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

1,5

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

Cf. A005153, A209312, A261627.

Sequence in context: A257564 A194509 A054716 * A325622 A060145 A358997

Adjacent sequences: A261638 A261639 A261640 * A261642 A261643 A261644

Keyword

nonn

Author

Zhi-Wei Sun, Aug 27 2015

Status

approved