A272082 -id:A272082 - cp's OEIS frontend

A272083 Irregular triangle read by rows: strictly decreasing positive integer sequences in lexicographic order with the property that the sum of inverses equals one.

1, 6, 3, 2, 12, 6, 4, 2, 15, 10, 3, 2, 15, 12, 10, 4, 2, 15, 12, 10, 6, 4, 3, 18, 9, 3, 2, 18, 12, 9, 4, 2, 18, 12, 9, 6, 4, 3, 18, 15, 10, 9, 6, 2, 18, 15, 12, 10, 9, 4, 3, 20, 5, 4, 2, 20, 6, 5, 4, 3, 20, 12, 6, 5, 2, 20, 15, 10, 5, 4, 3, 20, 15, 12, 10, 5

Offset: 1

Peter Kagey, Apr 19 2016

			First six rows:
[1]                   because 1/1 = 1.
[6, 3, 2]             because 1/6 + 1/3 + 1/2 = 1.
[12, 6, 4, 2]         because 1/12 + 1/6 + 1/4 + 1/2 = 1.
[15, 10, 3, 2]        because 1/15 + 1/10 + 1/3 + 1/2 = 1.
[15, 12, 10, 4, 2]    because 1/15 + 1/12 + 1/10 + 1/4 + 1/2 = 1.
[15, 12, 10, 6, 4, 3] because 1/15 + 1/12 + 1/10 + 1/6 + 1/4 + 1/3 = 1.

Peter Kagey, Table of n, a(n) for n = 1..1578 (All rows with first term less than or equal to 30.)
Wikipedia, Erdős-Graham problem.

Cf. A073546, A216975, A216993, A272020, A272036, A272081, A272082.

A272035 Numbers n such that the sum of the inverse of the exponents in the binary expansion of 2n is an integer.

Original entry on oeis.org

0, 1, 38, 39, 2090, 2091, 16902, 16903, 18954, 18955, 18988, 18989, 131334, 131335, 133386, 133387, 133420, 133421, 148258, 148259, 150284, 150285, 524314, 524315, 524348, 524349, 526386, 526387, 541212, 541213, 543250, 543251, 543284, 543285, 655644, 655645, 657682

Offset: 1

Michel Marcus, Apr 18 2016

nonn

That is, numbers such that A116416(n) equals 1.

2k is in this sequence if and only if 2k + 1 is. Therefore n + a(n) is odd for all n. - Peter Kagey, Apr 19 2016

			For n=39, 39_10=100111_2, and 1/1 + 1/2 + 1/3 + 1/6 = 2, an integer.

Peter Kagey, Table of n, a(n) for n = 1..450 (All terms less than 2^30)

Cf. A116416, A116417, A272034, A272036, A272082.

Select[Range[2^20], IntegerQ@ Total[1/Flatten@ Position[Reverse@ IntegerDigits[#, 2], 1]] &] (* Michael De Vlieger, Apr 18 2016 *)

isok(n) = {my(b = Vecrev(binary(n))); denominator(sum(k=1, #b, b[k]/k)) == 1;}

A272081 Irregular triangle read by rows: strictly decreasing positive integer sequences in lexicographic order with the property that the sum of inverses is the inverse of an integer.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 6, 3, 6, 3, 2, 7, 8, 9, 10, 11, 12, 12, 4, 12, 6, 12, 6, 4, 12, 6, 4, 2, 13, 14, 15, 15, 10, 15, 10, 3, 15, 10, 3, 2, 15, 10, 6, 15, 12, 10, 15, 12, 10, 4, 15, 12, 10, 4, 2, 15, 12, 10, 6, 4, 3, 16, 17, 18, 18, 9, 18, 9, 3, 18, 9, 3, 2, 18, 9

Offset: 1

Peter Kagey, Apr 19 2016

			First 18 rows:
  [1]           because 1 is self-inverse.
  [2]           because 1/2 is the inverse of an integer.
  [3]
  [4]           (...)
  [5]
  [6]
  [6, 3]        because 1/6 + 1/3              = 1/2.
  [6, 3, 2]     because 1/6 + 1/3 + 1/2        = 1/1.
  [7]
  [8]
  [9]           (...)
  [10]
  [11]
  [12]
  [12, 4]       because 1/12 + 1/4             = 1/3.
  [12, 6]       because 1/12 + 1/6             = 1/4.
  [12, 6, 4]    because 1/12 + 1/6 + 1/4       = 1/2.
  [12, 6, 4, 2] because 1/12 + 1/6 + 1/4 + 1/2 = 1/1.

Peter Kagey, Table of n, a(n) for n = 1..2170 (All rows with first term less than or equal to 30.)

Cf. A272020, A272034, A272082, A272083.

cp's OEIS Frontend

A272083 Irregular triangle read by rows: strictly decreasing positive integer sequences in lexicographic order with the property that the sum of inverses equals one.

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A272035 Numbers n such that the sum of the inverse of the exponents in the binary expansion of 2n is an integer.

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A272081 Irregular triangle read by rows: strictly decreasing positive integer sequences in lexicographic order with the property that the sum of inverses is the inverse of an integer.

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