A135941 -id:A135941 - cp's OEIS frontend

A265917 a(n) = floor(A070939(n)/A000120(n)) where A070939(n) is the binary length of n and A000120(n) is the binary weight of n.

1, 2, 1, 3, 1, 1, 1, 4, 2, 2, 1, 2, 1, 1, 1, 5, 2, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 6, 3, 3, 2, 3, 2, 2, 1, 3, 2, 2, 1, 2, 1, 1, 1, 3, 2, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 7, 3, 3, 2, 3, 2, 2, 1, 3, 2, 2, 1, 2, 1, 1, 1, 3, 2, 2, 1, 2, 1, 1

Offset: 1

Alex Ratushnyak, Dec 18 2015

1/a(n) gives a very rough approximation of the density of 1-bits in the binary representation (A007088) of n. This is 1 if more than half of the bits of n are 1. - Antti Karttunen, Dec 19 2015

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000120, A007088, A070939, A135941, A199238, A265918.

Table[Floor[IntegerLength[n, 2]/Total@ IntegerDigits[n, 2]], {n, 120}] (* Michael De Vlieger, Dec 21 2015 *)

a(n) = #binary(n)\hammingweight(n); \\ Michel Marcus, Dec 19 2015

for n in range(1, 88):
    print(str((len(bin(n))-2) // bin(n).count('1')), end=',')

A265918 a(n) = A070939(n) mod A000120(n), where A070939(n) is the binary length of n and A000120(n) is the binary weight of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 2, 1, 0, 0, 0, 2, 0, 2, 2, 1, 0, 2, 2, 1, 2, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 3, 2, 1, 1, 1, 3, 1, 3, 3

Offset: 1

Alex Ratushnyak, Dec 18 2015

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000120, A070939, A135941, A199238, A265917.

Table[Mod[IntegerLength[n, 2], Total@ IntegerDigits[n, 2]], {n, 120}] (* Michael De Vlieger, Dec 21 2015 *)

a(n) = #binary(n) % hammingweight(n); \\ Michel Marcus, Dec 19 2015

for n in range(1, 88): print((len(bin(n))-2) % bin(n).count('1'), end=', ')

A135993 a(0) = 0; a(n) = (floor(n/S2(n))) mod 2 for n >= 1, where S2(n) is the binary weight of n.

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1

Offset: 0

Ctibor O. Zizka, Mar 03 2008

			a(17) = floor(17/2) mod 2 = 0.
a(18) = floor(18/2) mod 2 = 1.

J.-P. Allouche, J. Shallit and J. Sondow, Summation of Series Defined by Counting Blocks of Digits, arXiv:math/0512399 [math.NT], 2005-2006.
J.-P. Allouche, J. Shallit and J. Sondow, Summation of series defined by counting blocks of digits, J. Number Theory 123 (2007), 133-143.
Jonathan Sondow and Petros Hadjicostas, The Generalized-Euler-Constant Function (z) and a Generalization of Somos's Quadratic Recurrence Constant, arXiv:math/0610499 [math.CA], 2006.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl. 332 (2007), 292-314.

Cf. A007953, A010060, A135941.

a(n) = if (n==0, 0, n\hammingweight(n) % 2); \\ Michel Marcus, Feb 04 2016

a(n) = A135941(n) mod 2 for n > 0. - Michel Marcus, Feb 04 2016

Converted references into links - R. J. Mathar, Oct 30 2009

A264749 a(n) = floor(n/BL(n)) where BL(n) = A070939(n) is the binary length of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 12

Offset: 0

Alex Ratushnyak, Nov 23 2015

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A070939, A135941, A253608.

a264749 n = div n $ a070939 n  -- Reinhard Zumkeller, Dec 05 2015

{0}~Join~Table[Floor[n/IntegerLength[n, 2]], {n, 84}] (* Michael De Vlieger, Dec 01 2015 *)

a(n) = if (n, n\#binary(n)); \\ Michel Marcus, Dec 01 2015

for n in range(88):  print(n // (len(bin(n))-2), end=', ')

cp's OEIS Frontend

A265917 a(n) = floor(A070939(n)/A000120(n)) where A070939(n) is the binary length of n and A000120(n) is the binary weight of n.

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A265918 a(n) = A070939(n) mod A000120(n), where A070939(n) is the binary length of n and A000120(n) is the binary weight of n.

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A135993 a(0) = 0; a(n) = (floor(n/S2(n))) mod 2 for n >= 1, where S2(n) is the binary weight of n.

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A264749 a(n) = floor(n/BL(n)) where BL(n) = A070939(n) is the binary length of n.

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