

A255679


Least number k such that pi(k*n)^2 = pi(i*n)^2 + pi(j*n)^2 for some 0 < i < j < k.


3



4, 6, 4, 12, 6, 17, 17, 6, 11, 6, 13, 14, 4, 10, 31, 35, 10, 8, 20, 20, 70, 20, 34, 23, 47, 17, 44, 25, 5, 46, 4, 26, 53, 13, 29, 33, 35, 45, 65, 10, 18, 64, 117, 21, 77, 19, 71, 101, 44, 80, 28, 93, 36, 183, 100, 98, 55, 4, 53, 81, 31, 120, 58, 40, 53, 56, 34, 49, 181, 218
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OFFSET

1,1


COMMENTS

Conjecture: a(n) exists for any n > 0. In other words, for each n = 1,2,3,... the sequence pi(k*n) (k = 1,2,3,...) contains a Pythagorean triple.


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..240
ZhiWei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.


EXAMPLE

a(1) = 4 since pi(4*1)^2 = 2^2 = pi(1*1)^2 + pi(3*1)^2.
a(21) = 70 since pi(70*21)^2 = pi(1470)^2 = 232^2 = 160^2 + 168^2 = pi(45*21)^2 + pi(48*21)^2.


MATHEMATICA

Do[k=0; Label[bb]; k=k+1; Do[If[PrimePi[k*n]^2==PrimePi[i*n]^2+PrimePi[j*n]^2, Goto[aa], Goto[cc]]; Label[cc]; Continue, {j, 1, k1}, {i, 1, j1}]; Goto[bb];
Label[aa]; Print[n, " ", k]; Continue, {n, 1, 70}]


PROG

(PARI) is_ok(n, k)={ r=0; for(i=1, k1, for(j=i+1, k1, if(primepi(k*n)^2 == primepi(i*n)^2 + primepi(j*n)^2, r=1; break(2)) )); return(r); }
a(n)={ k=0; while(!is_ok(n, k), k++); return(k); }
main(size)={ v=vector(size); i=1; m=1; while(i<=size, v[i]=a(i); i++); return(v); } /* Anders Hellström, Jul 11 2015 */


CROSSREFS

Cf. A000290, A000720, A255677.
Sequence in context: A143521 A278363 A123969 * A019188 A019244 A019189
Adjacent sequences: A255676 A255677 A255678 * A255680 A255681 A255682


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Jul 11 2015


STATUS

approved



