login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A235924 a(n) = |{0 < k < n: p = phi(k) + phi(n-k)/3 + 1, q = prime(p) - p + 1 and r = prime(q) - q + 1 are all prime}|, where phi(.) is Euler's totient function. 7
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 3, 0, 1, 0, 2, 3, 4, 3, 3, 1, 2, 3, 1, 6, 2, 9, 2, 5, 3, 4, 3, 8, 1, 4, 3, 9, 2, 3, 5, 6, 6, 7, 3, 8, 7, 6, 4, 4, 5, 7, 3, 6, 5, 1, 4, 6, 6, 2, 3, 4, 5, 4, 11, 4, 5, 4, 7, 2, 5, 5, 5, 2, 6, 2, 5, 5, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,13

COMMENTS

Conjecture: a(n) > 0 for all n > 37.

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

LINKS

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

Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

EXAMPLE

a(20) = 1 since phi(6) + phi(14)/3 + 1 = 5, prime(5) - 4 = 11 - 4 = 7 and prime(7) - 6 = 17 - 6 = 11 are all prime.

a(77) = 1 since phi(59) + phi(18)/3 + 1 = 61, prime(61) - 60 = 283 - 60 = 223 and prime(223) - 222 = 1409 - 222 = 1187 are all prime.

a(1471) = 1 since phi(25) + phi(1446)/3 + 1 = 181, prime(181) - 180 = 1087 - 180 = 907 and prime(907) - 906 = 7057 - 906 = 6151 are all prime.

MATHEMATICA

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

f[n_, k_]:=EulerPhi[k]+EulerPhi[n-k]/3+1

p[n_, k_]:=PrimeQ[f[n, k]]&&PrimeQ[q[f[n, k]]]&&PrimeQ[q[q[f[n, k]]]]

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

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

CROSSREFS

Cf. A000010, A000040, A234694, A234695.

Sequence in context: A082886 A287179 A236511 * A097304 A136745 A214157

Adjacent sequences:  A235921 A235922 A235923 * A235925 A235926 A235927

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jan 17 2014

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 3 13:49 EDT 2020. Contains 336198 sequences. (Running on oeis4.)