A359056 - OEIS

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A359056

Numbers k >= 3 such that 1/d(k - 2) + 1/d(k - 1) + 1/d(k) is an integer, d(i) = A000005(i).

3, 8, 15, 23, 39, 59, 159, 179, 383, 503, 543, 719, 879, 1203, 1319, 1383, 1439, 1623, 1823, 2019, 2559, 2579, 2859, 2903, 3063, 3119, 3779, 4283, 4359, 4443, 4679, 4703, 5079, 5099, 5583, 5639, 5703, 5939, 6339, 6639, 6663, 6719, 6999, 7419, 8223, 8783, 8819, 9183, 9663, 9903, 10079, 10839

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OFFSET

1,1

COMMENTS

The sets {2, 4, 4}, {2, 3, 6} and {3, 3, 3} including permutations of elements of the set are the solutions of this unit fraction. There is no k for which {d(k - 2), d(k - 1), d(k)} equals {3, 3, 3}. May the set {2, 3, 6} exist for some k?

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Thomas F. Bloom, On a density conjecture about unit fractions, arXiv:2112.03726 [math.NT], 2021.

Ernest S. Croot, III, On a coloring conjecture about unit fractions, Annals of Mathematics, Volume 157 (2003), 545-556.

EXAMPLE

k = 3:

1/d(1) + 1/d(2) + 1/d(3) = 1/1 + 1/2 + 1/2 = 2. Thus k = 3 is a term.

k = 8:

1/d(6) + 1/d(7) + 1/d(8) = 1/4 + 1/2 + 1/4 = 1. Thus k = 8 is a term.

MATHEMATICA

Select[Range[11000], IntegerQ[Total[1/DivisorSigma[0, # - {0, 1, 2}]]] &] (* Amiram Eldar, Dec 14 2022 *)

CROSSREFS