

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



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



KEYWORD

nonn


AUTHOR



STATUS

approved



