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%I #22 Feb 17 2024 23:43:41
%S 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,
%T 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,
%U 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
%N Number of ordered ways to write n as x^3 + y^2 + pi(z^2) (x >= 0, y >= 0 and z > 0) with z-1 or z+1 prime, where pi(m) denotes the number of primes not exceeding m.
%C Conjectures:
%C (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 z-1 or z+1 prime.
%C (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.
%C (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.
%C Compare these conjectures with the conjectures in A262746.
%H Zhi-Wei Sun, <a href="/A262887/b262887.txt">Table of n, a(n) for n = 1..10000</a>
%e a(22) = 2 since 22 = 0^3 + 4^2 + pi(4^2) = 0^3 + 2^2 + pi(8^2) with 4+1 = 5 and 8-1 = 7 both prime.
%e a(24) = 1 since 24 = 2^3 + 4^2 + pi(1^2) with 1+1 = 2 prime.
%e a(40) = 2 since 40 = 0^3 + 6^2 + pi(3^2) = 3^3 + 3^2 + pi(3^2) with 3-1 = 2 prime.
%e a(57) = 1 since 57 = 2^3 + 7^2 + pi(1^2) with 1+1 = 2 prime.
%e a(73) = 1 since 73 = 4^3 + 3^2 + pi(1^2) with 1+1 = 2 prime.
%t SQ[n_]:=IntegerQ[Sqrt[n]]
%t f[n_]:=PrimePi[n^2]
%t Do[r=0;Do[If[f[k]>n,Goto[aa]];If[PrimeQ[k-1]==False&&PrimeQ[k+1]==False,Goto[bb]];Do[If[SQ[n-f[k]-x^3],r=r+1],{x,0,(n-f[k])^(1/3)}];Label[bb];Continue,{k,1,n}];Label[aa];Print[n," ",r];Continue, {n,1,100}]
%Y Cf. A000040, A000290, A000578, A000720, A262311, A262746, A262785, A262813.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Oct 04 2015
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