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A233539 a(n) = |{0 < k < n-2: m - 1, m + 1, prime(m) - m and prime(m) + m are all prime with m = phi(k) + phi(n-k)/2}|, where phi(.) is Euler's totient function. 2
0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 3, 2, 4, 2, 4, 4, 2, 4, 3, 5, 1, 4, 2, 3, 1, 2, 2, 2, 1, 1, 0, 0, 1, 4, 0, 1, 2, 0, 5, 2, 4, 4, 1, 3, 3, 3, 2, 3, 8, 2, 2, 3, 5, 5, 4, 3, 5, 3, 4, 3, 1, 3, 8, 4, 5, 4, 2, 6, 0, 12, 2, 4, 1, 5, 0, 4, 1, 4, 3, 3, 2, 5, 4, 7, 5, 3, 11, 1, 5, 4, 3, 4, 6, 2, 2, 5, 5, 6, 4, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

Conjecture: (i) a(n) > 0 for all n > 794.

(ii) For any integer n > 59, there is a positive integer k < n such that m = phi(k) + phi(n-k)/4 is an integer with prime(m) - m and prime(m) + m both prime.

Clearly, part (i) of the conjecture implies that there are infinitely many positive integers m with m - 1, m + 1, prime(m) - m and prime(m) + m all prime.

LINKS

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

EXAMPLE

a(21) = 1 since phi(6) + phi(15)/2 = 6 with 6 - 1 = 5, 6 + 1 = 7, prime(6) - 6 = 7 and prime(6) + 6 = 19 all prime.

a(25) = 1 since phi(17) + phi(8)/2 = 18 with 18 - 1 = 17, 18 + 1 = 19, prime(18) - 18 = 43 and prime(18) + 18 = 79 all prime.

MATHEMATICA

q[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]&&PrimeQ[Prime[n]-n]&&PrimeQ[Prime[n]+n]

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

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

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

CROSSREFS

Cf. A000010, A000040, A014574, A064269, A064402, A232861, A235682.

Sequence in context: A256587 A249616 A307408 * A317223 A321576 A304101

Adjacent sequences:  A233536 A233537 A233538 * A233540 A233541 A233542

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jan 13 2014

STATUS

approved

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Last modified August 26 02:19 EDT 2019. Contains 326324 sequences. (Running on oeis4.)