A214841 Number of ways to write n=p+q/2, where p and q are practical numbers smaller than n.

0, 0, 1, 0, 1, 1, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 3, 3, 4, 1, 3, 2, 4, 2, 4, 2, 5, 3, 5, 4, 6, 2, 5, 2, 6, 3, 5, 2, 6, 3, 7, 4, 6, 1, 6, 3, 6, 4, 6, 1, 5, 3, 6, 6, 6, 2, 7, 2, 6, 5, 6, 2, 7, 3, 8, 6, 7, 1, 8, 3, 7, 6, 7, 1, 7, 3, 7, 8, 7, 2, 9, 2, 7, 7, 8, 3, 9, 3, 10, 8, 8, 2, 11, 3, 9, 8, 9

Offset

1,7

Comments

Conjecture: a(n)>0 for all n>4.

This has been verified for n up to 5*10^6.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

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(23)=1 since 23=20+6/2 with 6 and 20 practical and smaller than 23.

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[2k]==True&&pr[n-k]==True, 1, 0], {k, 1, (n-1)/2}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A005153.

Sequence in context: A156263 A109672 A279362 * A025917 A135689 A029438

Adjacent sequences: A214838 A214839 A214840 * A214842 A214843 A214844

Keyword

nonn

Author

Zhi-Wei Sun, Mar 08 2013

Status

approved