

A259764


Least prime p such that prime(p*n)1 is a square, or 0 if no such p exists.


2



3, 13, 41, 3, 11, 2, 241, 181, 5, 2927, 5, 523, 2, 4967, 3, 421, 33053, 8447, 17107, 20747, 1811, 5743, 20407, 99643, 165443, 769, 21269, 46099, 3121, 9883, 16301, 523, 10771, 41603, 17, 7, 48383, 455353, 711317, 1637, 3, 105397, 43, 12071, 186113, 56437, 303157, 211, 25951, 178817
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OFFSET

1,1


COMMENTS

Conjecture: a(n) > 0 for all n > 0.
This is stronger than the conjecture in A259731. It implies the wellknown conjecture that there are infinitely many primes of the form x^21 with x an integer.
I also conjecture that for any positive integer n there exists a prime p such that prime(p*n)+2 is a square.


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..500
ZhiWei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.


EXAMPLE

a(1) = 3 since 3 is prime and prime(3*1)1 = 2^2 is a square.
a(2) = 13 since 13 is prime and prime(13*2)1 = 10^2 is a square.


MATHEMATICA

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[k=0; Label[bb]; k=k+1; If[SQ[Prime[Prime[k]*n]1], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", Prime[k]]; Continue, {n, 1, 50}]


CROSSREFS

Cf. A000040, A000290, A002496, A028871, A259731.
Sequence in context: A290720 A289654 A095109 * A309139 A049167 A241527
Adjacent sequences: A259761 A259762 A259763 * A259765 A259766 A259767


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Jul 04 2015


STATUS

approved



