A105047 -id:A105047 - cp's OEIS frontend

A260580 Table read by rows: n-th row contains numbers not occurring earlier, that can be written as (p+q)/2 where p is the n-th odd prime, q <= p.

3, 4, 5, 6, 7, 8, 9, 11, 10, 12, 13, 14, 15, 17, 16, 18, 19, 20, 21, 23, 24, 26, 29, 22, 25, 27, 30, 31, 28, 33, 34, 37, 32, 35, 36, 39, 41, 40, 42, 43, 38, 44, 45, 47, 48, 50, 53, 51, 56, 59, 46, 49, 52, 54, 57, 60, 61, 55, 63, 64, 67, 62, 65, 66, 69, 71

Offset: 1

Reinhard Zumkeller, Aug 11 2015

Length of n-th row = A105047(n+1);
T(n,1) = A260485(n);
T(n,A105047(n)) = A065091(n).

			Let p(n) = A065091(n) = prime(n+1):
.   n | p(n) | T(n,*)
. ----+------+----------------- ------------------------------------------
.   1 |    3 | [3]              3
.   2 |    5 | [4,5]            (5+3)/2,5
.   3 |    7 | [6,7]            (7+5)/2,7
.   4 |   11 | [8,9,11]         (11+5)/2,(11+7)/2,11
.   5 |   13 | [10,12,13]       (13+7)/2,(13+11)/2,13
.   6 |   17 | [14,15,17]       (17+11)/2,(17+13)/2,17
.   7 |   19 | [16,18,19]       (19+13)/2,(19+17)/2,19
.   8 |   23 | [20,21,23]       (23+17)/2,(23+19)/2,23
.   9 |   29 | [24,26,29]       (29+19)/2,(29+17)/2,29
.  10 |   31 | [22,25,27,30,31] (31+13)/2,(31+19)/2,(31+23)/2,(31+29)/2,31
.  11 |   37 | [28,33,34,37]    (37+19)/2,(37+29)/2,(37+31)/2,37
.  12 |   41 | [32,35,36,39,41] (41+23)/2,(41+29)/2,(41+31)/2,(41+37)/2,41

Reinhard Zumkeller, Rows n = 1..1000 of triangle, flattened
Eric Weisstein's World of Mathematics, Goldbach Partition
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture

Cf. A065305, A105047, A065091.

import Data.List.Ordered (union); import Data.List ((\\))
a260580 n k = a260580_tabf !! (n-1) !! (k-1)
a260580_row n = a260580_tabf !! (n-1)
a260580_tabf = zipWith (\\) (tail zss) zss where
                            zss = scanl union [] a065305_tabl

A102696 Number of positive even integers that can be written as the sum of 2 of the first n odd primes (not necessarily distinct).

Original entry on oeis.org

1, 3, 5, 8, 11, 14, 17, 20, 23, 28, 32, 37, 40, 44, 47, 50, 57, 61, 66, 70, 73, 78, 83, 89, 94, 99, 103, 107, 110, 117, 122, 127, 134, 139, 144, 150, 154, 160, 165, 170, 177, 181, 187, 192, 196, 202, 207, 215, 220, 227, 231, 236, 242, 247, 250, 253, 261, 269, 274, 278

Offset: 1

Gabriel Cunningham (gcasey(AT)mit.edu), Feb 04 2005

A105047(n+2) = a(n+1) - a(n). - Reinhard Zumkeller, Aug 11 2015

			a(3) = 5 because with the primes {3, 5, 7} one can write 6 = 3+3, 8 = 3+5, 10 = 5+5, 12 = 5+7 and 14 = 7+7, for a total of 5 even numbers.
a(3) = 5 because with the primes {3, 5, 7} one can write 6 = 3+3, 8 = 3+5, 10 = 5+5 & 3+7, 12 = 5+7 and 14 = 7+7, for a total of 5 even numbers.

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

Cf. A105047 (first differences).

import Data.List (nub)
a102696 n = length $ nub
   [p + q | p <- take n a065091_list, q <- takeWhile (<= p) a065091_list]
-- Reinhard Zumkeller, Aug 11 2015

N:= 1000: # to get first N terms
Primes:= {seq(ithprime(i),i=2..N+1)}:
S:= {}:
for n from 1 to N do
S:= S union map(`+`,Primes[1..n],Primes[n]);
A[n]:= nops(S);
od:
seq(A[n],n=1..N); # Robert Israel, Sep 03 2014

f[n_] := Block[{tp = Table[ Prime[i], {i, 2, n + 1}]}, Length[ Union[ Flatten[ Table[tp[[i]] + tp[[j]], {i, n}, {j, i}]] ]]]; Table[ f[n], {n, 60}] (* Robert G. Wilson v, Feb 05 2005 *)

a(n)=my(P=prime(n+1),s); forstep(k=6,2*P,2, forprime(p=max(k-P,3), min(P,k/2), if(isprime(k-p), s++; break))); s \\ Charles R Greathouse IV, Sep 04 2014

list(n)=my(P=prime(n+1),u=vectorsmall(P),v=vector(n),k); forprime(p=3,P, forprime(q=3,p,u[(p+q)/2]=1); v[k++]=sum(i=1,p,u[i])); v \\ Charles R Greathouse IV, Sep 04 2014

More terms from Robert G. Wilson v, Feb 05 2005

cp's OEIS Frontend

A260580 Table read by rows: n-th row contains numbers not occurring earlier, that can be written as (p+q)/2 where p is the n-th odd prime, q <= p.

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