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A199920 Number of ways to write n = p+k with p, p+6, 6k-1 and 6k+1 all prime 10
0, 0, 0, 0, 0, 1, 1, 2, 1, 2, 0, 3, 1, 3, 2, 2, 2, 3, 2, 2, 1, 2, 3, 3, 3, 1, 1, 3, 2, 4, 1, 2, 2, 3, 3, 3, 2, 4, 2, 4, 3, 3, 5, 3, 3, 3, 3, 4, 5, 3, 3, 3, 3, 5, 4, 4, 3, 4, 3, 3, 2, 3, 6, 5, 4, 2, 1, 3, 5, 5, 5, 2, 2, 3, 5, 3, 5, 4, 5, 2, 3, 2, 5, 5, 6, 4, 2, 3, 3, 4, 3, 3, 5, 4, 3, 1, 1, 4, 5, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,8
COMMENTS
Conjecture: a(n)>0 for all n>11.
This implies that there are infinitely many twin primes and also infinitely many sexy primes. It has been verified for n up to 10^9. See also A199800 for a weaker version of this conjecture.
Zhi-Wei Sun also conjectured that any integer n>6 not equal to 319 can be written as p+k with p, p+6, 3k-2+(n mod 2) and 3k+2-(n mod 2) all prime.
LINKS
EXAMPLE
a(21)=1 since 21=11+10 with 11, 11+6, 6*10-1 and 6*10+1 all prime.
MATHEMATICA
a[n_]:=a[n]=Sum[If[PrimeQ[Prime[k]+6]==True&&PrimeQ[6(n-Prime[k])-1]==True&&PrimeQ[6(n-Prime[k])+1]==True, 1, 0], {k, 1, PrimePi[n]}]
Do[Print[n, " ", a[n]], {n, 1, 100}]
Table[Count[Table[{n-i, i}, {i, n-1}], _?(AllTrue[{#[[1]], #[[1]]+6, 6#[[2]]-1, 6#[[2]]+1}, PrimeQ]&)], {n, 100}] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 19 2015 *)
PROG
(PARI) a(n)=my(s, p=2, q=3); forprime(r=5, n+5, if(r-p==6 && isprime(6*n-6*p-1) && isprime(6*n-6*p+1), s++); if(r-q==6 && isprime(6*n-6*q-1) && isprime(6*n-6*q+1), s++); p=q; q=r); s \\ Charles R Greathouse IV, Jul 31 2016
CROSSREFS
Sequence in context: A274576 A257081 A271484 * A177995 A332104 A238735
KEYWORD
nonn,nice
AUTHOR
Zhi-Wei Sun, Dec 22 2012
STATUS
approved

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Last modified March 1 01:59 EST 2024. Contains 370428 sequences. (Running on oeis4.)