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A214841 Number of ways to write n=p+q/2, where p and q are practical numbers smaller than n. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 25 11:27 EDT 2020. Contains 334592 sequences. (Running on oeis4.)