A257938 -id:A257938 - cp's OEIS frontend

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

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}]

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.

Original entry on oeis.org

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 */

A260078 Least positive integer k such that prime(kn)-1+(prime(hn)-1) = prime(in)-1 and prime(kn)-1-(prime(hn)-1) = prime(jn)-1 for some positive integers h,i,j.

Original entry on oeis.org

3, 3, 15, 5, 25, 29, 32, 20, 41, 87, 17, 61, 18, 100, 58, 10, 82, 82, 45, 74, 166, 20, 28, 338, 18, 35, 159, 290, 64, 29, 353, 311, 75, 41, 42, 492, 107, 155, 77, 364, 100, 330, 145, 474, 502, 332, 227, 553, 238, 92, 121, 597, 338, 339, 452, 164, 239, 832, 221, 243

Offset: 1

Zhi-Wei Sun, Jul 15 2015

nonn

Conjecture: a(n) exists for any n > 0. In general, if m and n > 0 are integers with gcd(6,m) = 1, then the set {prime(k*n)+m: k = 1,2,3,...} contains two distinct elements x and y with x+y and x-y also in the set.

			a(2) = 3 since prime(3*2)-1+(prime(2*2)-1) = 12+6 = 18 = prime(4*2)-1, and prime(3*2)-1-(prime(2*2)-1) = 12-6 = 6 = prime(2*2)-1.
a(3) = 15 since prime(15*3)-1+(prime(12*3)-1) = 196+150 = 346 = prime(23*3)-1, and prime(15*3)-1-(prime(12*3)-1) = 196 -150 = 46 = prime(5*3)-1.
a(200) = 3319 since prime(3319*200)-1+(prime(2821*200)-1) = 9987120+8389110 = 18376230 = prime(5869*200)-1, and prime(3319*200)-1-(prime(2821*200)-1) = 9987120-8389110 = 1598010 = prime(605*200)-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, A257926, A257938.

f[n_]:=Prime[n]-1
PQ[n_,p_]:=PrimeQ[p]&&Mod[PrimePi[p],n]==0
Do[k=0;Label[bb];k=k+1;Do[If[PQ[n,f[k*n]+f[j*n]+1]&&PQ[n,f[k*n]-f[j*n]+1],Goto[aa]],{j,1,k-1}];Goto[bb];
Label[aa];Print[n," ",k];Continue,{n,1,60}]

A260080 Least positive integer k such that prime(kn)^2 - 2 = prime(in)prime(jn) for some integers 0 < i < j.

Original entry on oeis.org

5, 18, 18, 9, 115, 208, 69, 373, 68, 430, 8, 214, 57, 1887, 1255, 295, 880, 542, 5612, 767, 1562, 40, 853, 884, 753, 4332, 4750, 6077, 799, 1394, 639, 5442, 4785, 440, 7417, 1290, 15830, 27745, 3927, 5701, 1891, 22008, 8243, 6031, 9172, 5949, 43286, 20778, 9876, 12472

Offset: 1

Zhi-Wei Sun, Jul 15 2015

nonn

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

			a(1) = 5 since prime(5*1)^2-2 = 11^2-2 = 119 = 7*17 = prime(4*1)*prime(7*1).
a(66) = 149073 since prime(149073*66)^2-2 = 176365951^2-2 = 31104948672134399 = 3160879*9840600881 = prime(3448*66)*prime(9840600881*66).

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..105
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A062326, A253257, A257926, A257938, A260078.

Dv[n_]:=Divisors[Prime[n]^2-2]
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}]

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}]

A261385 Least positive integer k such that (prime(prime(k))-1)(prime(prime(kn))-1) = prime(p)-1 for some prime p.

Original entry on oeis.org

1, 3, 221, 15, 13, 137, 63, 103, 44, 2, 31, 3, 45, 3, 4, 104, 38, 237, 61, 19, 56, 183, 22, 11, 15, 374, 9, 5, 42, 97, 2, 47, 4, 19, 23, 399, 3, 103, 29, 10, 2, 109, 51, 1, 52, 80, 23, 64, 76, 2, 218, 3, 7, 98, 4, 145, 10, 12, 213, 87, 36, 181, 28, 169, 71, 25, 72, 71, 54, 50

Offset: 1

Zhi-Wei Sun, Aug 17 2015

nonn

Conjecture: Let d be any nonzero integer. Then each positive rational number r can be written as m/n, where m and n are positive integers with (prime(prime(m))+d)*(prime(prime(n))+d) = prime(p)+d for some prime p.

This conjecture implies that for any nonzero integer d the equation x*y = z with x,y,z in the set {prime(p)+d: p is prime} has infinitely many solutions.

			a(3) = 221 since (prime(prime(221))-1)*(prime(prime(221*3))-1) = (prime(1381)-1)*(prime(4957)-1) = 11446*48130 = 550895980 = prime(28890079)-1 with 28890079 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..1000
Zhi-Wei Sun, Checking the conjecture for d = -1 and r = a/b (a,b = 1..60)
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A257938, A261353.

f[n_]:=Prime[Prime[n]]-1
PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
Do[k=0;Label[bb];k=k+1;If[PQ[f[k]*f[k*n]+1],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,70}]

A258580 Least positive integer k such that (prime(jn)+prime(kn))/2 = prime(i*n)^2 for some integers i > 0 and 0 < j < k.

Original entry on oeis.org

3, 9, 4, 127, 98, 133, 55, 78, 65, 85, 375, 109, 251, 283, 105, 462, 681, 149, 156, 213, 525, 209, 205, 381, 757, 313, 252, 615, 61, 737, 478, 1754, 406, 1197, 131, 420, 492, 503, 127, 119, 549, 1748, 95, 442, 2740, 555, 677, 1258, 163, 816, 1649, 710, 203, 126, 628, 582, 1004, 135, 837, 1000

Offset: 1

Zhi-Wei Sun, Jul 15 2015

nonn

Conjecture: a(n) exists for any n > 0. In general, for any positive integers a, m and n, there are integers i,j,k > 0 with i > j such that (prime(i*n)+prime(j*n))/2 (or (prime(i*n)-prime(j*n))/2) is equal to a*prime(k*n)^m.

			a(1) = 3 since (prime(2*1)+prime(3*1))/2 = (3+5)/2 = 2^2 = prime(1*1)^2.
a(158) = 8405 since (prime(778*158)+prime(8405*158))/2 = (1625551+20967091)/2 = 3361^2 = prime(3*158)^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..160
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A257926, A257938, A260078, A260080, A260082.

PQ[n_,m_]:=PrimeQ[Sqrt[m]]&&Mod[PrimePi[Sqrt[m]],n]==0
Do[k=0;Label[bb];k=k+1;Do[If[PQ[n,(Prime[k*n]+Prime[j*n])/2],Goto[aa]];Continue,{j,1,k-1}];Goto[bb];
Label[aa];Print[n," ",k];Continue,{n,1,60}]

cp's OEIS Frontend

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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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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A260078 Least positive integer k such that prime(k*n)-1+(prime(h*n)-1) = prime(i*n)-1 and prime(k*n)-1-(prime(h*n)-1) = prime(j*n)-1 for some positive integers h,i,j.

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A260080 Least positive integer k such that prime(k*n)^2 - 2 = prime(i*n)*prime(j*n) for some integers 0 < i < j.

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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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A261385 Least positive integer k such that (prime(prime(k))-1)*(prime(prime(k*n))-1) = prime(p)-1 for some prime p.

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A258580 Least positive integer k such that (prime(j*n)+prime(k*n))/2 = prime(i*n)^2 for some integers i > 0 and 0 < j < k.

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

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.

A260078 Least positive integer k such that prime(kn)-1+(prime(hn)-1) = prime(in)-1 and prime(kn)-1-(prime(hn)-1) = prime(jn)-1 for some positive integers h,i,j.

A260080 Least positive integer k such that prime(kn)^2 - 2 = prime(in)prime(jn) for some integers 0 < i < j.

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

A261385 Least positive integer k such that (prime(prime(k))-1)(prime(prime(kn))-1) = prime(p)-1 for some prime p.

A258580 Least positive integer k such that (prime(jn)+prime(kn))/2 = prime(i*n)^2 for some integers i > 0 and 0 < j < k.