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A232442
a(n) = |{0 < k < n: m*prime(m) - 1 and m*prime(m) + 1 are both prime with m = sigma(k) + phi(n-k)}|, where sigma(k) is the sum of all positive divisors of k and phi(.) is Euler's totient function.
1
0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 2, 1, 1, 0, 0, 1, 1, 6, 1, 2, 2, 0, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 4, 1, 1, 0, 1, 2, 2, 2, 4, 0, 0, 1, 2, 0, 3, 3, 3, 2, 0, 1, 1, 2, 1, 2, 0, 1, 1, 14, 3, 2, 2, 2, 2, 3, 4, 5, 3, 2, 3, 1, 3, 3, 4, 6, 3, 0, 5, 3, 1, 0, 5, 2, 0, 3, 6, 1
OFFSET
1,23
COMMENTS
Conjecture: a(n) > 0 for all n > 214.
This implies that there are infinitely many twin prime pairs of the special form {m*prime(m) - 1, m*prime(m) + 1}.
We have verified the conjecture for n up to 10^5.
EXAMPLE
a(25) = 1 since sigma(6) + phi(19) = 12 + 18 = 30 with {30*prime(30) - 1, 30*prime(30) + 1} = {3389, 3391} a twin prime pair.
a(100) = 1 since sigma(75) + phi(25) = 124 + 20 = 144 with {144*prime(144) - 1, 144*prime(144) + 1} = {119087, 119089} a twin prime pair.
MATHEMATICA
sigma[n_]:=DivisorSigma[1, n]
q[n_]:=PrimeQ[n*Prime[n]-1]&&PrimeQ[n*Prime[n]+1]
f[n_, k_]:=sigma[k]+EulerPhi[n-k]
a[n_]:=Sum[If[q[f[n, k]], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 100}]
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 14 2014
STATUS
approved