

A035026


Number of times that i and 2ni are both prime, for i = 1, ..., 2n1.


16



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,4


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 firstcomefirstserved (FCFS) policy.
a(n) = Sum_{p prime < 2*n} A010051(2*n  p).  Reinhard Zumkeller, Oct 19 2011
For n > 1: length of nth 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


FORMULA

For n > 1, a(n) = 2*A045917(n)  A010051(n).
a(n) = A010051(n) + 2*A061357(n).  Wesley Ivan Hurt, Aug 21 2013


MAPLE

A035026 := proc(n)
local a, i ;
a := 0 ;
for i from 1 to 2*n1 do
if isprime(i) and isprime(2*ni) 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<=2n1, i++ If[PrimeQ[i]&&PrimeQ[2ni], 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 *)


PROG

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


CROSSREFS

Cf. A010051. Essentially the same as A002372.
Sequence in context: A231070 A230252 A002372 * A224962 A173540 A070770
Adjacent sequences: A035023 A035024 A035025 * A035027 A035028 A035029


KEYWORD

easy,nonn


AUTHOR

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


EXTENSIONS

Corrected by T. D. Noe, May 05 2002


STATUS

approved



