A257364 -id:A257364 - cp's OEIS frontend

A257928 Least prime p such that pi(pn) = pi(qn)pi(rn) for some primes q and r with p > q > r, where pi(x) denotes the number of primes not exceeding x.

13, 7, 13, 67, 19, 79, 47, 193, 107, 41, 229, 179, 383, 281, 173, 1327, 193, 701, 1429, 211, 113, 73, 1093, 83, 1447, 659, 197, 719, 331, 761, 1171, 2269, 467, 509, 863, 113, 643, 577, 563, 379, 607, 1291, 283, 3593, 2549, 881, 1523, 4663, 2657, 3583, 8807, 683, 2251, 863, 8929, 163, 6737, 2459, 4919, 6553

Offset: 1

Zhi-Wei Sun, Jul 13 2015

nonn

Conjecture: a(n) exists for any n > 0. Also, for each positive integer n there are distinct primes p, q and r such that pi(p*n) = pi(q*n) + pi(r*n).

			a(1) = 13 since 3, 5 and 13 are distinct primes with pi(13*1) = 6 = 2*3 = pi(3*1)*pi(5*1).
a(200) = 105227 since 19, 113 and 105227 are distinct primes with pi(105227*200) = 1332672 = 528*2524 = pi(19*200)*pi(113*200).

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..200
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A000720, A257364, A257938.

f[n_]:=PrimePi[n]
Do[k=0;Label[bb];k=k+1;Do[Do[If[f[Prime[k]*n]==f[Prime[i]*n]*f[Prime[j]*n],Goto[aa]];If[f[Prime[k]*n]

a(n)={my(i,j,k=3);while(1,for(j=2,k-1,for(i=1,j-1,if(primepi(prime(k)*n) == primepi(prime(i)*n)*primepi(prime(j)*n),break(3));));k++);return(prime(k));} main(size)={return(vector(size,n,a(n)));} /* Anders Hellström, Jul 13 2015 */

A260911 Least positive integer k < prime(n) such that there are 0 < i < j < k for which i^2 + j^2 = k^2 and i,j,k are all quadratic residues modulo prime(n), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 0, 0, 5, 0, 0, 0, 0, 25, 0, 34, 0, 41, 25, 25, 5, 5, 26, 5, 37, 0, 41, 0, 0, 65, 17, 34, 5, 61, 17, 5, 17, 25, 25, 29, 37, 26, 25, 41, 5, 5, 5, 25, 25, 53, 34, 17, 34, 5, 109, 5, 5, 5, 17, 37, 34, 41, 34, 53

Offset: 1

Zhi-Wei Sun, Aug 03 2015

nonn

Conjecture: (i) a(n) > 0 for all n > 25. In other words, for any prime p > 100, we have a^2 + b^2 = c^2 for some a,b,c in the set R(p) = {0

(ii) For any prime p > 50, we have a^2 + b^2 = c^2 for some a,b,c in the set N(p) = {0

(iii) For any prime p > 32, we have a^2 + b^2 = c^2 for some a,b in the set R(p) and c in the set N(p).

(iv) For any prime p > 72, we have a^2 + b^2 = c^2 for some a,b in the set N(p) and c in the set R(p).

I have verified the conjecture for primes p < 1.5*10^7.

			a(10) = 25 since 7^2 + 24^2 = 25^2, and 7, 24, 25 are all quadratic residues modulo prime(10) = 29.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A000290, A257364.

SQ[n_]:=IntegerQ[Sqrt[n]]
Do[Do[If[JacobiSymbol[k,Prime[n]]<1,Goto[bb]];Do[If[JacobiSymbol[j,Prime[n]]<1,Goto[cc]];
If[SQ[k^2-j^2]&&JacobiSymbol[Sqrt[k^2-j^2],Prime[n]]==1,Print[n," ",k];Goto[aa]];Label[cc];Continue,{j,1,k-1}];Label[bb];Continue,{k,1,Prime[n]-1}];
Print[n," ",0];Label[aa];Continue,{n,1,50}]

A260140 Least prime p such that pi(pn) = pi(qn)^2 for some prime q, where pi(x) denotes the number of primes not exceeding x.

Original entry on oeis.org

2, 5, 19, 3187, 11, 2251, 12149, 19, 239, 23761, 61, 157, 8419, 10973, 1117, 9601, 58741, 37, 53359, 14533, 1063, 934811, 78487, 27647, 1249, 720221, 1616077, 30091, 5501, 131627, 2003, 67, 677, 1313843, 45413, 273943, 127241, 19661, 188317, 811, 33863, 17789, 109073, 602269, 125201, 6424897, 441647, 2512897, 2909, 836471

Offset: 1

Zhi-Wei Sun, Jul 17 2015

nonn

Conjecture: a(n) exists for any n > 0. In general, for any integers a,b,c,n with a > 0 and n > 0, there are two elements x and y of the set {pi(p*n): p is prime} with a*x^2+b*x+c = y.

A supplement to the conjecture: For any integers b,c,n with b > 0 and n > 0, we have b*x+c = y for some elements x and y of the set {pi(p*n): p is prime}. - Zhi-Wei Sun, Aug 02 2015

			a(1) = 2 since pi(2*1) = 1^2 = pi(2*1)^2 with 2 prime.
a(4) = 3187 since pi(3187*4) = 1521 = 39^2 = pi(43*4)^2 with 43 and 3187 both prime.
a(72) = 25135867 since pi(25135867*72) = 89321401 = 9451^2 = pi(1367*72)^2 with 1367 and 25135867 both prime.
a(84) = 106788581 since pi(106788581*84) = 410224516 = 20254^2 = prime(2713*84)^2 with 2713 and 106788581 both prime.

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..150
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A000290, A000720, A257364, A257928, A260120.

SQ[n_]:=IntegerQ[Sqrt[n]]
f[n_]:=PrimePi[n]
Do[k=0;Label[bb];k=k+1;If[SQ[f[Prime[k]*n]]==False,Goto[bb]];Do[If[Sqrt[f[Prime[k]*n]]==f[Prime[j]*n],Goto[aa]];If[Sqrt[f[Prime[k]*n]]

A260232 Least prime p such that pi(pn) = npi(q*n) for some prime q.

Original entry on oeis.org

2, 5, 13, 67, 23, 19, 433, 443, 107, 41, 61, 251, 239, 1987, 541, 491, 1093, 499, 421, 179, 2137, 1297, 1097, 101, 103, 2411, 1283, 1847, 379, 4993, 8329, 5563, 4297, 5639, 9587, 1867, 5113, 6691, 3691, 1193, 4663, 2971, 27733, 7121, 593, 2273, 607, 6047, 4217, 2609

Offset: 1

Zhi-Wei Sun, Jul 20 2015

nonn

Conjecture: For any positive integer n, each rational number r > 0 can be written as pi(p*n)/pi(q*n) with p and q both prime.
For example, 4/7 = 416/728 = pi(479*6)/pi(919*6) with 479 and 919 both prime.
The conjecture holds trivially for n = 1 since pi(prime(m)*1) = m for all m = 1,2,3,.... Also, the conjecture implies that a(n) exists for any n > 0.

			a(4) = 67 since pi(67*4) = 56 = 4*14 = 4*pi(11*4) with 11 and 67 both prime.

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..300
Zhi-Wei Sun, Checking the conjecture for n = 1..10 and r = a/b with a,b = 1..100
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A000720, A257364, A257928, A260140, A260252.

f[n_]:=PrimePi[n]; Do[k=0;Label[bb];k=k+1;If[Mod[f[Prime[k]*n],n]>0,Goto[bb]];Do[If[f[Prime[k]n]==n*f[Prime[j]*n],Goto[aa]];If[f[Prime[k]n]

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A257928 Least prime p such that pi(p*n) = pi(q*n)*pi(r*n) for some primes q and r with p > q > r, where pi(x) denotes the number of primes not exceeding x.

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A260911 Least positive integer k < prime(n) such that there are 0 < i < j < k for which i^2 + j^2 = k^2 and i,j,k are all quadratic residues modulo prime(n), or 0 if no such k exists.

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A260140 Least prime p such that pi(p*n) = pi(q*n)^2 for some prime q, where pi(x) denotes the number of primes not exceeding x.

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A260232 Least prime p such that pi(p*n) = n*pi(q*n) for some prime q.

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A257928 Least prime p such that pi(pn) = pi(qn)pi(rn) for some primes q and r with p > q > r, where pi(x) denotes the number of primes not exceeding x.

A260140 Least prime p such that pi(pn) = pi(qn)^2 for some prime q, where pi(x) denotes the number of primes not exceeding x.

A260232 Least prime p such that pi(pn) = npi(q*n) for some prime q.