login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A237817 Number of primes p < n such that r = |{q <= n-p: q and q + 2 are both prime}| and r + 2 are both prime. 5
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 3, 3, 4, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 8, 7, 6, 6, 5, 5, 5, 5, 5, 5, 6, 6, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 4, 4, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,14

COMMENTS

Conjecture: (i) a(n) > 0 for all n > 12.

(ii) For any integer n > 2, there is a prime p < n such that r = |{q <= n-p: q and q + 2 are both prime}| is a square.

See also A237815 for a similar conjecture involving Sophie Germain primes.

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(13) = 1 since {q <= 13 - 2: q and q + 2 are both prime} = {3, 5, 11} has cardinality 3, and {3, 3 + 2} is a twin prime pair.

MATHEMATICA

TQ[n_]:=PrimeQ[n]&&PrimeQ[n+2]

sum[n_]:=Sum[If[PrimeQ[Prime[k]+2], 1, 0], {k, 1, PrimePi[n]}]

a[n_]:=Sum[If[TQ[sum[n-Prime[k]]], 1, 0], {k, 1, PrimePi[n-1]}]

Table[a[n], {n, 1, 80}]

CROSSREFS

Cf. A000040, A000290, A001359, A006512, A237705, A237706, A237769, A237815.

Sequence in context: A341151 A006546 A031268 * A306949 A189025 A236347

Adjacent sequences:  A237814 A237815 A237816 * A237818 A237819 A237820

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Feb 13 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 | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 1 13:38 EDT 2022. Contains 357149 sequences. (Running on oeis4.)