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A103151
Number of decompositions of 2n+1 into 2p+q, where p and q are both odd primes (A065091).
12
0, 0, 0, 1, 1, 2, 1, 3, 2, 2, 2, 3, 3, 4, 2, 4, 2, 4, 4, 4, 4, 5, 3, 4, 6, 5, 3, 6, 3, 3, 6, 6, 5, 7, 3, 4, 7, 6, 5, 8, 3, 7, 7, 7, 4, 10, 5, 6, 9, 5, 5, 11, 5, 6, 9, 7, 6, 10, 7, 5, 11, 8, 6, 10, 5, 6, 12, 8, 5, 12, 5, 9, 12, 8, 6, 13, 7, 6, 11, 9, 9, 16, 4, 8, 12, 9, 9, 13, 7, 6, 13, 11, 8, 16, 6
OFFSET
1,6
COMMENTS
Conjecture: all items for n>=4 are greater than or equal to 1. This is a stronger conjecture than the Goldbach conjecture.
LINKS
EXAMPLE
For 2*4+1 = 9 we have just one such composition: 9 = 2*3+3, so a(4)=1;
For 2*14+1 = 29 we have four such compositions: 29 = 2*3+23 = 2*5+19 = 2*11+7 = 2*13+3, so a(14)=4.
MAPLE
A103151 := proc(n)
local s, a, q;
a := 0 ;
s := 2*n+1 ;
for pi from 2 do
q := s-2*ithprime(pi) ;
if q <=2 then
return a ;
else
if isprime(q) then
a := a+1 ;
end if;
end if;
end do:
end proc: # R. J. Mathar, Feb 22 2014
MATHEMATICA
Do[m = 3; ct = 0; While[(m*2) < n, If[PrimeQ[m], cp = n - (2*m); If[ PrimeQ[cp], ct = ct + 1]]; m = m + 2]; Print[ct], {n, 9, 299, 2}]
PROG
(Scheme, with Aubrey Jaffer's SLIB Scheme library from http://www.swiss.ai.mit.edu/~jaffer/SLIB.html )
(define (A103151 n) (let loop ((i 2) (z 0)) (let ((p1 (A000040 i))) (cond ((>= p1 n) z) ((prime? (+ 1 (* 2 (- n p1)))) (loop (+ 1 i) (+ 1 z))) (else (loop (+ 1 i) z))))))
CROSSREFS
KEYWORD
nonn
AUTHOR
Lei Zhou, Feb 09 2005
EXTENSIONS
Edited and Scheme-code added by Antti Karttunen, Jun 19 2007
STATUS
approved