A259531 -id:A259531 - cp's OEIS frontend

A259712 Least positive integer k such that prime(k) + prime(k*n) is a square.

1, 10, 2, 1, 1126, 60, 55, 691, 1, 24, 15, 640, 5, 41, 1, 671, 261, 3, 8, 219, 103, 1, 1843, 128, 2240, 4, 664, 12, 111, 275, 19, 576, 166, 5, 3, 13, 7462, 243, 1, 1599, 228, 6297, 128, 853, 995, 49, 164, 1, 116, 10, 40, 3971, 1741, 32, 338, 11992, 3, 39, 20, 24, 2, 465, 352, 24, 138, 241, 343, 177, 32, 3

Offset: 1

Zhi-Wei Sun, Jul 03 2015

nonn

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

I have verified this conjecture for all those r = a/b with a,b = 1,...,700.

			a(2) = 10 since prime(10) + prime(10*2) = 29 + 71 = 10^2.
a(5) = 1126 since prime(1126) + prime(1126*5) = 9059 + 55457 = = 64516 = 254^2.

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..1000
Zhi-Wei Sun, Checking the conjecture for r = a/b with a,b = 1..200, 2015.
Zhi-Wei Sun, Checking the conjecture for r = a/b with 200 < a,b <= 700 and gcd(a,b) = 1, 2021.
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.
Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016, Springer Proc. in Math. & Stat., Vol. 220, Springer, New York, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT], 2012-2017.)

Cf. A000040, A000290, A257856, A259531.

SQ[n_]:=IntegerQ[Sqrt[n]]
Do[k=0;Label[bb];k=k+1;If[SQ[Prime[k]+Prime[k*n]],Goto[aa],Goto[bb]];Label[aa];Print[n," ",k];Continue,{n,1,70}]
lpi[n_]:=Module[{k=1},While[!IntegerQ[Sqrt[Prime[k]+Prime[k*n]]],k++];k]; Array[ lpi,70] (* Harvey P. Dale, Sep 07 2022 *)

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 */

A259678 Primes of the form p(k)^2 + p(m)^2, where k and m are positive integers, and p(.) is the partition function given by A000041.

Original entry on oeis.org

2, 5, 13, 29, 53, 229, 509, 709, 1021, 1789, 3137, 3257, 3361, 6829, 13337, 18229, 30977, 41177, 49201, 148229, 240101, 240109, 250301, 1004053, 1575029, 2511601, 3833989, 3851989, 5934121, 6314389, 9060109, 9148309, 13823549, 20842361, 31404937, 106714213, 116703973, 151536109, 221241901, 221743789, 230563789, 254416549, 323412629, 323799793, 332232629, 336996053, 468159773, 676780229, 676783361, 682714321

Offset: 1

Zhi-Wei Sun, Jul 03 2015

nonn

The conjecture in A259531 implies that the current sequence has infinitely many terms.

			a(1) = 2 since p(1)^2 + p(1)^2 = 2 is prime.
a(2) = 5 since p(1)^2 + p(2)^2 = 1^2 + 2^2 = 5 is prime.
a(3) = 13 since p(2)^2 + p(3)^2 = 2^2 + 3^2 = 13 is prime.
a(4) = 29 since p(2)^2 + p(4)^2 = 2^2 + 5^2 = 29 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..10000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A000041, A259531.

p[n_]:=p[n]=PartitionsP[n];
tab={};Do[s=p[m]^2+p[k]^2;If[PrimeQ[s],tab=Append[tab,s]],{m,1,1600},{k,1,m}];
LL:=LL=Sort[DeleteDuplicates[tab]];Do[Print[n,LL[[n]]],{n,1,10000}]
Take[Select[Total/@Tuples[PartitionsP[Range[2000]]^2,2],PrimeQ]//Union,50] (* Harvey P. Dale, Jan 24 2025 *)

Corrected and extended by Harvey P. Dale, Jan 24 2025

A261513 Least positive integer k with p(prime(k))+p(prime(k*n)) prime, where p(.) is the partition function given by A000041.

Original entry on oeis.org

1, 46, 1, 115, 1, 9, 4, 17, 1, 3, 12, 6, 5, 3, 2, 1253, 1035, 716, 4028, 6154, 9, 3, 1219, 94, 64, 195, 1545, 9909, 365, 52, 182, 76, 277, 135, 1321, 1619, 9693, 5485, 8001, 946, 1, 36, 7154, 10354, 1, 2157, 33, 1344, 1, 39, 1698, 732, 24505, 1, 637, 14, 8, 2127, 1460

Offset: 2

Zhi-Wei Sun, Aug 22 2015

nonn

Conjecture: Any positive rational number r not equal to one can be written as m/n, where m and n are positive integers with p(prime(m)) + p(prime(n)) prime.

This implies that there are infinitely many primes of the form p(q) + p(r) with q and r both prime.

			a(2) = 1 since p(prime(1)) + p(prime(1*2)) = p(2) + p(3) = 2 + 3 = 5 is prime.
a(3) = 46 since p(prime(46)) + p(prime(46*3)) = p(199) + p(787) = 3646072432125 + 3223934948277725160271634798 = 3223934948277728806344066923 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 = 2..100
Zhi-Wei Sun, Checking the conjecture for r = a/b with 1 <= a < b <= 37
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A000041, A259531, A261515.

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

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A259712 Least positive integer k such that prime(k) + prime(k*n) is a square.

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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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A259678 Primes of the form p(k)^2 + p(m)^2, where k and m are positive integers, and p(.) is the partition function given by A000041.

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A261513 Least positive integer k with p(prime(k))+p(prime(k*n)) prime, where p(.) is the partition function given by A000041.

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