A035026 Number of times that i and 2n-i are both prime, for i = 1, ..., 2n-1.

0, 1, 1, 2, 3, 2, 3, 4, 4, 4, 5, 6, 5, 4, 6, 4, 7, 8, 3, 6, 8, 6, 7, 10, 8, 6, 10, 6, 7, 12, 5, 10, 12, 4, 10, 12, 9, 10, 14, 8, 9, 16, 9, 8, 18, 8, 9, 14, 6, 12, 16, 10, 11, 16, 12, 14, 20, 12, 11, 24, 7, 10, 20, 6, 14, 18, 11, 10, 16, 14, 15, 22, 11, 10, 24, 8, 16, 22, 9, 16, 20, 10

Offset: 1

Author

Gordon R. Bower (siegmund(AT)mosquitonet.com)

Keywords

• easy
• nonn

Comments

a(n) is the convolution of terms 1 to 2n of the characteristic function of the primes, A010051, with itself. Related to Goldbach's conjecture that every even number can be expressed as the sum of two primes. - T. D. Noe, Aug 01 2002

The following sequences all appear to have the same parity (with an extra zero term at the start of A010051): A010051, A061007, A035026, A069754, A071574. - Jeremy Gardiner, Aug 09 2002

Total number of printer jobs in all possible schedules for n time slots in the first-come-first-served (FCFS) policy.

a(n) = Sum_{p prime < 2*n} A010051(2*n - p). - Reinhard Zumkeller, Oct 19 2011

For n > 1: length of n-th row of triangle A171637. - Reinhard Zumkeller, Mar 03 2014

a(n) = A001221(A238711(n)) = A238778(n) / n. - Reinhard Zumkeller, Mar 06 2014

From Robert G. Wilson v, Dec 15 2016: (Start)

First occurrence of k: 1, 2, 4, 5, 8, 11, 12, 17, 18, 37, 24, 53, 30, 89, 39, 71, 42, 101, 45, 179, 57, 137, 72, 193, 60, 233, ..., .

Conjectured last occurrence of k: 1, 3, 6, 19, 34, 31, 64, 61, 76, 79, 94, 83, 166, 199, 136, 181, 184, 229, 244, 271, 316, 277, 346, 313, 301, 293, ..., .

Conjectured number occurrences of k: 1, 2, 2, 3, 6, 3, 8, 4, 7, 5, 11, 5, 11, 8, 10, 3, 17, 7, 16, 3, 13, 8, 21, 4, 12, 3, 22, 7, 20, 8, 15, ..., .

Records: 0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 20, 24, 26, 28, 38, 42, 48, 54, 60, 64, 82, 88, 102, 104, 114, 116, 136, 146, 152, 166, 182, ..., .

(End)

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, Colored scatterplot of the first 500000 terms
• Index entries for sequences related to Goldbach conjecture

Crossrefs

Cf. A010051. Essentially the same as A002372.

Cf. A073610.

Programs

Haskell

a035026 n = sum $ map (a010051 . (2 * n -)) $
   takeWhile (< 2 * n) a000040_list
-- Reinhard Zumkeller, Oct 19 2011

Maple

A035026 := proc(n)
    local a,i ;
    a := 0 ;
    for i from 1 to 2*n-1 do
        if isprime(i) and isprime(2*n-i) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Jul 01 2013

Mathematica

For[lst={}; n=1, n<=100, n++, For[cnt=0; i=1, i<=2n-1, i++ If[PrimeQ[i]&&PrimeQ[2n-i], cnt++ ]]; AppendTo[lst, cnt]]; lst
f[n_] := Block[{c = Boole@ PrimeQ[ n/2], p = 2}, While[ 2p < n, If[ PrimeQ[n - p], c += 2]; p = NextPrime@ p]; c];; Array[ f[ 2#] &, 90] (* Robert G. Wilson v, Dec 15 2016 *)

Formula

For n > 1, a(n) = 2*A045917(n) - A010051(n).

a(n) = A010051(n) + 2*A061357(n). - Wesley Ivan Hurt, Aug 21 2013

a(n) = A073610(2*n). - Ridouane Oudra, Sep 06 2023

Extensions

Corrected by T. D. Noe, May 05 2002