A272034 -id:A272034 - cp's OEIS frontend

A272036 Numbers n such that the sum of the inverse of the exponents in the binary expansion of 2n is equal to 1.

1, 38, 2090, 16902, 18954, 18988, 131334, 133386, 133420, 148258, 150284, 524314, 524348, 526386, 541212, 543250, 543284, 655644, 657682, 657716, 672568, 674580, 8388742, 8390794, 8390828, 8405666, 8407692, 8520098, 8522124, 8536962, 8536996, 8539048, 8913052, 8915090

Offset: 1

Michel Marcus, Apr 18 2016

nonn

That is, numbers such that both A116416(n) and A116417(n) are equal to 1.
Intersection of A272034 and A272035.
A number m with an exponent k in the binary sum must have another power of 2 having an exponent at least A275288(k). - David A. Corneth, Apr 01 2017

			For n=38, 2*38_10 = 2^6 + 2^3 + 2^2 = 1001100_2, and 1/2 + 1/3 + 1/6 = 1.

Robert Israel, Table of n, a(n) for n = 1..1655 (first 200 terms from Peter Kagey)

Cf. A116416, A116417, A272034, A272035, A272083, A275288.

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

is(n) = my(b = Vecrev(binary(n))); sum(k=1, #b, b[k]/k) == 1;

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

A272036 Numbers n such that the sum of the inverse of the exponents in the binary expansion of 2n is equal to 1.

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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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Mathematica

PARI

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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Author

Keywords

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