A255679 -id:A255679 - cp's OEIS frontend

A257364 Least prime p such that pi(pn)^2 = pi(qn)^2 + pi(r*n)^2 for some primes q and r, where pi(x) denotes the number of primes not exceeding x.

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

Offset: 1

Zhi-Wei Sun, Jul 11 2015

nonn

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.

			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.

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.

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.

Cf. A000040, A000720, A255679, A257928.

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

A255677 Least integer k > 1 such that pi(k)^2 + pi(k*n)^2 is a square, where pi(.) is the prime-counting function given by A000720.

Original entry on oeis.org

5, 30, 8458, 18, 252, 25, 1407, 476, 9098, 108, 1814, 1868, 153, 1005, 67, 26532, 1592, 200, 963, 99, 833, 1356, 3869, 981, 531, 127, 4961, 366, 1192, 1873, 41308, 409, 21756, 194664, 180, 27071, 7433, 160179, 2076, 544, 211, 10639, 19571, 33483, 603, 68380, 1517, 47529, 35923

Offset: 2

Zhi-Wei Sun, Jul 10 2015

nonn

Conjecture: Each positive rational number r < 1 can be written as m/n with 1 < m < n such that pi(m)^2 + pi(n)^2 is a square. Also, any rational number r > 1 can be written as m/n with m > n > 1 such that pi(m)^2 - pi(n)^2 is a square.

For example, 23/24 = 19947716/20815008 with pi(19947716)^2 + pi(20815008)^2 = 1267497^2 + 1319004^2 = 1829295^2, and 7/3 = 26964/11556 with pi(26964)^2 - pi(11556)^2 = 2958^2 - 1392^2 = 2610^2.

			a(2) = 5 since pi(5)^2 + pi(5*2)^2 = 3^2 + 4^2 = 5^2.
a(3) = 30 since pi(30)^2 + pi(30*3)^2 = 10^2 + 24^2 = 26^2.
a(68) = 6260592 since pi(6260592)^2 + pi(6260592*68)^2 = 429505^2 + 22632876^2 = 22636951^2.
a(95) = 7955506 since pi(7955506)^2 + pi(7955506*95)^2 = 536984^2 + 38985687^2 = 38989385^2.

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.

Zhi-Wei Sun, Table of n, a(n) for n = 2..100
Zhi-Wei Sun, Checking the conjecture for r = a/b, b/a with 1 <= a < b <= 50
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000720, A000290, A255679, A259531, A259789.

SQ[n_]:=IntegerQ[Sqrt[n]]
Do[k=1;Label[aa];k=k+1;If[SQ[PrimePi[k]^2+PrimePi[k*n]^2],Goto[bb],Goto[aa]];Label[bb];Print[n," ",k];Continue,{n,2,50}]

a(n)={ k=2; while(!issquare(primepi(k)^2 + primepi(k*n)^2),k++); return(k);}
main(size)={ v=vector(size); for(i=2, size+1, v[i-1]=a(i)); return(v);} /* Anders Hellström, Jul 11 2015 */

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A257364 Least prime p such that pi(pn)^2 = pi(qn)^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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A255677 Least integer k > 1 such that pi(k)^2 + pi(k*n)^2 is a square, where pi(.) is the prime-counting function given by A000720.

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cp's OEIS Frontend

A257364 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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A255677 Least integer k > 1 such that pi(k)^2 + pi(k*n)^2 is a square, where pi(.) is the prime-counting function given by A000720.

Views

Author

Keywords

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Examples

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PARI

A257364 Least prime p such that pi(pn)^2 = pi(qn)^2 + pi(r*n)^2 for some primes q and r, where pi(x) denotes the number of primes not exceeding x.