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A261528 Least positive integer k such that both k and k*n belong to the set {m>0: prime(m)+2 is prime with prime(prime(m)+2) = prime(prime(m))+6}. 2
2, 891, 81002, 814812, 86050, 5917, 65527, 109853, 2563344, 25379, 2640232, 266076, 775889, 67387, 68111, 37950, 353416, 347139, 56390, 11299, 89491, 545458, 910786, 353416, 1913477, 9025, 111569, 511796, 1456228, 37909, 1494675, 212092, 69352, 107769, 300657, 1155675, 391972, 1073031, 55074, 49892 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Conjecture: Any positive rational number r can be written as m/n with m and n in the set {k>0: prime(k)+2 is prime with prime(prime(k)+2) = prime(prime(k))+6}.

This implies that there are infinitely many twin prime pairs {p, p+2} with prime(p+2) - prime(p) = 6.

Note that if prime(n+2)-prime(n) = 6 then prime(n+1)-prime(n) = 2 or 4.

REFERENCES

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..100

Zhi-Wei Sun, Checking the conjecture for r = a/b with a,b = 1..20

Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

EXAMPLE

a(1) = 2 since 2*1 = 2, and prime(2)+2 = 3+2 = 5 is prime with prime(5)-prime(3) = 11-5 = 6.

a(2) = 891 since prime(891)+2 = 6947 + 2 = 6949 is prime with prime(6949)-prime(6947) = 70123-70117 = 6, and prime(891*2)+2 = 15269 + 2 = 15271 is prime with prime(15271)-prime(15269) = 167119-167113 = 6.

MATHEMATICA

f[n_]:=Prime[n]

PQ[k_]:=PrimeQ[f[k]+2]&&f[f[k]+2]-f[f[k]]==6

Do[k=0; Label[bb]; k=k+1; If[PQ[k]&&PQ[k*n], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k]; Continue, {n, 1, 40}]

CROSSREFS

Cf. A000040, A001359, A006512, A023201, A046117, A259487, A259540, A261533.

Sequence in context: A050247 A159705 A229678 * A265881 A203609 A265617

Adjacent sequences:  A261525 A261526 A261527 * A261529 A261530 A261531

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Aug 23 2015

STATUS

approved

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Last modified January 28 04:30 EST 2022. Contains 350654 sequences. (Running on oeis4.)