cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A259789 Least integer k > 1 such that pi(k)*pi(k*n) is a square, where pi(.) is the prime-counting function given by A000720.

Original entry on oeis.org

2, 27, 8, 2, 2, 9, 3, 5, 96, 10, 9, 2, 2, 2, 28, 4, 9, 11, 8, 195, 3, 3, 723, 28, 573, 225, 2, 2, 2, 35, 46, 132, 4, 4, 65, 14, 58, 11, 8, 967, 311, 10, 98, 3, 3, 21, 94, 20, 2, 2, 28, 23, 30, 16, 29, 3419, 134, 4, 251, 7
Offset: 1

Views

Author

Zhi-Wei Sun, Jul 05 2015

Keywords

Comments

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 pi(m)*pi(n) a positive square.
For example, 25/32 = 13102500/16771200 with pi(13102500)*pi(16771200) = 855432*1077512 = 921738245184 = 960072^2, and 49/58 = 1076068567/1273713814 with pi(1076068567)*pi(1273713814) = 54511776*63975626 = 3487424993971776 = 59054424^2.

Examples

			a(1) = 2 since pi(2)*pi(2*1) = 1^2.
a(2) = 27 since pi(27)*pi(27*2) = 9*16 = 12^2.
a(8) = 5 since pi(5)*pi(5*8) = 3*12 = 6^2.
a(9) = 96 since pi(96)*pi(96*9) = 24*150 = 60^2.
a(675) = 1465650 since pi(1465650)*pi(1465650*675) = 111747*50331648 = 5624410669056 = 2371584^2.
a(946) = 1922745 since pi(1922745)*pi(1922745*946) = 143599*89749375 = 12887920500625 = 3589975^2.
		

References

  • 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.

Crossrefs

Programs

  • Mathematica
    SQ[n_]:=IntegerQ[Sqrt[n]]
    Do[k=1; Label[bb]; k=k+1; If[SQ[PrimePi[k]*PrimePi[k*n]], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k]; Continue,{n,1,60}]
    liks[n_]:=Module[{k=2},While[!IntegerQ[Sqrt[PrimePi[k]PrimePi[k*n]]],k++];k]; Array[liks,60] (* Harvey P. Dale, Jul 12 2024 *)
  • PARI
    main(size) = {v=vector(size);for(t=1,size,v[t]=1;until(issquare(primepi(v[t])*primepi(t*v[t])),v[t]++));return(v);} \\ Anders Hellström, Jul 05 2015

A259731 Least positive integer k such that prime(k*n)-1 is a square, or 0 if no such k exists.

Original entry on oeis.org

1, 6, 1, 3, 9, 2, 1, 181, 5, 459, 5, 1, 2, 18, 3, 421, 35, 14, 183, 3274, 12, 143, 501, 422, 1407, 1, 170, 9, 55, 153, 2044, 426, 274, 74, 17, 7, 68, 452, 1084, 1637, 3, 6, 43, 1141, 1, 8218, 1860, 211, 42, 1582, 53, 813, 2, 85, 1, 5714, 61, 1379, 296, 1457, 57, 1022, 4, 213, 1331, 137, 525, 37, 167, 1130
Offset: 1

Views

Author

Zhi-Wei Sun, Jul 04 2015

Keywords

Comments

Conjecture: a(n) > 0 for all n > 0.
This is stronger than the well-known conjecture that there are infinitely many primes of the form x^2+1 with x an integer.

Examples

			a(1) = 1 since prime(1*1)-1 = 2-1 = 1^2.
a(2) = 6 since prime(6*2)-1 = 37-1 = 6^2.
		

References

  • 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.

Crossrefs

Programs

  • Mathematica
    SQ[n_]:=IntegerQ[Sqrt[n]]
    Do[k=0;Label[bb];k=k+1;If[SQ[Prime[k*n]-1],Goto[aa],Goto[bb]];Label[aa];Print[n," ",k];Continue,{n,1,70}]
    lpi[n_]:=Module[{k=1},While[!IntegerQ[Sqrt[Prime[k*n]-1]],k++];k]; Array[ lpi,70] (* Harvey P. Dale, Apr 18 2019 *)

A257856 Least positive integer k such that prime(k*n) - prime(k) is a square.

Original entry on oeis.org

1, 1, 5, 2, 1, 291, 4, 31, 4, 131, 66, 8, 113, 4, 1770, 19, 122, 27, 509, 61, 138, 1484, 1, 508, 118, 1033, 48, 314, 78, 1522, 4, 8, 169, 341, 650, 37, 3456, 1172, 221, 21, 119, 105, 34, 670, 196, 19, 30, 4, 1, 88, 496, 30, 1460, 90, 12, 1270, 812, 2096, 311, 131, 95, 241, 198, 34, 19, 63, 8, 75, 2, 10413
Offset: 1

Views

Author

Zhi-Wei Sun, Jul 12 2015

Keywords

Comments

Conjecture: a(n) exists for any n > 0. In general, every rational number r > 1 can be written as m/n with m > n > 0 such that prime(m) - prime(n) is a square.
This conjecture is a supplement to the conjecture in A259712.

Examples

			a(3) = 5 since prime(5*3) - prime(5) = 47 - 11 = 6^2.
a(70) = 10413 since prime(10413*70) - prime(10413) = 11039173 - 109537 = 3306^2.
a(1133) = 697092 since prime(697092*1133) - prime(697092) = 17813555143 - 10523959 = 133428^2.
		

References

  • 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.

Crossrefs

Programs

  • Mathematica
    SQ[n_]:=IntegerQ[Sqrt[n]]
    Do[k=0;Label[bb];k=k+1;If[SQ[Prime[n*k]-Prime[k]],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,70}]
    lpi[n_]:=Module[{k=1,sq},sq=Prime[k*n]-Prime[k];While[!IntegerQ[ Sqrt[ sq]], k++;sq=Prime[k*n]-Prime[k]];k]; Array[lpi, 70] (* Harvey P. Dale, Oct 15 2015 *)

A257663 Least positive integer k such that prime(k*n) - prime(k+n) is a square.

Original entry on oeis.org

2, 24, 57, 32, 388, 37, 15, 28, 97, 67, 112, 137, 654, 8, 37, 33, 1092, 1479, 3390, 15, 77, 63, 3, 676, 36, 183, 9, 2, 252, 341, 5, 17, 3, 260, 2, 7, 193, 166, 7, 3, 1102, 7, 297, 122, 2, 807, 387, 493, 41, 1029, 189, 746, 79, 28850, 467, 4, 93, 559, 2026
Offset: 2

Views

Author

Zhi-Wei Sun, Jul 12 2015

Keywords

Comments

Conjecture: a(n) exists for any n > 1. Also, for any n > 0 there is a number k > 0 such that prime(k*n) + prime(k+n) is a square.

Examples

			a(2) = 2 since prime(2*2) - prime(2+2) = 7 - 7 = 0^2.
a(3) = 24 since prime(24*3) - prime(24+3) = 359 - 103 = 16^2.
		

References

  • 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.

Crossrefs

Programs

  • Mathematica
    SQ[n_]:=IntegerQ[Sqrt[n]]
    Do[k=0; Label[bb]; k=k+1; If[SQ[Prime[k*n]-Prime[k+n]], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k]; Continue,{n,2,60}]
    lpi[n_]:=Module[{k=1},While[!IntegerQ[Sqrt[Prime[k*n]-Prime[k+n]]], k++]; k]; Array[lpi,60,2] (* Harvey P. Dale, Mar 12 2019 *)
Showing 1-4 of 4 results.