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A257364
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Least prime p such that pi(p*n)^2 = pi(q*n)^2 + pi(r*n)^2 for some primes q and r, where pi(x) denotes the number of primes not exceeding x.
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5
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11, 59, 47, 211, 23, 233, 181, 257, 109, 109, 13, 311, 929, 47, 389, 757, 1747, 13, 67, 2389, 1087, 569, 311, 853, 103, 5569, 1399, 3203, 10891, 3673, 3793, 1873, 4357, 41, 2297, 131, 3253, 6737, 2621, 5113, 2879, 953, 6379, 3539, 12343, 4337, 6067, 11939, 43441, 5179
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OFFSET
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1,1
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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