OFFSET
1,1
COMMENTS
Conjecture: a(n) exists for any n > 0. In other words, for each fixed positive integer n the sequence pi(p*n) with p prime contains a Pythagorean triple.
This is stronger than the conjecture in A255679.
REFERENCES
Zhi-Wei 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 China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..100
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.
EXAMPLE
a(1) = 11 since 5, 7 and 11 are primes with pi(5*1)^2 + pi(7*1)^2 = 3^2 + 4^2 = 5^2 = pi(11*1)^2.
a(45) = 12343 since 4337, 11311 and 12343 are primes with pi(4337*45)^2 + pi(11311*45)^2 = 17590^2 + 42216^2 = 45734^2 = pi(12343*45)^2.
a(49) = 43441 since 15427, 39839 and 43441 are primes with pi(15427*49)^2 + pi(39839*49)^2 = 60685^2 + 145644^2 = 157781^2 = pi(43441*49)^2.
MATHEMATICA
f[n_]:=PrimePi[n]
Do[k=0; Label[bb]; k=k+1; Do[Do[If[f[Prime[k]*n]^2==f[Prime[i]*n]^2+f[Prime[j]*n]^2, Goto[aa]]; If[f[Prime[k]*n]^2<f[Prime[i]*n]^2+f[Prime[j]*n]^2, Goto[cc]]; Continue, {i, 1, j-1}]; Label[cc]; Continue, {j, 1, k-1}]; Goto[bb];
Label[aa]; Print[n, " ", Prime[k]]; Continue, {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jul 11 2015
STATUS
approved