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Number of nonnegative integers k<n with n+k^3 practical.
5

%I #14 Dec 05 2018 06:13:34

%S 1,1,1,2,2,1,2,4,2,2,2,3,2,2,4,5,4,2,3,7,5,1,2,7,4,2,7,5,6,1,5,9,4,4,

%T 6,9,9,2,5,12,9,3,5,6,8,5,6,13,4,2,8,6,11,6,11,14,8,2,4,7,4,5,7,29,8,

%U 3,5,8,11,4,13,16,13,2,7,12,13,6,10,16,10,6,15,9,13,3,9,20,11,8,11,20,9,2,8,22,14,6,15,15

%N Number of nonnegative integers k<n with n+k^3 practical.

%C Conjecture: a(n)>0 for all n>0. Moreover, if n>0 is different from 74, 138, 166, 542, then n+k^3 is practical for some 0<=k<=sqrt(n)*log(n); if n is not equal to 102, then n+k and n+k^3 are both practical for some k=0,...,n-1.

%C Zhi-Wei Sun also conjectured that any integer n>1 can be written as x^3+y (x,y>0) with 2x and 4xy both practical.

%H Zhi-Wei Sun, <a href="/A210531/b210531.txt">Table of n, a(n) for n = 1..5000</a>

%H G. Melfi, <a href="http://dx.doi.org/10.1006/jnth.1996.0012">On two conjectures about practical numbers</a>, J. Number Theory 56 (1996) 205-210 [<a href="http://www.ams.org/mathscinet-getitem?mr=1370203">MR96i:11106</a>].

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588 [math.NT], 2012-2017.

%e a(22)=1 since 22+2^3=30 is practical.

%t f[n_]:=f[n]=FactorInteger[n]

%t Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])

%t Con[n_]:=Con[n]=Sum[If[Part[Part[f[n], s+1], 1]<=DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]]+1, 0, 1], {s, 1, Length[f[n]]-1}]

%t pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0)

%t a[n_]:=a[n]=Sum[If[pr[n+k^3]==True,1,0],{k,0,n-1}]

%t Do[Print[n," ",a[n]],{n,1,100}]

%Y Cf. A005153, A185636, A208243, A208244, A208246, A208249, A209236, A209253, A209254, A209312, A209315, A209320, A210528.

%K nonn

%O 1,4

%A _Zhi-Wei Sun_, Jan 28 2013