A257928 -id:A257928 - cp's OEIS frontend

A257938 Least positive integer k such that prime(kn) - 1 = (prime(in)-1)(prime(jn)-1) for some integers 0 < i < j < k.

6, 3, 8, 71, 12, 14, 105, 221, 24, 499, 261, 612, 1341, 175, 917, 549, 1351, 2303, 2273, 4767, 364, 1395, 1390, 1431, 6481, 2479, 918, 2412, 17783, 3178, 2994, 7538, 3409, 1361, 9645, 3454, 9197, 7074, 10418, 6059, 36235, 182, 1910, 4648, 1130, 695, 3973, 10839, 8647, 7942

Offset: 1

Zhi-Wei Sun, Jul 13 2015

nonn

Conjecture: a(n) exists for any n > 0. In general, for any nonzero integer m and positive integer n, the set {prime(k*n)+m: k = 1,2,3,...} always contains three distinct elements x, y and z with x*y = z.

			a(1) = 6 since prime(6*1)-1 = 12 = 2*6 = (prime (2*1)-1)*(prime(4*1)-1).
a(4) = 71 since prime(71*4)-1 = 1860 = 6*310 = (prime(1*4)-1)*(prime(16*4)-1).
a(41) = 36235 since prime(36235*41)-1 = 23634312 = 676*34962 = (prime(3*41)-1)*(prime(91*41)-1).
a(69) = 64999 since prime(64999*69)-1 = 76643820 = 4590*16698 = (prime(9*69)-1)*(prime(28*69)-1).
a(77) = 137789 since prime(137789*77)-1 = 191037600 = 2028*94200 = (prime(4*77)-1)*(prime(118*77)-1).
a(99) = 167708 since prime(167708*99)-1 = 306849088 = 10528*29146 = (prime(13*99)-1)*(prime(32*99)-1).
a(189) = 951492 since prime(951492*189)-1 = 3776304996 = 4126*915246 = (prime(3*189)-1)*(prime(383*189)-1).

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, A257928.

Dv[n_]:=Divisors[Prime[n]-1]
L[n_]:=Length[Dv[n]]
P[k_,n_,i_]:=PrimeQ[Part[Dv[k*n],i]+1]&&Mod[PrimePi[Part[Dv[k*n],i]+1],n]==0
Do[k=0;Label[bb];k=k+1; Do[If[P[k,n,i]&&P[k,n,L[k*n]-i+1],Goto[aa]],{i,2,L[k*n]/2}];Goto[bb];Label[aa];Print[n, " ", k];Continue,{n,1,50}]

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

A257926 Least positive integer k such that prime(kn)+2 = prime(in)prime(jn) for some 0 < i < j.

Original entry on oeis.org

6, 4, 10, 8, 451, 426, 622, 175, 1424, 500, 33, 703, 1761, 4428, 1563, 959, 8147, 7055, 5948, 250, 7517, 12706, 8405, 2948, 2610, 1949, 10424, 2214, 6722, 1963, 3335, 16382, 15687, 17591, 15073, 7818, 32202, 31169, 2248, 14899, 69955, 7580, 2393, 39295, 42352, 5884, 9367, 3630, 14090, 1305

Offset: 1

Zhi-Wei Sun, Jul 14 2015

nonn

Conjecture: a(n) exists for any n > 0.

This is much stronger than Chen's famous result that there are infinitely many Chen primes.

			a(1) = 6 since prime(6*1)+2 = 15 = 3*5 = prime(2*1)*prime(3*1).
a(3) = 10 since prime(10*3)+2 = 115 = 5*23 = prime(1*3)*prime(3*3).
a(149) = 1476387 since prime(1476387*149)+2 = 4666119529 = 8311*561439 = prime(7*149)*prime(310*149).

Jing-run Chen, On the representation of a large even integer as the sum of a prime and a product of at most two primes, Sci. Sinica 16(1973), 157-176.
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, A109611, A257928, A257938.

Dv[n_]:=Divisors[Prime[n]+2]
L[n_]:=Length[Dv[n]]
P[k_,n_]:=L[k*n]==4&&PrimeQ[Part[Dv[k*n],2]]&&Mod[PrimePi[Part[Dv[k*n],2]],n]==0&&PrimeQ[Part[Dv[k*n],3]]&&Mod[PrimePi[Part[Dv[k*n],3]],n]==0
Do[k=0;Label[bb];k=k+1;If[P[k,n],Goto[aa]];Goto[bb];Label[aa];Print[n," ", k];Continue,{n,1,50}]

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.

Original entry on oeis.org

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

A260082 Least positive integer k such that (prime(kn)-1)^2 = (prime(in)-1)(prime(jn)-1) for some integers 0 < i < j.

Original entry on oeis.org

2, 2, 2, 21, 9, 10, 12, 14, 47, 32, 32, 171, 177, 175, 64, 187, 330, 206, 77, 467, 4, 126, 127, 355, 279, 982, 249, 1930, 105, 109, 659, 801, 269, 777, 703, 125, 819, 1347, 904, 1153, 549, 2344, 757, 1301, 1793, 303, 105, 3168, 2645, 3055, 110, 1619, 1580, 2423, 220, 965, 1397, 84, 988, 322

Offset: 1

Zhi-Wei Sun, Jul 15 2015

nonn

Conjecture: a(n) exists for any n > 0. In general, for any nonzero integer m and positive integer n there are distinct positive integers i,j,k such that (prime(i*n)+m)*(prime(j*n)+m) = (prime(k*n)+m)^2.

			a(4) = 21 since (prime(21*4)-1)^2 = 432^2 = 18*10368 = (prime(2*4)-1)*(prime(318*4)-1).
a(61) = 15160 since (prime(15160*61)-1)^2 = 14242116^2 = 47316*4286876916 = (prime(80*61)-1)*(prime(3326491*61)-1).

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, A257926, A257928, A257938, A260078, A260080.

Dv[n_]:=Divisors[(Prime[n]-1)^2]
L[n_]:=Length[Dv[n]]
P[k_,n_,i_]:=PrimeQ[Part[Dv[k*n],i]+1]&&Mod[PrimePi[Part[Dv[k*n],i]+1],n]==0
Do[k=0;Label[bb];k=k+1; Do[If[P[k,n,i]&&P[k,n,L[k*n]-i+1],Goto[aa]],{i,1,L[k*n]/2}];Goto[bb];Label[aa];Print[n, " ", k];Continue,{n,1,60}]

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]

cp's OEIS Frontend

A257938 Least positive integer k such that prime(k*n) - 1 = (prime(i*n)-1)*(prime(j*n)-1) for some integers 0 < i < j < k.

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A257926 Least positive integer k such that prime(k*n)+2 = prime(i*n)*prime(j*n) for some 0 < i < j.

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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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A260082 Least positive integer k such that (prime(k*n)-1)^2 = (prime(i*n)-1)*(prime(j*n)-1) for some integers 0 < i < j.

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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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A257938 Least positive integer k such that prime(kn) - 1 = (prime(in)-1)(prime(jn)-1) for some integers 0 < i < j < k.

A257926 Least positive integer k such that prime(kn)+2 = prime(in)prime(jn) for some 0 < i < j.

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.

A260082 Least positive integer k such that (prime(kn)-1)^2 = (prime(in)-1)(prime(jn)-1) for some integers 0 < i < j.

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.