

A262887


Number of ordered ways to write n as x^3 + y^2 + pi(z^2) (x >= 0, y >= 0 and z > 0) with z1 or z+1 prime, where pi(m) denotes the number of primes not exceeding m.


3



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

1,1


COMMENTS

Conjecture: (i) a(n) > 0 for all n > 0. Also, each natural number can be written as x^2 + y^2 + pi(z^2) (0 <= x <= y and z > 0) with z1 or z+1 prime.
(ii) Any integer n > 1 can be written as x^3 + y^2 + pi(z^2) with x >= 0, y >= 0 and z > 0 such that y or z is prime.
(iii) Any integer n > 1 can be written as x^3 + pi(y^2) + pi(z^2) (x >= 0, y > 0 and z > 0) with y or z prime. Also, each integer n > 1 can be written as x^2 + pi(p^2) + pi(q^2) (x >= 0 and p >= q > 0) with p prime.
Compare this conjecture with the conjecture in A262746.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000


EXAMPLE

a(22) = 2 since 22 = 0^3 + 4^2 + pi(4^2) = 0^3 + 2^2 + pi(8^2) with 4+1 = 5 and 81 = 7 both prime.
a(24) = 1 since 24 = 2^3 + 4^2 + pi(1^2) with 1+1 = 2 prime.
a(40) = 2 since 40 = 0^3 + 6^2 + pi(3^2) = 3^3 + 3^2 + pi(3^2) with 31 = 2 prime.
a(57) = 1 since 57 = 2^3 + 7^2 + pi(1^2) with 1+1 = 2 prime.
a(73) = 1 since 73 = 4^3 + 3^2 + pi(1^2) with 1+1 = 2 prime.


MATHEMATICA

SQ[n_]:=IntegerQ[Sqrt[n]]
f[n_]:=PrimePi[n^2]
Do[r=0; Do[If[f[k]>n, Goto[aa]]; If[PrimeQ[k1]==False&&PrimeQ[k+1]==False, Goto[bb]]; Do[If[SQ[nf[k]x^3], r=r+1], {x, 0, (nf[k])^(1/3)}]; Label[bb]; Continue, {k, 1, n}]; Label[aa]; Print[n, " ", r]; Continue, {n, 1, 100}]


CROSSREFS

Cf. A000040, A000290, A000578, A000720, A262311, A262746, A262785, A262813.
Sequence in context: A227177 A132944 A210568 * A106432 A227923 A029836
Adjacent sequences: A262884 A262885 A262886 * A262888 A262889 A262890


KEYWORD

nonn,hard


AUTHOR

ZhiWei Sun, Oct 04 2015


STATUS

approved



