A316997 Number of 1's in the first n digits of the binary expansion of sqrt(n).
0, 1, 1, 2, 1, 2, 4, 3, 5, 2, 5, 5, 9, 7, 11, 13, 1, 7, 9, 9, 12, 9, 11, 14, 10, 2, 13, 13, 16, 12, 16, 12, 16, 19, 18, 15, 2, 21, 18, 20, 19, 25, 19, 20, 25, 26, 19, 24, 26, 3, 20, 25, 25, 31, 28, 36, 30, 33, 33, 37, 38
Offset: 0
Examples
For n = 7 we have sqrt(7) = 2.64575131... with binary expansion 10.1010010.... Of the first 7 digits there are a(7) = 3 digits equal to 1.
Links
- Rainer Rosenthal, Table of n, a(n) for n = 0..1000
Programs
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Maple
zaehle := proc(n) local e, p, c, i, z, m; Digits := n+5; e := evalf(sqrt(n)); p := [op(convert(e, binary))]; c := convert(p[1], base, 10); z := 0; m := min(n, nops(c)); for i to m do if c[-i] = 1 then z := z+1; fi; od; return z; end: seq(zaehle(n), n=0..60); # Rainer Rosenthal, Dec 14 2018 a := n -> StringTools:-CountCharacterOccurrences(convert(convert(evalf(sqrt(n), n+5), binary, n), string), "1"): seq(a(n),n=0..60); # Peter Luschny, Dec 15 2018
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Mathematica
a[n_] := Count[RealDigits[Sqrt[n], 2, n][[1]], 1]; Array[a, 60, 0] (* Amiram Eldar, Dec 14 2018 *)
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PARI
a(n)=my(v=concat(binary(sqrt(n))));hammingweight(v[1..n]) \\ Hugo Pfoertner, Dec 16 2018
Formula
a(n^2) = A000120(n). - Michel Marcus, Dec 15 2018