A210531 Number of nonnegative integers k<n with n+k^3 practical.

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, 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, 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

Offset

1,4

Comments

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.

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.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000

G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106].

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.

Example

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

Mathematica

f[n_]:=f[n]=FactorInteger[n]
Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
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}]
pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
a[n_]:=a[n]=Sum[If[pr[n+k^3]==True, 1, 0], {k, 0, n-1}]
Do[Print[n, " ", a[n]], {n, 1, 100}]

Crossrefs

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

Sequence in context: A366308 A347089 A354825 * A066954 A144925 A029262

Adjacent sequences: A210528 A210529 A210530 * A210532 A210533 A210534

Keyword

nonn

Author

Zhi-Wei Sun, Jan 28 2013

Status

approved