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A235912 a(n) = |{0 < k < n - 2: 2*m + 1, m*(m-1) - prime(m) and m*(m+1) - prime(m) are all prime with m = phi(k) + phi(n-k)/2}|, where phi(.) is Euler's totient function. 3
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 3, 2, 4, 2, 6, 5, 6, 7, 4, 8, 7, 8, 8, 11, 7, 12, 9, 9, 12, 5, 14, 10, 9, 9, 9, 9, 7, 8, 11, 9, 8, 7, 14, 8, 6, 9, 5, 5, 9, 11, 3, 9, 6, 13, 8, 8, 6, 7, 6, 5, 4, 3, 1, 5, 5, 5, 6, 5, 7, 7, 4, 7, 11, 8, 3, 5, 3, 10, 4, 4, 3, 9, 2, 4, 4, 5, 8, 12, 13, 4, 9, 5, 11, 5, 12, 7, 4, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,12

COMMENTS

Conjecture: a(n) > 0 for all n > 11.

This implies that there are infinitely many odd primes p = 2*m + 1 with q = m*(m-1) - prime(m) and r = m*(m+1) - prime(m) both prime. Note that r - q = 2*m.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

EXAMPLE

a(10) = 1 since phi(5) + phi(5)/2 = 6 with 2*6 + 1 = 13, 5*6 - prime(6) = 30 - 13 = 17 and 6*7 - prime(6) = 42 - 13 = 29 all prime.

MATHEMATICA

PQ[n_]:=n>0&&PrimeQ[n]

p[n_]:=PrimeQ[2n+1]&&PQ[n(n-1)-Prime[n]]&&PQ[n(n+1)-Prime[n]]

f[n_, k_]:=EulerPhi[k]+EulerPhi[n-k]/2

a[n_]:=Sum[If[p[f[n, k]], 1, 0], {k, 1, n-3}]

Table[a[n], {n, 1, 100}]

CROSSREFS

Cf. A000010, A000040, A235592, A235728.

Sequence in context: A322587 A058973 A155520 * A339749 A277859 A308566

Adjacent sequences:  A235909 A235910 A235911 * A235913 A235914 A235915

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jan 16 2014

STATUS

approved

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Last modified December 5 12:24 EST 2021. Contains 349557 sequences. (Running on oeis4.)