A261641 Number of practical numbers q such that n+(n mod 2)-q and n-(n mod 2)+q are both practical numbers.

1, 0, 1, 1, 2, 1, 2, 1, 2, 3, 2, 3, 3, 2, 1, 3, 3, 3, 3, 4, 3, 4, 4, 6, 4, 2, 2, 4, 3, 5, 4, 5, 4, 4, 5, 8, 5, 2, 3, 5, 3, 6, 4, 7, 4, 2, 5, 11, 6, 1, 4, 7, 3, 7, 4, 7, 5, 4, 6, 11, 4, 2, 3, 8, 5, 8, 3, 9, 5, 2, 5, 13, 6, 2, 2, 7, 3, 9, 4, 9

Offset: 1

Author

Zhi-Wei Sun, Aug 27 2015

Keywords

• nonn

Comments

Conjecture: a(n) > 0 except for n = 2. Also, for any integer n > 3, there is a practical number q such that n-(n mod 2)-q and n+(n mod 2)+q are both practical numbers.

This is an analog of the author's conjecture in A261627, and it is stronger than Margenstern's conjecture proved by Melfi in 1996.

Examples

			a(15) = 1 since 4, 15-(4-1) = 12 and 15+(4-1) = 18 are all practical.
a(2206) = 1 since 2106, 2206-2106 = 100 and 2206+2106 = 4312 are all practical.
		

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], 2012-2015.

Crossrefs

Cf. A005153, A209312, A261627.

Programs

Mathematica

f[n_]:=FactorInteger[n]
Pow[n_, i_]:=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
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_]:=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
Do[r=0;Do[If[pr[q]&&pr[n+Mod[n,2]-q]&&pr[n-Mod[n,2]+q],r=r+1],{q,1,n}];Print[n," ",r];Continue,{n,1,80}]