A075311 -id:A075311 - cp's OEIS frontend

A180094 Number of steps to reach 0 or 1, starting with n and applying the map k -> (number of 1's in binary expansion of k) repeatedly.

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

Offset: 0

Joerg Arndt, Jan 15 2011

nonn

The number of 1's in binary expansion of n is called the binary weight (or Hamming weight) of n, A000120(n).
a(n)=0 for n=0 and n=1;  a(n)=1 for powers of 2.
Records appear for n = 2, 3, 7, 127=2^7-1, 2^127-1, ... (terms of A007013).
It appears that the indices of the even terms for n>0 are sequence A075311.

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

Cf. A000120, A072086.

One less than A078627.

a180094 n = snd $ until ((< 2) . fst) (\(x, c) -> (a000120 x, c+1)) (n,0)
-- Reinhard Zumkeller, Apr 22 2011

Countbits:=func< n | &+Intseq(n, 2) >;
StepsTo01:=function(n); s:=0; k:=n; while k gt 1 do k:=Countbits(k); s+:=1; end while; return s; end function;
[ StepsTo01(n): n in [0..105] ]; // Klaus Brockhaus, Jan 15 2011

a:= n-> `if`(n<2, 0, 1 + a(add(i, i=convert(n, base, 2)))):
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 15 2011

Table[Length[NestWhileList[DigitCount[#,2,1]&,n,#>1&]]-1,{n,0,100}] (* Harvey P. Dale, Jul 27 2012 *)

bitcount(x)=
{ /* Return Hamming weight of x, i.e. A000120(x) */
    local(p);  p = 0;
    while ( x, p+=bitand(x, 1); x>>=1; );
    return( p );
}
X(n)=
{ /* Return how many iterations of bitcount() are needed to reach 0 or 1 */
    if ( n<=1, return(0) );
    return( 1+X(bitcount(n)) );
}
{ for (n=0, 100, print1(X(n),", ") ); } /* print terms of sequence */

def a(n):
    c = 1 if n > 1 else 0
    while (n:=n.bit_count()) > 1:
        c += 1
    return c
print([a(n) for n in range(101)]) # Michael S. Branicky, Mar 12 2025

A217122 a(n) = (number of 1's in binary expansion of n)th positive number not among the previous terms.

Original entry on oeis.org

1, 2, 4, 3, 6, 7, 9, 5, 10, 11, 13, 12, 15, 16, 18, 8, 17, 19, 21, 20, 23, 24, 26, 22, 27, 28, 30, 29, 32, 33, 35, 14, 31, 34, 37, 36, 39, 40, 42, 38, 43, 44, 46, 45, 48, 49, 51, 41, 50, 52, 54, 53, 56, 57, 59, 55, 60, 61, 63, 62, 65, 66, 68, 25, 58, 64, 69, 67, 71, 72, 74, 70, 75, 76, 78, 77, 80, 81, 83, 73, 82, 84, 86, 85, 88, 89, 91, 87, 92

Offset: 1

Paul Tek, Mar 16 2013

This is a permutation of the positive numbers.

			3 has 2 1's in its binary expansion, and the 2nd positive number not among the previous terms is 4; hence a(3)=4

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A000120: number of 1's in binary expansion of n.

Cf. A225589 (inverse), A075311.

import Data.List (delete)
a217122 n = a217122_list !! (n-1)
a217122_list = f 1 [0..] where
   f x zs = y : f (x + 1) (delete y zs) where
     y = zs !! a000120 x
-- Reinhard Zumkeller, May 11 2013

A075517 Created by removing all integers which take an odd number of nested digit sums to reach <10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 28, 29, 37, 38, 39, 46, 47, 48, 49, 55, 56, 57, 58, 59, 64, 65, 66, 67, 68, 69, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 98, 99, 109, 118, 119, 127, 128, 129, 136, 137, 138, 139, 145, 146, 147

Offset: 0

Jon Perry, Oct 11 2002

			18 -> 1+8 = 9. This takes 1 step to be reduced to a single integer, hence is not in the sequence. 99 -> 9+9 = 18, takes 2 steps and so is in the sequence.

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

Cf. A007953, A075311.

a075517 n = a075517_list !! n
a075517_list = [0..9] ++ f 1 [0..9] where
   f x ys = if a007953 x `elem` ys then f (x + 1) ys
                                   else x : f (x + 1) (x : ys)
-- Reinhard Zumkeller, Sep 29 2014, Apr 22 2012

ondQ[n_]:=OddQ[Length[NestWhileList[Total[IntegerDigits[#]]&,n,#>9&]]]; Select[Range[0,200],ondQ] (* Harvey P. Dale, Dec 22 2016 *)

sumdigits(n)=local(c); c=0; while (n>0,c=c+n%10; n=n-n%10; n=n/10); c checkSieve(n)=local(c); c=0; while(n>9, n=sumdigits(n); c++); 1-c%2 for (n=1,2000,if (checkSieve(n),print1(n,",")))

Offset fixed by Reinhard Zumkeller, Apr 22 2012

Added a(0) = 0. - Jon Perry, Nov 28 2012

A079495 Numbers k such that the sum of the squares of the digits of k in base 3 is 0 (mod 3).

Original entry on oeis.org

0, 13, 14, 16, 17, 22, 23, 25, 26, 31, 32, 34, 35, 37, 38, 39, 42, 46, 47, 48, 51, 58, 59, 61, 62, 64, 65, 66, 69, 73, 74, 75, 78, 85, 86, 88, 89, 91, 92, 93, 96, 100, 101, 102, 105, 109, 110, 111, 114, 117, 126, 136, 137, 138, 141, 144, 153, 166, 167, 169, 170, 172, 173

Offset: 1

Carlos Alves, Jan 20 2003

In base 2 this gives the "Evil Numbers" (cf. A001969) and slope 2. One may conjecture that in base b the asymptotic slope will be b and might suspect asymptotic density 1/b for each result (mod b). For nonprime b larger variations occur and "very big" numbers must be considered to believe in the conjecture (1 million or more...). (Related to A006287, here mod b is considered)

			59 is a member because 59 = 2013_3 and 2^2+0^2+1^2+1^2 = 6 = 0 (mod 3).

Cf. A001969, A006287, A075311.

Ev = Function[{b, x}, vx = IntegerDigits[x, b]; Mod[vx.vx, b]]; Seq = Function[{b, n}, Flatten[Position[Table[Ev[b, k], {k, 1, n}], 0]]]; Seq[3, 1000]

Revised by Sean A. Irvine, Aug 17 2025

cp's OEIS Frontend

A180094 Number of steps to reach 0 or 1, starting with n and applying the map k -> (number of 1's in binary expansion of k) repeatedly.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Python

A217122 a(n) = (number of 1's in binary expansion of n)th positive number not among the previous terms.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

A075517 Created by removing all integers which take an odd number of nested digit sums to reach <10.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Extensions

A079495 Numbers k such that the sum of the squares of the digits of k in base 3 is 0 (mod 3).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions