

A262408


Positive integers m such that pi(m^2) = pi(j^2) + pi(k^2) for no 0 < j <= k < m.


12



1, 2, 5, 10, 21, 23, 46, 103, 105, 193, 222, 232, 285, 309, 345, 392, 404, 476, 587, 670, 779, 912, 1086, 1162, 1249, 2508, 2592, 2852, 2964, 3362, 3673, 3895, 4218, 4732, 5452, 6417, 7667, 7759, 8430, 8796, 9606, 11096, 11953, 12014, 12125, 13956, 14474, 15018, 17854, 18861, 18879, 19307, 22843, 28106, 29423, 31576, 37182
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OFFSET

1,2


COMMENTS

Conjecture: (i) There are infinitely many positive integers m such that pi(m^2) = pi(x^2) + pi(y^2) for some 0 < x <= y < m. Also, the current sequence has infinitely many terms.
(ii) For every n = 4,5,... the equation pi(x^n) + pi(y^n) = pi(z^n) has no integral solution with 0 < x <= y < z.
It is interesting to compare this conjecture with Fermat's Last Theorem. See also A262409 for the equation pi(x^3) + pi(y^3) = pi(z^3).


REFERENCES

ZhiWei 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 ChinaJapan Seminar (Fukuoka, Oct. 28  Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169187.


LINKS

Table of n, a(n) for n=1..57.
ZhiWei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.


EXAMPLE

a(3) = 5 since pi(5^2) = 9 cannot be written as pi(j^2) + pi(k^2) with 0 < j <= k < 5. Note that pi(1^2) = 0, pi(2^2) = 2, pi(3^2) = 4 and pi(4^2) = 6 are all even.


MATHEMATICA

f[n_]:=PrimePi[n^2]
T[n_]:=Table[f[k], {k, 1, n}]
n=0; Do[Do[If[MemberQ[T[m1], f[m]f[k]], Goto[aa]], {k, 1, m1}]; n=n+1; Print[n, " ", m]; Label[aa]; Continue, {m, 1, 32000}]


CROSSREFS

Cf. A000720, A038107, A262403, A262409.
Sequence in context: A212951 A051109 A124146 * A344378 A032468 A327286
Adjacent sequences: A262405 A262406 A262407 * A262409 A262410 A262411


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Sep 21 2015


STATUS

approved



