

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.


9



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
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OFFSET

1,1


COMMENTS

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.


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

ZhiWei Sun, Table of n, a(n) for n = 1..300
ZhiWei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.


EXAMPLE

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.


MATHEMATICA

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}]


CROSSREFS

Cf. A000290, A000720, A038107, A262408, A262409.
Sequence in context: A190208 A061570 A096679 * A287552 A275191 A279776
Adjacent sequences: A262440 A262441 A262442 * A262444 A262445 A262446


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Sep 23 2015


STATUS

approved



