

A261339


Least positive integer k such that both k and k*n belong to the set {m>0: m+1, m^2+1 and m^2+prime(m)^2 are all prime}.


2



1, 1, 47500, 20440, 2, 124560, 17850, 2730, 185550, 1, 518910, 429180, 10, 687480, 81030, 36, 1568340, 2, 1165750, 7410, 10, 6780, 481140, 10, 10, 5430, 240, 2730, 72660, 2080, 18700, 291720, 295080, 52860, 5430, 1, 81030, 56400, 12490, 43590, 124560, 40030, 5170, 278700, 2091850, 131320, 184110, 11206510, 12910, 1245780
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


COMMENTS

Conjecture: a(n) exists for any n > 0. In general, any positive rational number r can be written as m/n with m and n in the set {k>0: k+1, k^2+1 and k^2+prime(k)^2 are all prime}.
For example, 5/8 = 3567600/5708160 with 3567600+1, 3567600^2+1 = 12727769760001, 3567600^2 + prime(3567600)^2 = 3567600^2 + 60098671^2 = 3624578025726241, 5708160+1, 5708160^2+1 = 32583090585601, and 5708160^2 + prime(5708160)^2 = 5708160^2 + 99018553^2 = 9837256928799409 all prime.
The conjecture implies that there are infinitely many primes p with (p1)^2+1 and (p1)^2+prime(p1)^2 both prime.
We also guess that any positive rational number can be written as m/n, where m and n are positive integers with m^2+prime(m)^2, m^2+prime(n)^2, n^2+prime(m)^2 and n^2+prime(n)^2 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..1000
ZhiWei Sun, Checking the conjecture for r = a/b with a,b = 1..60
ZhiWei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.


EXAMPLE

a(3) = 47500 since 47501, 47500^2 + 1 = 2256250001, 47500^2 + prime(47500)^2 = 47500^2 + 578827^2 = 337296945929, 47500*3 + 1 = 142501, (47500*3)^2 + 1 = 20306250001, and (47500*3)^2 + prime(47500*3)^2 = 142500^2 + 1907023^2 = 3657042972529 are all prime.


MATHEMATICA

PQ[n_]:=PrimeQ[n+1]&&PrimeQ[n^2+1]&&PrimeQ[n^2+Prime[n]^2]
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, 50}]


CROSSREFS

Cf. A000040, A005574, A236193, A259492, A259540, A260120.
Sequence in context: A234708 A069370 A241221 * A166003 A221017 A254800
Adjacent sequences: A261336 A261337 A261338 * A261340 A261341 A261342


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Aug 15 2015


STATUS

approved



