|
|
A239430
|
|
Number of ways to write n = k + m with k > 0 and m > 0 such that pi(2*k) - pi(k) is prime and pi(2*m) - pi(m) is a square, where pi(x) denotes the number of primes not exceeding x.
|
|
2
|
|
|
0, 0, 0, 0, 1, 1, 2, 2, 4, 3, 3, 3, 2, 4, 2, 5, 3, 4, 5, 1, 5, 3, 6, 7, 5, 9, 3, 7, 5, 4, 7, 5, 9, 5, 5, 4, 2, 4, 2, 5, 4, 6, 7, 5, 9, 6, 9, 8, 7, 10, 8, 10, 6, 7, 6, 6, 7, 6, 5, 6, 7, 5, 5, 6, 7, 8, 7, 10, 11, 12, 11, 7, 6, 9, 10, 8, 7, 6, 7, 5
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,7
|
|
COMMENTS
|
Conjecture: a(n) > 0 for all n > 4, and a(n) = 1 only for n = 5, 6, 20.
|
|
LINKS
|
|
|
EXAMPLE
|
a(5) = 1 since 5 = 4 + 1 with pi(2*4) - pi(4) = 4 - 2 = 2 prime and pi(2*1) - pi(1) = 1^2.
a(20) = 1 since 20 = 8 + 12 with pi(2*8) - pi(8) = 6 - 4 = 2 prime and pi(2*12) - pi(12) = 9 - 5 = 2^2.
|
|
MATHEMATICA
|
SQ[n_]:=IntegerQ[Sqrt[n]]
s[n_]:=SQ[PrimePi[2n]-PrimePi[n]]
p[n_]:=PrimeQ[PrimePi[2n]-PrimePi[n]]
a[n_]:=Sum[If[p[k]&&s[n-k], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 80}]
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|