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 1-6 of 6 results.

A262746 Number of ordered ways to write n as x^2 + y^2 + pi(z^2) with 0 <= x <= y and z > 0 such that 2*x*y + 3 is prime, where pi(m) denotes the number of primes not exceeding m.

Original entry on oeis.org

1, 2, 1, 3, 2, 3, 2, 3, 3, 3, 4, 2, 3, 2, 3, 3, 3, 3, 4, 5, 1, 4, 4, 3, 3, 6, 5, 2, 4, 4, 6, 3, 2, 5, 6, 3, 1, 6, 4, 4, 4, 4, 4, 4, 4, 2, 6, 4, 3, 7, 5, 5, 4, 6, 5, 7, 2, 3, 8, 3, 5, 3, 4, 6, 7, 5, 4, 7, 4, 6, 7, 3, 4, 8, 7, 4, 3, 4, 4, 11, 3, 4, 9, 4, 4, 6, 7, 2, 9, 6, 3, 6, 4, 6, 7, 3, 5, 8, 5, 5
Offset: 1

Views

Author

Zhi-Wei Sun, Sep 29 2015

Keywords

Comments

Conjectures:
(i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 3, 21, 37, 117, 184. Also, any integer n > 8 can be written as x^2 + y^2 + pi(z^2), where x, y and z are integers with x+y (or z) odd.
(ii) Each n = 8,9,... can be written as p^2 + pi(x^2) + pi(y^2), where p is prime, and x and y are positive integers.
(iii) Every n = 8,9,... can be written as pi(p^2) + pi(x^2) + pi(y^2), where p is prime, and x and y are positive integers.
Note that pi(x^2) > n if x > n > 0. We have verified that a(n) > 0 for all n = 1,...,10^6.

Examples

			a(1) = 1 since 1 = 0^2 + 1^2 + pi(1^2) with 2*0*1 + 3 = 3 prime.
a(2) = 2 since 2 = 0^2 + 0^2 + pi(2^2) = 1^2 + 1^2 + pi(1^2) with 2*0*0 + 3 = 3 and 2*1*1 + 3 = 5 both prime.
a(3) = 1 since 3 = 0^2 + 1^2 + pi(2^2) with 2*0*1 + 3 = 3 prime.
a(21) = 1 since 21 = 1^2 + 4^2 + pi(3^2) with 2*1*4 + 3 = 11 prime.
a(37) = 1 since 37 = 1^2 + 5^2 + pi(6^2) with 2*1*5 + 3 = 13 prime.
a(117) = 1 since 117 = 0^2 + 5^2 + pi(22^2) with 2*0*5 + 3 = 3 prime.
a(184) = 1 since 184 = 0^2 + 13^2 + pi(7^2) with 2*0*13 + 3 = 3 prime.

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.

Links

Crossrefs

Cf. A000040, A000290, A000720, A001481, A038107, A262408, A262447, A262698, A262731.

Programs

  • Mathematica
    pi[n_]:=PrimePi[n^2]
SQ[n_]:=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[pi[z]>n,Goto[aa]];Do[If[SQ[n-pi[z]-y^2]&&PrimeQ[2y*Sqrt[n-pi[z]-y^2]+3],r=r+1],{y,0,Sqrt[(n-pi[z])/2]}];Continue,{z,1,n}];Label[aa];Print[n," ",r];Continue,{n,1,100}]

A262707 Positive integers m such that pi(k^2)*pi(m^2) is a square for some 1 < k < m, where pi(x) denotes the number of primes not exceeding x.

Original entry on oeis.org

5, 8, 10, 14, 16, 19, 23, 31, 35, 39, 45, 63, 65, 66, 68, 71, 74, 82, 87, 92, 94, 115, 130, 145, 151, 162, 172, 204, 250, 279, 292, 304, 334, 391, 413, 415, 418, 449, 451, 454, 461, 499, 514, 524, 552, 557, 626, 664, 676, 683, 691, 706, 708, 724, 763, 766, 846, 848, 858, 866
Offset: 1

Views

Author

Zhi-Wei Sun, Sep 27 2015

Keywords

Comments

Conjecture: The sequence has infinitely many terms.
See also A262700 for a related conjecture.

Examples

			a(2) = 8 since pi(8^2)*pi(2^2) = 18*2 = 6^2.
a(3) = 10 since pi(10^2)*pi(3^2) = 25*4 = 10^2.

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.

Links

Crossrefs

Cf. A000290, A000720, A262408, A262443, A262447, A262462, A262698, A262700.

Programs

  • Mathematica
    f[n_]:=PrimePi[n^2]
SQ[n_]:=IntegerQ[Sqrt[n]]
n=0;Do[Do[If[SQ[f[x]*f[y]],n=n+1;Print[n," ",y];Goto[aa]],{x,2,y-1}];Label[aa];Continue,{y,1,870}]

A262730 Primes p such that p^2 = pi(x^3) + pi(y^3) for some positive integers x and y, where pi(m) denotes the number of primes not exceeding m.

Original entry on oeis.org

2, 3, 23, 83, 199, 331, 487, 1069, 1289, 1697, 2467, 3463, 3617, 3733, 5153, 5449, 6221, 9203, 9811, 9967, 12473, 13883, 14723, 15791, 16001, 18919, 33589, 33827, 46093, 58321, 59051, 59921, 60289, 71249, 84349, 85133, 88211, 124309, 126047, 126359, 127541, 145679, 146807, 153247, 165233
Offset: 1

Views

Author

Zhi-Wei Sun, Sep 28 2015

Keywords

Comments

Conjecture: (i) The sequence has infinitely many terms.
(ii) There are infinitely many primes p such that p^2 = pi(x^3+y^3) for some positive integers x and y.
See also A262731 for a related conjecture.

Examples

			a(1) = 2 since pi(1^3)+pi(2^3) = 0+4 = 2^2 with 2 prime.
a(3) = 23 since pi(9^3)+pi(14^3) = pi(729)+pi(2744) = 129+400 = 529 = 23^2 with 23 prime.
a(20) = 9967 since pi(841^3)+pi(1109^3) = pi(594823321)+pi(1363938029) = 31068537+68272552 = 99341089 = 9967^2 with 9967 prime.
a(38) = 124309 since pi(5773^3)+pi(5779^3) = pi(192399824917)+pi(193000344139) = 7714808769+7737918712 = 15452727481 = 124309^2 with 124309 prime.
a(45) = 165233 since pi(6924^3)+pi(7148^3) = pi(331948857024)+pi(365219225792) = 13025048890+14276895399 = 27301944289 = 165233^2 with 165233 prime.

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.

Links

Crossrefs

Cf. A000040, A000290, A000578, A000720, A262409, A262462, A262536, A262698, A262722, A262731.

Programs

  • Mathematica
    f[n_]:=PrimePi[n^3]
T[1]:={0}
T[n_]:=Union[T[n-1],{f[n]}]
n=0;Do[Do[If[f[x]>Prime[y]^2,Goto[aa]];If[MemberQ[T[Prime[y]],Prime[y]^2-f[x]],n=n+1;Print[n," ",Prime[y]];Goto[aa]];Continue,{x,1,Prime[y]}];
Label[aa];Continue,{y,1,15111}]

A262731 Primes p in the form pi(q^2)+pi(r^2) with q and r both prime, where pi(x) denotes the number of primes not exceeding x.

Original entry on oeis.org

11, 13, 17, 19, 41, 43, 101, 103, 223, 293, 313, 331, 359, 401, 409, 439, 491, 521, 523, 571, 613, 677, 709, 821, 883, 947, 1009, 1039, 1061, 1193, 1283, 1291, 1303, 1373, 1409, 1427, 1453, 1471, 1487, 1543, 1553, 1609, 1669, 1697, 1811, 1861, 1879, 1907, 1949, 1999, 2039, 2063, 2143, 2213, 2239, 2251, 2267, 2287, 2309, 2381
Offset: 1

Views

Author

Zhi-Wei Sun, Sep 29 2015

Keywords

Comments

Conjecture: The sequence has infinitely many terms. In general, for each n = 2,3,4,... there are infinitely many primes p in the form pi(q^n)+pi(r^n) with q and r both prime.
Compare this conjecture with the well-known result that there are infinitely many primes p in the form x^2+y^2 with x and y positive integers (such a prime p is either 2 or congruent to 1 modulo 4).

Examples

			a(1) = 11 since 11 = 2 + 9 = pi(2^2) + pi(5^2) with 11, 2 and 5 all prime.
a(60) = 2381 since 2381 = 1000 + 1381 = pi(89^2) + pi(107^2) with 2381, 89 and 107 all prime.

Links

Crossrefs

Cf. A000040, A000290, A000720, A237687, A262447, A262698, A262730.

Programs

  • Mathematica
    f[n_]:=PrimePi[Prime[n]^2]
T[1]:={f[1]}
T[n_]:=Union[T[n-1],{f[n]}]
n=0;Do[Do[If[f[x]>Prime[y],Goto[aa]];If[MemberQ[T[y],Prime[y]-f[x]],n=n+1;Print[n," ",Prime[y]];Goto[aa]];Continue,{x,1,y}];
Label[aa];Continue,{y,1,353}]

A262700 Primes p such that pi(p^2)*pi(q^2) is a square for some prime q < p, where pi(x) denotes the number of primes not exceeding x.

Original entry on oeis.org

5, 19, 31, 151, 691, 1181, 1489, 1511, 1601, 2579, 3037, 7297, 9661, 10993, 11699, 20407, 25657, 33937, 65099, 96419, 102911, 133157, 251789, 411841, 417271, 670729, 808211, 1179907, 1671277
Offset: 1

Views

Author

Zhi-Wei Sun, Sep 27 2015

Keywords

Comments

Conjecture: (i) The sequence has infinitely many terms.
(ii) The Diophantine equation pi(x^n)*pi(y^n) = z^n with n > 2 and x,y,z > 0 has no solution.

Examples

			a(1) = 5 since pi(5^2)*pi(3^2) = 9*4 = 6^2 with 5 and 3 both prime.
a(2) = 19 since pi(19^2)*pi(2^2) = 72*2 = 12^2 with 19 and 2 both prime.
a(21) = 102911 since pi(102911^2)*pi(919^2) = pi(10590673921)*pi(844561) = 480670430*67230 = 32315473008900 = 5684670^2 with 102911 and 919 both prime.
a(22) = 133157 since pi(133157^2)*pi(19^2) = pi(17730786649)*pi(361) = 786299168*72 = 56613540096 = 237936^2 with 133157 and 19 both prime.
a(23) = 251789 since pi(251789^2)*pi(10513^2) = pi(63397700521)*pi(110523169) = 2660789341*6331444 = 16846638708338404 = 129794602^2 with 251789 and 10513 both prime.

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.

Links

Crossrefs

Cf. A000040, A000290, A000720, A262408, A262443, A262447, A262462, A262698, A262707.

Programs

  • Mathematica
    f[n_]:=PrimePi[Prime[n]^2]
SQ[n_]:=IntegerQ[Sqrt[n]]
n=0;Do[Do[If[SQ[f[k]*f[m]],n=n+1;Print[n, " ", Prime[m]];Goto[aa]],{k,1,m-1}];Label[aa];Continue,{m,2,22200}]

Extensions

a(24)-a(29) from Chai Wah Wu, Aug 21 2019

A262722 Positive integers m such that pi(k^3+m^3) is a cube for some k = 1..m, where pi(x) denotes the number of primes not exceeding x.

Original entry on oeis.org

1, 41, 56, 74, 103, 157, 384, 491, 537, 868, 1490, 1710, 4322, 4523, 4877, 4942, 5147, 5407, 7564, 17576, 67722, 131455, 220641, 438895, 443475, 553878, 571473, 625611
Offset: 1

Views

Author

Zhi-Wei Sun, Sep 28 2015

Keywords

Comments

Conjecture: (i) There are infinitely many distinct primes p,q,r such that pi(p^2+q^2) = r^2.
(ii) The Diophantine equation pi(x^3+y^3) = z^3 with 0 < x <= y and z > 0 only has the following 13 solutions: (x,y,z) = (1,1,1), (5,41,19), (47,56,29), (28,74,33), (2,103,44), (3,103,44), (6,157,65), (235,384,160), (266,491,198), (91,537,206), (359,868,331), (783,1490,565), (1192,1710,677).
(iii) The Diophantine equation pi(x^n+y^n) = z^n with n > 3 and x,y,z > 0 has no solution.
Part (ii) of the conjecture implies that the current sequence only has 12 terms as shown here.
Conjecture (ii) is false as there are more terms beyond 1710. It is likely the sequence has an infinite number of terms. (x,y,z) for 1710 < y <= 7564: (1429, 4322, 1514), (1974, 4523, 1604), (3361, 4877, 1840), (3992, 4942, 1949), (3253, 5147, 1902), (971, 5407, 1859), (935, 7564, 2563). - Chai Wah Wu, Apr 12 2021
Solutions (x,y,z) for 7564 < y <= 67722: (3484, 17576, 5783), (25184, 67722, 21604). - Chai Wah Wu, Apr 17 2021
Solutions (x,y,z) for 67722 < y <= 625611: (61021, 131455, 41715), (93577, 220641, 68507), (394510, 438895, 155930), (3086, 443475, 131933), (338485, 553878, 175133), (239982, 571473, 172855), (610794, 625611, 228409). - Chai Wah Wu, Apr 26 2021

Examples

			a(2) = 41 since pi(5^3+41^3) = pi(125+68921) = pi(69046) = 6859 = 19^3.

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.

Links

Crossrefs

Cf. A000040, A000290, A000578, A000720, A262408, A262409, A262443, A262462, A262536, A262698, A262700, A262707.

Programs

  • Mathematica
    f[x_,y_]:=PrimePi[x^3+y^3]
CQ[n_]:=IntegerQ[n^(1/3)]
n=0;Do[Do[If[CQ[f[x,y]],n=n+1;Print[n," ",y];Goto[aa]],{x,1,y}];Label[aa];Continue,{y,1,1800}]

Extensions

a(13)-a(19) from Chai Wah Wu, Apr 12 2021
a(20)-a(21) from Chai Wah Wu, Apr 17 2021
a(22)-a(28) from Chai Wah Wu, Apr 26 2021
Showing 1-6 of 6 results.

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