQ[p], k++]; eList=Delete[eList,k]; lastP=p, {999}]] (* T. D. Noe, Mar 04 2011 *)
s={3,3}; ev={6}; a=3; Do[k=2; While[!FreeQ[ev,(b=a+(p=Prime[k]))],k++]; a=p; AppendTo[ev,b]; AppendTo[s,a], {3000}]; s (* Zak Seidov, Mar 03 2011 *)

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A173520 Partial sums of A118371.

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A187072 Prime numbers chosen such that the even numbers that are the sum of two consecutive terms occur only once and occur as early as possible.

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