A209312 Number of practical numbers p

0, 0, 1, 2, 2, 2, 2, 1, 2, 3, 2, 4, 1, 2, 3, 3, 3, 4, 2, 4, 3, 4, 3, 6, 3, 2, 3, 4, 4, 6, 3, 5, 3, 4, 5, 8, 3, 2, 5, 5, 4, 7, 4, 7, 4, 2, 4, 11, 3, 1, 4, 7, 4, 7, 6, 7, 3, 4, 5, 12, 3, 2, 4, 8, 7, 8, 5, 9, 4, 2, 6, 14, 5, 2, 6, 7, 7, 9, 5, 9, 4, 4, 5, 14, 8, 2, 5, 8, 7, 10, 6, 9, 6, 2, 8, 15, 5, 3, 5, 8

Offset: 1

Author

Zhi-Wei Sun, Jan 19 2013

Keywords

• nonn

Comments

Conjecture: a(n)>0 for all n>2.

This has been verified for n up to 10^7.

Except for p=1, all practical numbers are even. Thus, (n-p,n+p) prime is possible only if n is odd, and (n-p,n+p) can be practical only if n is even (except for p=1). - M. F. Hasler, Jan 19 2013

Examples

			a(8)=1 since 4, 8-4 and 8+4 are all practical.
a(13)=1 since 6 is practical, and 13-6 and 13+6 are both prime.
		

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.

Crossrefs

Cf. A005153, A208243, A208244, A208246, A208249, A209253, A209254.

Cf. A209321: Indices for which a(n)=2.

Programs

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[p]==True&&((PrimeQ[n-p]==True&&PrimeQ[n+p]==True)||(pr[n-p]==True&&pr[n+p]==True)),1,0],{p,1,n-1}]
Do[Print[n," ",a[n]],{n,1,100}]

PARI

A209312(n)=sum(p=1,n-1, is_A005153(p) && ((is_A005153(n-p) && is_A005153(n+p)) || (isprime(n-p) && isprime(n+p)))) \\ (Could be made more efficient by separating the case of odd and even n.) - M. F. Hasler, Jan 19 2013