A259492 -id:A259492 - cp's OEIS frontend

A259487 Least positive integer m with prime(m)+2 and prime(prime(m))+2 both prime such that prime(mn)+2 and prime(prime(mn))+2 are both prime.

2, 1860, 408, 25011, 51312, 37977, 695, 4071, 10970, 3621, 17671, 12005, 1230, 19494, 542, 577, 408, 2476, 584, 542, 469, 34229, 343, 24078, 3011, 25749, 20706, 24198, 2478, 3926, 1030, 1030, 13857, 3621, 343, 13380, 2476, 4922, 2476, 296, 19176, 29175, 34737, 13, 625, 2956, 408, 572, 7, 469, 15604, 9699, 26515, 2167, 5302, 9773, 54254, 1410, 4524, 4351

Offset: 1

Zhi-Wei Sun, Jun 28 2015

nonn

Conjecture: Any positive rational number r can be written as m/n with m and n terms of A259488.
This implies that there are infinitely many primes p with p+2 and prime(p)+2 both prime.
I have verified the conjecture for all those r = a/b with a,b = 1,...,400. - Zhi-Wei Sun, Jun 29 2015

			a(1) = 2 since prime(2)+2 = 3+2 = 5 and prime(prime(2))+2 = prime(3)+2 = 7 are both prime, but prime(1)+2 = 4 is composite.
a(49) = 7 since prime(7)+2 = 17+2 = 19, prime(prime(7))+2 = prime(17)+2 = 59+2 = 61, prime(49*7)+2 = 2309+2 = 2311 and prime(prime(49*7))+2 = prime(2309)+2 = 20441+2 = 20443 are all 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..5000
Zhi-Wei Sun, Checking the conjecture for r = a/b with a,b = 1..400
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A001359, A006512, A236458, A258803, A258836, A259488, A259492.

PQ[k_]:=PrimeQ[Prime[k]+2]&&PrimeQ[Prime[Prime[k]]+2]
Do[k=0;Label[bb];k=k+1;If[PQ[k]&&PQ[n*k], Goto[aa], Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,60}]

A259540 Least positive integer k such that k and k*n are terms of A259539.

Original entry on oeis.org

60, 326940, 728700, 115020, 375258, 70920, 33150, 297990, 2340, 72870, 858, 1416210, 284130, 78978, 91368, 9438, 5547000, 767760, 1182918, 30468, 485208, 60, 7908810, 916188, 21522, 823968, 87720, 390210, 3252, 72870, 7878, 1823010, 1179990, 98010, 3462, 7878, 280590, 6870, 60, 434460

Offset: 1

Zhi-Wei Sun, Jun 30 2015

nonn

Conjecture: Any positive rational number r can be written as m/n with m and n terms of A259539.

For example, 4/5 = 11673840/14592300 with 11673840 and 14592300 terms of A259539.

			a(22) = 60 since 60 and 60*22 = 1320 are terms of A259539. In fact, 60-1 = 59, 60+1 = 61, prime(60)+2 = 283, 1320-1 = 1319, 1320+1 = 1321 and prime(1320)+2 = 10861 are all 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..2000
Zhi-Wei Sun, Checking the conjecture for 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, A014574, A258803, A258836, A259487, A259492, A259539.

PQ[k_]:=PrimeQ[Prime[k]+2]&&PrimeQ[Prime[Prime[k]+1]+2]
QQ[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]&&PrimeQ[Prime[n]+2]
Do[k=0;Label[bb];k=k+1;If[PQ[k]&&QQ[n*(Prime[k]+1)], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", Prime[k]+1];Continue,{n, 1, 40}]

A259531 Least positive integer k such that p(k)^2 + p(k*n)^2 is prime, where p(.) is the partition function given by A000041, or 0 if no such k exists.

Original entry on oeis.org

1, 1, 14, 11, 6, 31, 2, 34, 2, 76, 1, 100, 71, 38, 1, 7, 62, 1128, 1, 180, 123, 15, 174, 128, 4, 111, 110, 2, 4, 2, 2241, 21, 144, 416, 397, 31, 11, 8, 15, 5, 91, 56, 53, 23, 89, 18, 25, 341, 12, 1, 66, 454, 159, 36, 573, 26, 2, 488, 72, 416, 802, 440, 28, 30, 595, 17, 236, 947, 1289, 1287, 1000, 367, 80, 407, 1, 77, 938, 150, 36, 1

Offset: 1

Zhi-Wei Sun, Jul 02 2015

nonn

Conjecture: Any positive rational number r can be written as m/n, where m and n are positive integers with p(m)^2 + p(n)^2 prime.
For example, 4/5 = 124/155, and the number p(124)^2 + p(155)^2 = 2841940500^2 + 66493182097^2 = 4429419891190341567409 is prime.
We also guess that any positive rational number can be written as m/n, where m and n are positive integers with p(m)+p(n) (or p(m)*p(n)-1, or p(m)*p(n)+1) prime.

			a(5) = 6 since p(6)^2 + p(6*5)^2 = 11^2 + 5604^2 = 31404937 is 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..200
Zhi-Wei Sun, Checking the conjecture for 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, A000041, A258803, A258836, A259487, A259492, A259540.

Do[k=0;Label[bb];k=k+1;If[PrimeQ[PartitionsP[k]^2+PartitionsP[k*n]^2],Goto[aa],Goto[bb]];Label[aa];Print[n," ",k]; Continue,{n,1,80}]

A261339 Least positive integer k such that both k and k*n belong to the set {m>0: m+1, m^2+1 and m^2+prime(m)^2 are all prime}.

Original entry on oeis.org

1, 1, 47500, 20440, 2, 124560, 17850, 2730, 185550, 1, 518910, 429180, 10, 687480, 81030, 36, 1568340, 2, 1165750, 7410, 10, 6780, 481140, 10, 10, 5430, 240, 2730, 72660, 2080, 18700, 291720, 295080, 52860, 5430, 1, 81030, 56400, 12490, 43590, 124560, 40030, 5170, 278700, 2091850, 131320, 184110, 11206510, 12910, 1245780

Offset: 1

Zhi-Wei Sun, Aug 15 2015

nonn

Conjecture: a(n) exists for any n > 0. In general, any positive rational number r can be written as m/n with m and n in the set {k>0: k+1, k^2+1 and k^2+prime(k)^2 are all prime}.
For example, 5/8 = 3567600/5708160 with 3567600+1, 3567600^2+1 = 12727769760001, 3567600^2 + prime(3567600)^2 = 3567600^2 + 60098671^2 = 3624578025726241, 5708160+1, 5708160^2+1 = 32583090585601,  and 5708160^2 + prime(5708160)^2 = 5708160^2 + 99018553^2 = 9837256928799409 all prime.
The conjecture implies that there are infinitely many primes p with (p-1)^2+1 and (p-1)^2+prime(p-1)^2 both prime.
We also guess that any positive rational number can be written as m/n, where m and n are positive integers with m^2+prime(m)^2, m^2+prime(n)^2, n^2+prime(m)^2 and n^2+prime(n)^2 all prime.

			a(3) = 47500 since 47501, 47500^2 + 1 = 2256250001, 47500^2 + prime(47500)^2 = 47500^2 + 578827^2 = 337296945929, 47500*3 + 1 = 142501, (47500*3)^2 + 1 = 20306250001, and (47500*3)^2 + prime(47500*3)^2 = 142500^2 + 1907023^2 = 3657042972529 are all 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..1000
Zhi-Wei Sun, Checking the conjecture for r = a/b with a,b = 1..60
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A005574, A236193, A259492, A259540, A260120.

PQ[n_]:=PrimeQ[n+1]&&PrimeQ[n^2+1]&&PrimeQ[n^2+Prime[n]^2]
Do[k=0;Label[bb];k=k+1;If[PQ[k]&&PQ[k*n],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,50}]

A261382 Least positive integer k such that (k-1)^2+(kn)^2, k^2+(kn-1)^2, (k+1)^2+(kn)^2 and k^2+(kn+1)^2 are all prime.

Original entry on oeis.org

2, 2510, 15, 30, 5, 510, 730, 440, 195, 6230, 2040, 2760, 20, 1010, 12570, 31340, 1625, 1650, 725, 2480, 2160, 520, 1055, 60, 5, 20, 1260, 25800, 6185, 6240, 10, 1180, 12600, 7500, 5330, 390, 325, 2880, 11655, 32670, 5850, 43110, 3230, 1470, 7680, 4950, 255, 202650, 10530, 450, 2445, 11670, 8745, 103350, 80, 6890, 135, 18930, 80, 245040

Offset: 1

Zhi-Wei Sun, Aug 17 2015

nonn

Conjecture: a(n) exists for any n > 0. In general, any positive rational number r can be written as m/n, where m and n are positive integers with (m-1)^2+n^2, m^2+(n-1)^2, (m+1)^2+n^2 and m^2+(n+1)^2 all prime.

It is easy to prove that if m and n are positive integers with (m-1)^2+n^2, m^2+(n-1)^2, (m+1)^2+n^2 and m^2+(n+1)^2 all prime, then either m = n = 2 or m == n == 0 (mod 5).

			a(2) = 2510 since (2510-1)^2+(2510*2)^2 = 31495481, 2510^2+(2510*2-1)^2 = 31490461, (2510+1)^2+(2510*2)^2 = 31505521 and 2510^2+(2510*2+1)^2 = 31510541 are all 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..1000
Zhi-Wei Sun, Checking the conjecture for r = a/b with a,b = 1..60
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A000290, A236193, A259492, A261339.

PQ[p_]:=PrimeQ[p]
q[m_,n_]:=PQ[(m-1)^2+n^2]&&PQ[m^2+(n-1)^2]&&PQ[(m+1)^2+n^2]&&PQ[m^2+(n+1)^2]
Do[k=0;Label[bb];k=k+1;If[q[k,k*n],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,60}]

is_ok(k,n)=isprime((k-1)^2+(k*n)^2)&&isprime(k^2+(k*n-1)^2)&&isprime((k+1)^2+(k*n)^2)&&isprime(k^2+(k*n+1)^2)
first(m)=my(v=vector(m),k=1);for(i=1,m,while(!is_ok(k,i),k++);v[i]=k;k++;);v; \\ Anders Hellström, Aug 17 2015

A261387 Number of ways to write n = k + m with 0 < k < m < n such that prime(k) is a primitive root modulo prime(m) and also prime(m) is a primitive root modulo prime(k).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 0, 2, 1, 3, 3, 1, 1, 2, 1, 2, 7, 4, 2, 1, 1, 1, 4, 3, 4, 2, 4, 3, 3, 4, 7, 3, 3, 5, 5, 5, 5, 4, 3, 6, 7, 5, 5, 5, 3, 7, 7, 5, 2, 7, 6, 4, 5, 5, 7, 10, 9, 8, 8, 4, 7, 5, 11, 14, 7, 12, 11, 9, 6

Offset: 1

Zhi-Wei Sun, Aug 27 2015

nonn

Conjecture: (i) a(n) > 0 except for n = 1, 2, 8.

(ii) Any positive rational number r not equal to 1 can be written as m/n, where m and n are positive integers such that prime(m) is a primitive root modulo prime(n) and also prime(n) is a primitive root modulo prime(m).

			a(7) = 2 since 7 = 1+6 = 3+4, prime(1) = 2 is a primitive root modulo prime(6) = 13 and 13 is a primitive root modulo 2, also prime(3) = 5 is a primitive root modulo prime(4) = 7 and 7 is a primitive root modulo 5.
a(22) = 1 since 22 = 4+18, prime(4)= 7 is a primitive root modulo prime(18) = 61 and 61 is a primitive root modulo 7.

Zhi-Wei Sun, Table of n, a(n) for n = 1..3000
Zhi-Wei Sun, Checking part (ii) of the conjecture for r = a/b with 1 <= a < b <= 100
Zhi-Wei Sun, New observations on primitive roots modulo primes, arXiv:1405.0290 [math.NT], 2014.

Cf. A000040, A242748, A259492.

f[n_]:=Prime[n]
Dv[n_]:=Divisors[n]
LL[n_]:=Length[Dv[n]]
Do[r=0;Do[Do[If[Mod[f[k]^(Part[Dv[f[n-k]-1],i])-1,f[n-k]]==0,Goto[bb]],{i,1,LL[f[n-k]-1]-1}];Do[If[Mod[f[n-k]^(Part[Dv[f[k]-1],i])-1,f[k]]==0,Goto[bb]],{i,1,LL[f[k]-1]-1}];
r=r+1;Label[bb];Continue,{k,1,(n-1)/2}];Print[n," ",r];Continue,{n,1,70}]

cp's OEIS Frontend

A259487 Least positive integer m with prime(m)+2 and prime(prime(m))+2 both prime such that prime(m*n)+2 and prime(prime(m*n))+2 are both prime.

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A259540 Least positive integer k such that k and k*n are terms of A259539.

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A259531 Least positive integer k such that p(k)^2 + p(k*n)^2 is prime, where p(.) is the partition function given by A000041, or 0 if no such k exists.

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A261339 Least positive integer k such that both k and k*n belong to the set {m>0: m+1, m^2+1 and m^2+prime(m)^2 are all prime}.

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A261382 Least positive integer k such that (k-1)^2+(k*n)^2, k^2+(k*n-1)^2, (k+1)^2+(k*n)^2 and k^2+(k*n+1)^2 are all prime.

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A261387 Number of ways to write n = k + m with 0 < k < m < n such that prime(k) is a primitive root modulo prime(m) and also prime(m) is a primitive root modulo prime(k).

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A259487 Least positive integer m with prime(m)+2 and prime(prime(m))+2 both prime such that prime(mn)+2 and prime(prime(mn))+2 are both prime.

A261382 Least positive integer k such that (k-1)^2+(kn)^2, k^2+(kn-1)^2, (k+1)^2+(kn)^2 and k^2+(kn+1)^2 are all prime.