login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A261353 Least positive integer k such that prime(prime(k))*prime(prime(k*n)) = prime(p)+2 for some prime p. 4
11, 2, 1, 606, 350, 166, 53, 1865, 7, 45, 1308, 68, 215, 61, 256, 13, 248, 90, 1, 1779, 796, 1, 4, 444, 650, 55, 157, 303, 82, 84, 25, 3, 1912, 621, 128, 205, 164, 1091, 61, 12, 337, 1, 303, 15, 23, 418, 212, 23, 2494, 1, 472, 771, 1, 36, 8, 46, 8, 18, 264, 22, 725, 85, 65, 231, 606, 3, 1, 43, 144, 164 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Conjecture: a(n) exists for any n > 0. In general, any positive rational number r can be written as m/n, where m and n are positive integers such that prime(prime(m))*prime(prime(n)) = prime(p)+2 for some prime p.

This implies that the sequence A261352 has infinitely many terms.

REFERENCES

Jing-run Chen, On the representation of a large even integer as the sum of a prime and a product of at most two primes, Sci. Sinica 16(1973), 157-176.

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..382

Zhi-Wei Sun, Checking the conjecture for r = a/b with a,b = 1,...,31

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

EXAMPLE

a(1) = 11 since prime(prime(11))*prime(prime(11*1)) = prime(31)^2 = 127^2 = 16129 = prime(1877)+2 with 1877 prime.

a(4) = 606 since prime(prime(606))*prime(prime(606*4)) = prime(4457)*prime(21589) = 42643*244471 = 10424976853 = prime(473490161)+2 with 473490161 prime.

MATHEMATICA

f[n_]:=Prime[Prime[n]]

PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]

Do[k=0; Label[bb]; k=k+1; If[PQ[f[k]*f[k*n]-2], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k]; Continue, {n, 1, 70}]

CROSSREFS

Cf. A000040, A109611, A261282, A261352.

Sequence in context: A193925 A010190 A323454 * A087774 A322562 A040118

Adjacent sequences:  A261350 A261351 A261352 * A261354 A261355 A261356

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Aug 15 2015

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 December 12 22:06 EST 2019. Contains 329963 sequences. (Running on oeis4.)