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A261641 Number of practical numbers q such that n+(n mod 2)-q and n-(n mod 2)+q are both practical numbers. 2
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 (list; graph; refs; listen; history; text; internal format)
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 A257806

Adjacent sequences:  A261638 A261639 A261640 * A261642 A261643 A261644

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Aug 27 2015

STATUS

approved

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Last modified October 18 03:22 EDT 2019. Contains 328135 sequences. (Running on oeis4.)