login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A262409 Positive integers m such that pi(m^3) = pi(j^3) + pi(k^3) for some 0 < j <= k < m. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

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

Table of n, a(n) for n=1..42.

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

EXAMPLE

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).

MATHEMATICA

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}]

CROSSREFS

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

Sequence in context: A065754 A244013 A235858 * A183880 A220318 A220341

Adjacent sequences:  A262406 A262407 A262408 * A262410 A262411 A262412

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Sep 22 2015

EXTENSIONS

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 5 10:45 EST 2019. Contains 329751 sequences. (Running on oeis4.)