OFFSET
1,5
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.
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.
EXAMPLE
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.
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Aug 27 2015
STATUS
approved