A262408 -id:A262408 - 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

A262443 Positive integers m such that pi(m^2) = pi(j^2)*pi(k^2) for some 0 < j < k < m, where pi(x) denotes the number of primes not exceeding x.

Original entry on oeis.org

8, 11, 14, 19, 20, 36, 38, 45, 66, 87, 91, 115, 139, 143, 152, 155, 201, 220, 227, 279, 357, 383, 391, 415, 418, 452, 476, 480, 489, 496, 500, 514, 521, 524, 549, 552, 557, 588, 595, 632, 653, 676, 706, 708, 749, 753, 761, 766, 820, 846, 863, 877, 922, 1009, 1038, 1041, 1044, 1052, 1057, 1080

Offset: 1

Zhi-Wei Sun, Sep 23 2015

nonn

Conjecture: (i) The sequence has infinitely many terms. Also, there are infinitely many positive integers m such that pi(m^2) = pi(j^2)*pi(k^2) for no 0 < j <= k < m.

(ii) For any integer n > 2, the equation pi(x^n)*pi(y^n) = pi(z^n) has no solution with 0 < x <= y < z.

			 a(1) = 8 since pi(8^2) = pi(64) = 18 = 2*9 = pi(2^2)*pi(5^2) with 0 < 2 < 5 < 8.
a(4) = 19 since pi(19^2) = pi(361) = 72 = 4*18 = pi(3^2)*pi(8^2) with 0 < 3 < 8 < 19.

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..300
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000290, A000720, A038107, A262408, A262409.

f[n_]:=PrimePi[n^2]
T[n_]:=Table[f[k],{k,1,n}]
Dv[n_]:=Divisors[f[n]]
Le[n_]:=Length[Dv[n]]
n=0;Do[Do[If[MemberQ[T[m],Part[Dv[m],i]]&&MemberQ[T[m],Part[Dv[m],Le[m]-i+1]],n=n+1;Print[n," ",m];Goto[aa]],{i,2,(Le[m]-1)/2}];Label[aa];Continue,{m,1,1080}]

A262409 Positive integers m such that pi(m^3) = pi(j^3) + pi(k^3) for some 0 < j <= k < m.

Original entry on oeis.org

4, 89, 97, 101, 110, 196, 237, 372, 410, 1457, 2522, 3327, 4244, 4437, 5684, 5777, 7647, 8827, 9608, 9680, 9807, 10744, 17563, 19146, 21208, 23188, 27153, 28286, 34086, 35443, 40057, 49338, 49613, 54425, 55360, 56906, 61304, 69147, 69515, 73694, 84508, 95674

Offset: 1

Zhi-Wei Sun, Sep 22 2015

nonn

Conjecture: The Diophantine equation pi(x^3) + pi(y^3) = pi(z^3) with 0 < x <= y < z has infinitely many solutions.
The 25 terms we have found yield the following 25 solutions to the equation: (x,y,z) = (3,3,4), (54,80,89), (63,85,97), (27,100,101), (47,106,110), (80,190,196), (122,223,237), (229,335,372), (151,401,410), (263,1453,1457), (1302,2382,2522), (879,3301,3327), (2190,4011,4244), (498,4434,4437), (3792,4991,5684), (4496,4584,5777), (3113,7442,7647), (5239,8090,8827), (6904,8116,9608), (5659,8910,9680), (5323,9187,9807), (5527,10168,10744), (7395,17050,17563), (11637,17438,19146), (4486,21125,21208).
See also the conjecture in A262408 involving the n-th powers with n = 2,4,5,....
Solution triples (x,y,z) corresponding to a(n) for n = 26..42: (16440, 19774, 23188), (4775, 27091, 27153), (10708, 27687, 28286), (25272, 28248, 34086), (6302, 35360, 35443), (3941, 40040, 40057), (16336, 48639, 49338), (33631, 43365, 49613), (6206, 54390, 54425), (6741, 55317, 55360), (28160, 54247, 56906), (25339, 59637, 61304), (41473, 63300, 69147), (27684, 67825, 69515), (29690, 71841, 73694), (65989, 67172, 84508), (55781, 88294, 95674) - Chai Wah Wu, May 24 2018

			a(1) = 4 since pi(4^3) = pi(64) = 18 = 9 + 9 = pi(27) + pi(27) = pi(3^3) + pi(3^3).
a(2) = 89 since pi(89^3) = 56924 = 14479 + 42445 = pi(157464) + pi(512000) = pi(54^3) + pi(80^3).
a(22) = 10744 since pi(10744^3) = pi(1240217910784) = 46266787130 = 6805722064 + 39461065066 = pi(168837298183) + pi(1051251461632) = pi(5527^3) + pi(10168^3).
a(23) = 17563 since pi(17563^3) = pi(5417464872547) = 191548794617 = 15745791385 + 175803003232 = pi(404403154875) + pi(4956477625000) = pi(7395^3) + pi(17050^3).
a(24) = 19146 since pi(19146^3) = pi(7018336124136) = 245897610272 = 58267274193 + 187630336079 = pi(1575879851853) + pi(5302614071672) = pi(11637^3) + pi(17438^3).
a(25) = 21208 since pi(21208^3) = pi(9538918630912) = 330649999352 = 3733416265 + 326916583087 = pi(90277143256) + pi(9427361328125) = pi(4486^3) + pi(21125^3).

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.

Cf. A000720, A000578, A262403, A262408, A262443.

f[n_]:=PrimePi[n^3]
T[1]:={0}
T[n_]:=Union[T[n-1],{f[n]}]
Do[n=0;Do[If[MemberQ[T[m-1],f[m]-f[k]],n=n+1;Print[n," ",m];Goto[aa]],{k,1,m-1}];Label[aa];Continue,{m,1,21350}]

a(26)-a(42) from Chai Wah Wu, May 24 2018

A262403 Number of ways to write pi(T(n)) = pi(T(k)) + pi(T(m)) with 1 < k < m < n, where T(x) is the triangular number x*(x+1)/2, and pi(x) is the number of primes not exceeding x.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 2, 2, 1, 2, 1, 3, 4, 4, 4, 3, 3, 3, 3, 5, 4, 3, 4, 6, 4, 5, 2, 3, 6, 4, 1, 5, 8, 3, 2, 6, 1, 4, 5, 4, 2, 7, 2, 4, 5, 5, 5, 3, 4, 9, 9, 4, 5, 4, 8, 7, 6, 9, 4, 7, 5, 6, 2, 5, 9, 3, 8, 5, 6, 8, 5, 4, 3, 8, 4, 8, 7, 8, 5, 7, 8, 7, 4, 6, 2, 7, 7, 8, 7, 4, 5, 6, 4, 6, 4, 6, 4, 6, 6

Offset: 1

Zhi-Wei Sun, Sep 21 2015

nonn

Conjecture: (i) a(n) > 0 for all n > 4, and a(n) = 1 only for n = 5, 6, 7, 10, 12, 32, 38, 445, 727.
(ii) All those numbers pi(T(n)) (n = 1,2,3,...) are pairwise distinct. Moreover, if sum_{i=j,...,k}1/pi(T(i)) and sum_{r=s,...,t}1/pi(T(r)) with 1 < j <= k and j <= s <= t have the same fractional part but the ordered pairs (j,k) and (s,t) are different, then j = 2, k = 5 and s = t = 4.
Clearly, part (i) is related to addition chains, and the first assertion in part (ii) is an analog of Legendre's conjecture that pi(n^2) < pi((n+1)^2) for all n = 1,2,3,....
See also A262408 and A262409 for related conjectures involving powers.

			a(5) = 1 since pi(T(5)) = pi(15) = 6 = 2 + 4 = pi(3) + pi(10) = pi(T(2)) + pi(T(4)).
a(6) = 1 since pi(T(6)) = pi(21) = 8 = 2 + 6 = pi(3) + pi(15) = pi(T(2)) + pi(T(5)).
a(7) = 1 since pi(T(7)) = pi(28) = 9 = 3 + 6 = pi(6) + pi(15) = pi(T(3)) + pi(T(5)).
a(10) = 1 since pi(T(10)) = pi(55) = 16 = 2 + 14 = pi(3) + pi(45) = pi(T(2)) + pi(T(9)).
a(12) = 1 since pi(T(12)) = pi(78) = 21 = 3 + 18 = pi(6) + pi(66) = pi(T(3)) + pi(T(11)).
a(32) = 1 since pi(T(32)) = pi(528) = 99 = 9 + 90 = pi(28) + pi(465) = pi(T(7)) + pi(T(30)).
a(38) = 1 since pi(T(38)) = pi(741) = 131 = 32 + 99 = pi(136) + pi(528) = pi(T(16)) + pi(T(32)).
a(445) = 1 since pi(T(445)) = pi(99235) = 9526 = 2963 + 6563 = pi(27028) + pi(65703) = pi(T(232)) + pi(T(362)).
a(727) = 1 since pi(T(727)) = pi(264628) = 23197 = 10031 + 13166 = pi(105111) + pi(141778) = pi(T(458)) + pi(T(532)).

R. K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. (Cf. Section C6 on addition chains.)
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. A000217, A000720, A111208, A262408, A262409, A262439, A262446.

f[n_]:=PrimePi[n(n+1)/2]
T[m_,n_]:=Table[f[k],{k,m,n}]
Do[r=0;Do[If[MemberQ[T[k+1,n-1],f[n]-f[k]],r=r+1];Continue,{k,2,n-2}];Print[n," ",r];Continue,{n,1,100}]

A262447 Primes p such that pi(p^2) = pi(q^2) + pi(r^2) for some distinct primes q and r.

Original entry on oeis.org

13, 53, 73, 131, 199, 277, 281, 283, 313, 353, 641, 643, 647, 701, 773, 839, 887, 977, 1033, 1103, 1117, 1163, 1187, 1223, 1259, 1409, 1433, 1439, 1487, 1489, 1583, 1721, 1913, 1931, 2239, 2243, 2269, 2309, 2371, 2441, 2473, 2477, 2621, 2683, 2707, 2797, 2843, 2851, 2953, 3049, 3137, 3257, 3307, 3499, 3511, 3613, 3659, 3769, 3779, 3911

Offset: 1

Zhi-Wei Sun, Sep 23 2015

nonn

Conjecture: The sequence has infinitely many terms.

See also A262408 and A262443 for related conjectures.

			a(1) = 13 since pi(13^2) = pi(169) = 39 = 9 + 30 = pi(5^2) + pi(11^2) with 13, 5 and 11 distinct primes.

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..10000 (n = 1..500 from Zhi-Wei Sun)
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A000290, A000720, A262408, A262443.

f[n_]:=PrimePi[Prime[n]^2]
T[n_]:=Table[f[k],{k,1,n}]
n=0;Do[Do[If[2*f[k]>=f[m],Goto[aa]];If[MemberQ[T[m-1],f[m]-f[k]],n=n+1;Print[n," ",Prime[m]];Goto[aa]];Continue,{k,1,m-1}];Label[aa];Continue,{m,1,541}]

A262462 Positive integers k with pi(k^3) a square, where pi(x) denotes the number of primes not exceeding x.

Original entry on oeis.org

1, 2, 3, 14, 1122

Offset: 1

Zhi-Wei Sun, Sep 23 2015

Conjecture: (i) The Diophantine equation pi(x^2) = y^2 with x > 0 and y > 0 has infinitely many solutions.
(ii) The only solutions to the Diophantine equation pi(x^m) = y^n with {m,n} = {2,3}, x > 0 and y > 0 are as follows:
pi(89^2) = 10^3, pi(2^3) = 2^2, pi(3^3) = 3^2, pi(14^3) = 20^2 and pi(1122^3) = 8401^2.
(iii) For m > 1 and n > 1 with m + n > 5, the equation pi(x^m) = y^n with x > 0 and y > 0 has no integral solution.
The conjecture seems reasonable in view of the heuristic arguments.
Part (ii) of the conjecture implies that the only terms of the current sequence are 1, 2, 3, 14 and 1122.

			a(1) = 1 since pi(1^3) = 0^2.
a(2) = 2 since pi(2^3) = 2^2.
a(3) = 3 since pi(3^3) = 3^2.
a(4) = 14 since pi(14^3) = pi(2744) = 400 = 20^2.
a(5) = 1122 since pi(1122^3) = pi(1412467848) = 70576801 = 8401^2.

Cf. A000290, A000578, A000720, A064523, A262408, A262409, A262443.

SQ[n_]:=IntegerQ[Sqrt[n]]
f[n_]:=PrimePi[n^3]
n=0;Do[If[SQ[f[k]],n=n+1;Print[n," ",k]],{k,1,1200}]
Select[Range[1200],IntegerQ[Sqrt[PrimePi[#^3]]]&] (* Harvey P. Dale, Aug 21 2024 *)

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}]

A262536 Positive integers z such that pi(x^3+y^3) = pi(z^3) for some 0 < x <= y < z, where pi(m) denotes the number of primes not exceeding m.

Original entry on oeis.org

7, 9, 11, 12, 34, 46, 49, 65, 89, 95, 103, 127, 144, 150, 163, 172, 206, 236, 249, 258, 275, 288, 300, 309, 312, 385, 492, 495, 505, 577, 641, 683, 729, 738, 751, 796, 835, 873, 904, 990, 995, 1010, 1154, 1210, 1297, 1312, 1403, 1458, 1476, 1502, 1544, 1626, 1661, 1731, 1808, 1852, 1985, 1988, 2020, 2059, 2107, 2214, 2304, 2316, 2370, 2448, 2594, 2833, 2840, 2883, 2920, 3073, 3088, 3097

Offset: 1

Zhi-Wei Sun, Sep 24 2015

nonn

The sequence has infinitely many terms. In fact, for any integer t > 0, we have (6*t^2)^3 + (6*t^3-1)^3 = (6*t^3+1)^3 - 2 and hence pi((6*t^2)^3+(6*t^3-1)^3) = pi((6*t^3+1)^3) since neither (6*t^3+1)^3 nor (6*t^3+1)^3-1 is prime.
Concerning the equation pi(x^3+y^3) = pi(z^3) with 0 < x <= y < z, there are exactly 70 solutions with z <= 2700. They are (x,y,z) = (5,6,7),(6,8,9),(7,10,11),(9,10,12),(15,33,34),(23,44,46),(24,47,49),(43,58,65),(41,86,89),(47,91,95),(64,94,103),(95,106,127),(71,138,144),(73,144,150),(54,161,163),(135,138,172),(128,188,206),(55,235,236),(135,235,249),(197,212,258),(159,256,275),(142,276,288),(146,288,300),(192,282,309),(161,297,312),(96,383,385),(252,345,385),(390,391,492),(334,438,495),(372,426,505),(426,486,577),(297,619,641),(353,650,683),(242,720,729),(244,729,738),(150,749,751),(602,659,796),(161,833,835),(470,825,873),(566,823,904),(668,876,990),(514,947,995),(744,852,1010),(791,812,1010),(509,1120,1154),(852,972,1154),(236,1207,1210),(216,1295,1297),(459,1293,1312),(915,1259,1403),(484,1440,1458),(488,1458,1476),(300,1498,1502),(368,1537,1544),(511,1609,1626),(420,1652,1661),(1278,1458,1731),(1132,1646,1808),(1033,1738,1852),(1241,1808,1985),(1010,1897,1988),(1582,1624,2020),(294,2057,2059),(237,2106,2107),(732,2187,2214),(575,2292,2304),(577,2304,2316),(1518,2141,2370),(1611,2189,2448),(432,2590,2594).
Recall Fermat's Last Theorem, which asserts that the Diophantine equation x^n + y^n = z^n with n > 2 and x,y,z > 0 has no solution. In 1936 K. Mahler discovered that
    (9*t^3+1)^3 + (9*t^4)^3 - (9*t^4+3*t)^3 = 1.
Conjecture: (i) For any integers n > 3 and x,y,z > 0 with {x,y} not equal to {1,z}, we have |x^n+y^n-z^n| >= 2^n-2, unless n = 5, {x,y} = {13,16} and z = 17.
(ii) For any integer n > 3 and x,y,z > 0 with {x,y} not containing z, there is a prime p with x^n+y^n < p < z^n or z^n < p < x^n+y^n, unless n = 5, {x,y} = {13,16} and z = 17.
(iii) For any integers n > 3, x > y >= 0 and z > 0 with x not equal to z, there always exists a prime p with x^n-y^n < p < z^n or z^n < p < x^n-y^n.
We have verified part (i) of the conjecture for n = 4..10 and 0 < x,y,z <= 1700.

			a(1) = 7 since pi(5^3+6^3) = pi(125+216) = pi(341) = 68 = pi(343) = pi(7^3).
a(2) = 9 since pi(6^3+8^3) = pi(216+512) = pi(728) = 129 = pi(729) = pi(9^3).
a(50) = 1502 since pi(300^3+1498^3) = pi(27000000+3361517992) = pi(3388517992) = 162202081 = pi(3388518008) = pi(1502^3).

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.

Chai Wah Wu, Table of n, a(n) for n = 1..446
M. Beck, E. Pine, W. Tarrant and K. Y. Jensen, New integer representations as the sum of three cubes, Math. Comp. 76(2007), 1683-1690.
V. L. Gardiner, R. B. Lazarus, P. R. Stein, Solutions of the diophantine equation x^3+y^3=z^3-d, Math. Comp. 18 (1964) 408-413
K. Mahler, Note on hypothesis K of Hardy and Littlewood, J. London Math. Soc. 11(1936), 136-138.
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.
A. J. Wiles, Modular elliptic curves and Fermat's last theorem, Ann. Math. 141 (1995), 443-551.

Cf. A000578, A000720, A019590, A262408, A262409, A262443, A262462.

pi[n_]:=PrimePi[n]
n=0;Do[Do[If[pi[x^3+y^3]==pi[z^3],n=n+1;Print[n," ",z];Goto[aa]],{x,1,z-1},{y,x,z-1}];Label[aa];Continue,{z,1,2700}]

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}]

cp's OEIS Frontend

A038107 Number of primes < n^2.

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A262443 Positive integers m such that pi(m^2) = pi(j^2)*pi(k^2) for some 0 < j < k < m, where pi(x) denotes the number of primes not exceeding x.

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A262409 Positive integers m such that pi(m^3) = pi(j^3) + pi(k^3) for some 0 < j <= k < m.

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A262403 Number of ways to write pi(T(n)) = pi(T(k)) + pi(T(m)) with 1 < k < m < n, where T(x) is the triangular number x*(x+1)/2, and pi(x) is the number of primes not exceeding x.

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A262447 Primes p such that pi(p^2) = pi(q^2) + pi(r^2) for some distinct primes q and r.

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A262462 Positive integers k with pi(k^3) a square, where pi(x) denotes the number of primes not exceeding x.

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