A161764 -id:A161764 - cp's OEIS frontend

A199238 n mod (number of ones in binary representation of n).

0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 2, 3, 0, 1, 0, 1, 0, 0, 1, 3, 0, 1, 2, 3, 1, 1, 2, 1, 0, 1, 0, 2, 0, 1, 2, 3, 0, 2, 0, 3, 2, 1, 2, 2, 0, 1, 2, 3, 1, 1, 2, 0, 2, 1, 2, 4, 0, 1, 2, 3, 0, 1, 0, 1, 0, 0, 1, 3, 0, 1, 2, 3, 1, 1, 2, 4, 0, 0, 1, 3, 0, 1, 2

Offset: 1

Reinhard Zumkeller, Nov 04 2011

nonn

a(A049445(n)) = 0;
a(n) = n - A161764(n);
a(A199262(n)) = n and a(m) <> n for m < A199262(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n

Cf. A000120, A070635, A049445, A161764, A199262.

a199238 n = a199238_list !! (n-1)
a199238_list = zipWith mod [1..] $ tail a000120_list

Mod[#,DigitCount[#,2,1]]&/@Range[90] (* Harvey P. Dale, Nov 08 2011 *)

A199238(n)=n%norml2(binary(n))  \\ M. F. Hasler, Oct 09 2012

A161765 a(n) is the smallest multiple of {the number of 1's in the binary representation of n} that is >= n.

Original entry on oeis.org

1, 2, 4, 4, 6, 6, 9, 8, 10, 10, 12, 12, 15, 15, 16, 16, 18, 18, 21, 20, 21, 24, 24, 24, 27, 27, 28, 30, 32, 32, 35, 32, 34, 34, 36, 36, 39, 39, 40, 40, 42, 42, 44, 45, 48, 48, 50, 48, 51, 51, 52, 54, 56, 56, 55, 57, 60, 60, 60, 60, 65, 65, 66, 64, 66, 66, 69, 68, 69, 72, 72, 72

Offset: 1

Leroy Quet, Jun 18 2009

			11 (decimal) in binary is 1011. There are three 1's. Because 12 is the smallest multiple of 3 that is >= 11, then a(11) = 12.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A161764, A000120.

a := proc (n) local n2, s, j: n2 := convert(n, base, 2): s := add(n2[i], i = 1 .. nops(n2)): for j while j*s < n do end do: j*s end proc: seq(a(n), n = 1 .. 80); # Emeric Deutsch, Jun 24 2009

Table[d=DigitCount[n,2,1];d*Ceiling[n/d],{n,80}] (* Harvey P. Dale, Aug 23 2013 *)

a(n) = my(nb = hammingweight(n)); nb*ceil(n/nb); \\ Michel Marcus, Nov 11 2018

a(n) = A000120(n)*ceiling(n/A000120(n)). - Michel Marcus, Nov 11 2018

Extended by Emeric Deutsch, Jun 24 2009

A358139 Numbers k > 0 sorted by k/A000120(k) in increasing order. A000120 is the binary weight of k. If k/A000120(k) yields equal values, the smaller k will appear first.

Original entry on oeis.org

1, 3, 2, 7, 5, 6, 11, 15, 4, 13, 9, 14, 10, 23, 12, 31, 19, 27, 21, 29, 22, 30, 8, 25, 17, 26, 18, 28, 47, 39, 20, 63, 43, 55, 45, 46, 35, 59, 24, 61, 37, 62, 38, 51, 53, 54, 41, 42, 57, 58, 44, 60, 79, 95, 16, 49, 33, 50, 34, 52, 87, 71, 36, 127, 91, 111, 93, 56

Offset: 1

Thomas Scheuerle, Oct 31 2022

A permutation of the positive integers.

This permutation satisfies a weak ordering: If b = a(c*d) and e = a(c) and f = a(d) then b > e and b > f with c,d > 1.

Thomas Scheuerle, a(1..25000) as scatter plot.
Thomas Scheuerle, a(1..1000) as scatter plot colored by binary weight and a(n)/A000120(a(n)) plotted as blue line.
Index entries for sequences that are permutations of the natural numbers

Cf. A000120, A049445, A161764, A199238, A235602, A348416.

f(x) = x/hammingweight(x);
cmpb(x, y) = my(hx=f(x), hy=f(y)); if (hx != hy, return(sign(hx-hy))); return(sign(x-y));
lista(nn) = Vec(vecsort([1..2*nn], cmpb, 1), nn); \\ Michel Marcus, Nov 05 2022

a(2^n) = 2^(n+1) - 1.

abs(a(n)-n) < n.

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A199238 n mod (number of ones in binary representation of n).

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A161765 a(n) is the smallest multiple of {the number of 1's in the binary representation of n} that is >= n.

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A358139 Numbers k > 0 sorted by k/A000120(k) in increasing order. A000120 is the binary weight of k. If k/A000120(k) yields equal values, the smaller k will appear first.

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