A261641 - OEIS

%I #9 Aug 28 2015 10:44:20

%S 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,

%T 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,

%U 9,5,2,5,13,6,2,2,7,3,9,4,9

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

%C 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.

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

%H Zhi-Wei Sun, <a href="/A261641/b261641.txt">Table of n, a(n) for n = 1..10000</a>

%H G. Melfi, <a href="http://dx.doi.org/10.1006/jnth.1996.0012">On two conjectures about practical numbers</a>, J. Number Theory 56 (1996) 205-210 [<a href="http://www.ams.org/mathscinet-getitem?mr=1370203">MR96i:11106</a>].

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588 [math.NT], 2012-2015.

%e a(15) = 1 since 4, 15-(4-1) = 12 and 15+(4-1) = 18 are all practical.

%e a(2206) = 1 since 2106, 2206-2106 = 100 and 2206+2106 = 4312 are all practical.

%t f[n_]:=FactorInteger[n]

%t Pow[n_, i_]:=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])

%t 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}]

%t pr[n_]:=n>0&&(n<3||Mod[n, 2]+Con[n]==0)

%t 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}]

%Y Cf. A005153, A209312, A261627.

%K nonn

%O 1,5

%A _Zhi-Wei Sun_, Aug 27 2015