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A236460
Number of ordered ways to write n = k + m with k > 0 and m > 0 such that p = phi(k) + phi(m)/2 - 1, prime(p) + 4 and prime(p) + 6 are all prime, where phi(.) is Euler's totient function.
4
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 2, 1, 0, 1, 2, 2, 2, 1, 1, 2, 0, 4, 1, 2, 1, 5, 2, 1, 3, 1, 1, 3, 2, 6, 3, 0, 2, 5, 5, 6, 3, 4, 5, 3, 4, 4, 4, 6, 3, 2, 6, 2, 3, 2, 10, 2, 3, 1, 6, 1, 4, 0, 2, 3, 4, 2, 4, 0, 4, 0, 3, 2, 3, 0, 4, 0, 1, 1, 4
OFFSET
1,32
COMMENTS
Conjecture: a(n) > 0 for all n > 211.
This implies that there are infinitely many primes p with {prime(p), prime(p) + 4, prime(p) + 6} a prime triple. See A236462 for such primes p.
EXAMPLE
a(30) = 1 since 30 = 13 + 17 with phi(13) + phi(17)/2 - 1 = 19, prime(19) + 4 = 67 + 4 = 71 and prime(19) + 6 = 73 all prime.
a(831) = 1 since 831 = 66 + 765 with phi(66) + phi(765)/2 - 1 = 20 + 192 - 1 = 211, prime(211) + 4 = 1297 + 4 = 1301 and prime(211) + 6 = 1303 all prime.
MATHEMATICA
p[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]+4]&&PrimeQ[Prime[n]+6]
f[n_, k_]:=EulerPhi[k]+EulerPhi[n-k]/2-1
a[n_]:=Sum[If[p[f[n, k]], 1, 0], {k, 1, n-3}]
Table[a[n], {n, 1, 100}]
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 26 2014
STATUS
approved