

A163577


Count of indices x in [0,n] that satisfy the equation A000120(x) + A000120(nx) = A000120(n) + 2.


4



0, 0, 0, 0, 2, 0, 1, 0, 2, 4, 1, 0, 5, 2, 2, 0, 2, 4, 5, 8, 5, 2, 4, 0, 5, 10, 4, 4, 10, 4, 4, 0, 2, 4, 5, 8, 9, 10, 12, 16, 5, 10, 6, 4, 12, 8, 8, 0, 5, 10, 12, 20, 12, 8, 12, 8, 10, 20, 8, 8, 20, 8, 8, 0, 2, 4, 5, 8, 9, 10, 12, 16, 9, 18, 14, 20, 20, 24, 24, 32, 5, 10, 14, 20, 14, 12, 16, 8, 12, 24
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OFFSET

0,5


COMMENTS

For every solution x, binomial(n,x) is 4 times an odd integer.


LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000
V. Shevelev, Binomial predictors, arXiv:0907.3302 [math.NT], 2009.
L. Spiegelhofer, M. Wallner, Divisibility of binomial coefficients by powers of two, arXiv:1710.10884


EXAMPLE

For n=8, there are a(8)=2 solutions, namely x=2 and x=6.
For n=9, there are a(9)=4 solutions, namely x=2, 3, 6 and 7.


MAPLE

read("transforms") ; A000120 := proc(n) wt(n) ; end:
A163577 := proc(n) local a, x ; a := 0 ; for x from 0 to n do if A000120(x)+A000120(nx) = A000120(n)+2 then a := a+1; fi; od: a; end:
seq(A163577(n), n=0..130) ; # R. J. Mathar, Jul 08 2009


MATHEMATICA

a120[n_] := DigitCount[n, 2, 1]; a[n_] := Count[Range[0, n], x_ /; a120[x] + a120[nx] == a120[n]+2]; Array[a, 90, 0] (* JeanFrançois Alcover, Jul 10 2017 *)


CROSSREFS

Cf. A000120, A007814.
A001316 and A163000 count binomial coefficients with 2adic valuation 0 and 1. A275012 gives a measure of complexity of these sequences.  Eric Rowland, Mar 15 2017
Sequence in context: A182936 A072662 A030010 * A132178 A039655 A103775
Adjacent sequences: A163574 A163575 A163576 * A163578 A163579 A163580


KEYWORD

nonn,look


AUTHOR

Vladimir Shevelev, Jul 31 2009


EXTENSIONS

Extended beyond a(22), examples added by R. J. Mathar, Jul 08 2009


STATUS

approved



