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A236573
Number of ordered ways to write n = k + m (k > 0, m > 0) such that p = 2*k + phi(m) - 1, prime(p + 2) + 2 and 2*n - p are all prime, where phi(.) is Euler's totient function.
1
0, 0, 0, 1, 2, 2, 1, 2, 3, 0, 2, 3, 0, 1, 0, 0, 2, 1, 2, 0, 2, 3, 1, 4, 3, 3, 8, 3, 2, 5, 5, 4, 3, 1, 2, 7, 6, 0, 8, 4, 2, 8, 4, 4, 7, 4, 4, 3, 6, 3, 5, 3, 1, 4, 6, 4, 9, 2, 4, 11, 2, 1, 5, 2, 4, 4, 1, 2, 9, 4, 0, 3, 2, 2, 5, 2, 4, 4, 1, 4, 1, 1, 1, 4, 0, 0, 3, 2, 5, 5, 0, 1, 2, 2, 1, 2, 1, 2, 2, 1
OFFSET
1,5
COMMENTS
Conjecture: a(n) > 0 for all n > 712.
This implies the conjecture in A236566.
EXAMPLE
a(100) = 1 since 100 = 10 + 90 with 2*10 + phi(90) - 1 = 20 + 24 - 1 = 43, prime(43 + 2) + 2 = 197 + 2 = 199 and 2*100 - 43 = 157 all prime.
a(1727) = 1 since 1727 = 956 + 771 with 2*956 + phi(771) - 1 = 1912 + 512 - 1 = 2423, prime(2423 + 2) + 2 = 21599 + 2 = 21601 and 2*1727 - 2423 = 1031 all prime.
MATHEMATICA
p[n_]:=PrimeQ[n]&&PrimeQ[Prime[n+2]+2]
f[n_, k_]:=2k+EulerPhi[n-k]-1
a[n_]:=Sum[If[p[f[n, k]]&&PrimeQ[2n-f[n, k]], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 100}]
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 29 2014
STATUS
approved