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A215461
Number of decompositions of 2n into ordered sums of one prime and one nonprime.
1
0, 0, 2, 2, 2, 0, 4, 4, 2, 4, 6, 4, 4, 6, 8, 6, 12, 6, 4, 16, 10, 8, 14, 12, 8, 12, 16, 10, 18, 16, 8, 24, 14, 10, 28, 16, 14, 22, 20, 12, 26, 24, 12, 26, 28, 10, 30, 28, 18, 36, 24, 18, 32, 30, 22, 32, 28, 18, 34, 36, 10, 44, 38, 18, 48, 32, 26, 40, 42, 32, 38, 36, 22, 44
OFFSET
0,3
COMMENTS
A002372(n) + a(n) + A215462(n) = n.
Note: a(n) always even.
Conjecture: a(n) is never zero for n > 5, verified to 10^9.
Goldbach conjecture: a(n) + A215462(n) < n for all n > 2.
FORMULA
a(n) = convolve(p,c) + convolve(c,p) = 2*convolve(p,c) where p(n) = 1 if 2n+1 is prime and 0 otherwise, and c(n) = 1 if 2n+1 is nonprime and 0 otherwise.
EXAMPLE
n=15, 2*n=30, 2*n = { 3+27, 5+25, 29+1; 1+29, 25+5, 27+3 }, a(15) = 6
n=18, 2*n=36, 2*n = { 3+33, 11+25; 11+25, 33+3 }, a(18) = 4
CROSSREFS
Sequence in context: A000091 A155123 A125938 * A158851 A151930 A356907
KEYWORD
nonn
AUTHOR
Peter A. Lawrence, Aug 11 2012
STATUS
approved