

A261541


Least positive integer m such that both m and m*n belong to the set {k>0: prime(k)+2, prime(k)+6, prime(k)+8 are all prime}.


1



3, 358712, 34772, 79631, 1822685, 22865, 2066, 2593722, 26, 3418900, 26, 711611, 286, 1493190, 882854, 513312, 1707237, 788232, 913695, 1980985, 7147, 443152, 479580, 2589105, 865432, 265243, 103641, 160536, 398360, 851672
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

Conjecture: (i) Each positive rational number r can be written as m/n with m and n in the set {k>0: prime(k)+2, prime(k)+6 and prime(k)+8 are all prime}.
(ii) Any positive rational number r can be written as m/n with m and n in the set {k>0: prime(k)+4, prime(k)+6 and prime(k)+10 are all prime}.
For example, 3/4 = 20723892/27631856, and prime(20723892)+2 = 387875561+2 = 387875563, prime(20723892)+6 = 387875567, prime(20723892)+8 = 387875569, prime(27631856)+2 = 525608591+2 =525608593, prime(27631856)+6 = 525608597, prime(27631856)+8 = 525608599 are all prime. Also, 3/4 = 599478/799304, and prime(599478)+4 = 8951857+4 = 8951861, prime(599478)+6 = 8951863, prime(599478)+10 = 8951867, prime(799304)+4 = 12183943+4 = 12183947, prime(799304)+6 = 12183949, prime(799304)+10 = 12183953 are all prime.
Part (i) of the conjecture implies that there are infinitely many primes p with p+2, p+6 and p+8 all prime, while part (ii) implies that there are infinitely many primes p with p+4, p+6 and p+10 all prime.


REFERENCES

ZhiWei 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 ChinaJapan Seminar (Fukuoka, Oct. 28  Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169187.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..100
ZhiWei Sun, Checking part (i) of the conjecture for r = a/b with a,b = 1..25
ZhiWei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.


EXAMPLE

a(1) = 3 since 3*1 = 3, and prime(3)+2 = 5+2 =7, prime(3)+6 = 11 and prime(3)+8 = 13 are all prime.
a(2) = 358712 since prime(358712)+2 = 5158031+2 = 5158033, prime(358712)+6 = 5158037, prime(358712)+8 = 5158039, prime(358712*2)+2 = 10852601+2 = 10852603, prime(358712*2)+6 = 10852607 and prime(358712*2)+8 = 10852609 are all prime.


MATHEMATICA

f[n_]:=Prime[n]
PQ[k_]:=PrimeQ[f[k]+2]&&PrimeQ[f[k]+6]&&PrimeQ[f[k]+8]
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, 30}]


CROSSREFS

Cf. A000040, A007530, A052378, A236511, A259487, A259540.
Sequence in context: A033982 A326618 A154824 * A309225 A137131 A229725
Adjacent sequences: A261538 A261539 A261540 * A261542 A261543 A261544


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Aug 24 2015


STATUS

approved



