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A235682
Number of ways to write n = k + m with k > 0 and m > 2 such that p = phi(k) + phi(m)/2 + 1, prime(p) - p + 1 and p*(p+1) - prime(p) are all prime, where phi(.) is Euler's totient function.
5
0, 0, 0, 1, 2, 1, 1, 3, 2, 1, 3, 1, 1, 2, 2, 3, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 3, 0, 1, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 4, 6, 3, 6, 0, 6, 4, 5, 3, 1, 3, 4, 2, 3, 4, 1, 8, 6, 4, 8, 8
OFFSET
1,5
COMMENTS
Conjecture: a(n) > 0 for all n > 84.
Clearly, this implies that there are infinitely many primes p with prime(p) - p + 1 and p*(p+1) - prime(p) both prime.
EXAMPLE
a(10) = 1 since 10 = 1 + 9 with phi(1) + phi(9)/2 + 1 = 5, prime(5) - 5 + 1 = 7 and 5*6 - prime(5) = 19 all prime.
a(95) = 1 since 95 = 62 + 33 with phi(62) + phi(33)/2 + 1 = 41, prime(41) - 41 + 1 = 139 and 41*42 - prime(41) = 1543 all prime.
a(421) = 1 since 421 = 289 + 132 with phi(289) + phi(132)/2 + 1 = 293, prime(293) - 293 + 1 = 1621 and 293*294 - prime(293) = 84229 all prime.
MATHEMATICA
PQ[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]-n+1]&&PrimeQ[n(n+1)-Prime[n]]
f[n_, k_]:=EulerPhi[k]+EulerPhi[n-k]/2+1
a[n_]:=Sum[If[PQ[f[n, k]], 1, 0], {k, 1, n-3}]
Table[a[n], {n, 1, 100}]
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 13 2014
STATUS
approved