A262447 -id:A262447 - cp's OEIS frontend

A038107 Number of primes < n^2.

0, 0, 2, 4, 6, 9, 11, 15, 18, 22, 25, 30, 34, 39, 44, 48, 54, 61, 66, 72, 78, 85, 92, 99, 105, 114, 122, 129, 137, 146, 154, 162, 172, 181, 191, 200, 210, 219, 228, 240, 251, 263, 274, 283, 295, 306, 319, 329, 342, 357, 367, 378, 393, 409, 421, 434, 445, 457, 474

Offset: 0

Joe K. Crump (joecr(AT)carolina.rr.com)

nonn

Also number of primes <= n^2 since n^2 is not prime.
Also the number of primes contained within an n X n square spiral. - William A. Tedeschi, Mar 03 2008
For large n, these numbers closely approximate the sum of primes less than n. For example, n = 10^10, sum of primes < n = 2220822432581729238. The number of primes < (10^10)^2 = 10^20 = 2220819602560918840. The error is 0.0000012743... The derivation of this is in the link Sum of Primes. - Cino Hilliard, Jun 09 2008
a(n) - A000720(n) = A073882(n) - A010051(n) = A117490(n). - Reinhard Zumkeller, May 20 2010
A061265(a(n)) = 1 for n > 1. - Reinhard Zumkeller, Apr 15 2013
From Zhi-Wei Sun, Feb 17 2014: (Start)
Conjecture:
(i) The sequence a(n)^(1/n) (n = 3, 4, ...) is strictly decreasing (to the limit 1).
(ii) If n > 0 is not among 25, 35, 44, 46, 105, then the interval [a(n), a(n+1)] contains at least one prime. (End)
A classical conjecture of Legendre asserts that a(n) < a(n+1) for all n > 0.
Conjecture: All the numbers Sum_{i=j,...,k} 1/a(i) with 1 < j <= k have pairwise distinct fractional parts. - Zhi-Wei Sun, Sep 24 2015

			a(2)=2 because the only primes < 4 are 2 and 3.

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. (See Conjectures 2.14-2.16.)

T. D. Noe, Table of n, a(n) for n = 0..1000
Cino Hilliard, Sum of Primes.
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.
Wikipedia, Legendre's conjecture.

Cf. A014085 (first differences), A111208, A194189, A262408, A262443, A262447, A262462.

a038107 0 = 0
a038107 n = a000720 $ a000290 n
-- Reinhard Zumkeller, Apr 15 2013, Nov 01 2011

A038107 := proc(n) numtheory[pi]( n^2) ; end: seq(A038107(n),n=0..100) ; # R. J. Mathar, Jun 22 2009

Table[PrimePi[n^2], {n, 0, 100}] (* Ray Chandler, Oct 22 2005 *)

a(n)=primepi(n^2) \\ Charles R Greathouse IV, Apr 26 2012

[prime_pi(n^2) for n in range(0, 59)] # Zerinvary Lajos, Jun 06 2009

a(n) = A000720(A000290(n)).

a(n) ~ 1/2 * n^2/log n. - Charles R Greathouse IV, Apr 26 2012

Extended by Ray Chandler, Oct 22 2005

A262698 Positive integers m such that pi(k^3) + pi(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, 2, 4, 24, 41, 51, 88, 95, 99, 179, 183, 663, 782, 829, 1339, 2054, 2816, 7918, 8474, 13264, 16664, 27415, 39514, 48606, 51145, 187222, 200906, 261980, 353209, 375162, 396967, 400469

Offset: 1

Zhi-Wei Sun, Sep 27 2015

Conjecture: (i) There are infinitely many distinct primes p,q,r such that pi(p^2) + pi(q^2) = r^2.
(ii) The Diophantine equation pi(x^3) + pi(y^3) = z^3 with 0 < x <= y and z >= 0 only has the following 17 solutions: (x,y,z) = (1,1,0), (2,2,2), (3,4,3), (16,24,13), (3,41,19), (37,51,26), (53,88,41), (18,95,41), (45,99,44), (108,179,79), (149,183,87), (8,663,251), (243,782,297), (803,829,385), (100,1339,489), (674,2054,745), (1519,2816,1047).
(iii) The Diophantine equation pi(x^n) + pi(y^n) = z^n with n > 3 and x,y,z > 0 has no solution.
a(26) > 10^5, if it exists. Conjecture (ii) above is false since these further solutions exist: (1339, 7918, 2682), (3360, 8474, 2922), (8443, 13264, 4764), (15590, 16664, 6696), (15883, 27415, 9431), (9719, 39514, 12689), (22265, 48606, 15933), (38606, 51145, 18297). - Giovanni Resta, Jun 14 2020
Further solutions: (79522, 187222, 58554), (65281, 200906, 61833), (222863, 261980, 92917), (226465, 353209, 114585), (41559, 375162, 112168), (244967, 396967, 127399), (291034, 400469, 133443) - Chai Wah Wu, Apr 13 2021

			a(4) = 24 since pi(16^3) + pi(24^3) = pi(4096) + pi(13824) = 564 + 1633 = 2197 = 13^3.

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, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2016.

Cf. A000578, A000720, A019590, A262408, A262409, A262447, A262462, A262536.

f[n_]:=PrimePi[n^3]
CQ[n_]:=IntegerQ[n^(1/3)]
n=0;Do[Do[If[CQ[f[x]+f[y]],n=n+1;Print[n," ",y];Goto[aa]],{x,1,y}];Label[aa];Continue,{y,1,3000}]

lista(nn) = {my(c=0, v=vector(nn)); for(m=1, nn, forprime(p=(m-1)^3+1, m^3, c++); v[m]=c; if(sum(k=1, m, ispower(v[k]+v[m], 3)), print1(m, ", "))); } \\ Jinyuan Wang, Jun 13 2020

a(18)-a(25) from Giovanni Resta, Jun 14 2020
a(26)-a(29) from Chai Wah Wu, Apr 05 2021
a(30)-a(32) from Chai Wah Wu, Apr 09 2021

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 2xy + 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

Zhi-Wei Sun, Sep 29 2015

nonn

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.

			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.

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

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

Zhi-Wei Sun, Sep 27 2015

nonn

Conjecture: The sequence has infinitely many terms.

See also A262700 for a related conjecture.

			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.

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.

Chai Wah Wu, Table of n, a(n) for n = 1..1000 (n = 1..200 from Zhi-Wei Sun)
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

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

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}]

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

Zhi-Wei Sun, Sep 29 2015

nonn

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).

			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.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..3000 from Zhi-Wei Sun)

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

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

Zhi-Wei Sun, Sep 27 2015

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.

			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.

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, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

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

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}]

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

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A038107 Number of primes < n^2.

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A262698 Positive integers m such that pi(k^3) + pi(m^3) is a cube for some k = 1,...,m, where pi(x) denotes the number of primes not exceeding x.

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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.

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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.

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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.

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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.

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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 2xy + 3 is prime, where pi(m) denotes the number of primes not exceeding m.