A078627 -id:A078627 - 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

A381962 Irregular triangle read by rows, where row n lists the iterates of f(x), starting at x = n until f(x) <= 1, where f(x) is the Hamming weight of x (A000120).

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 1, 4, 1, 5, 2, 1, 6, 2, 1, 7, 3, 2, 1, 8, 1, 9, 2, 1, 10, 2, 1, 11, 3, 2, 1, 12, 2, 1, 13, 3, 2, 1, 14, 3, 2, 1, 15, 4, 1, 16, 1, 17, 2, 1, 18, 2, 1, 19, 3, 2, 1, 20, 2, 1, 21, 3, 2, 1, 22, 3, 2, 1, 23, 4, 1, 24, 2, 1, 25, 3, 2, 1, 26, 3, 2, 1

Offset: 0

Paolo Xausa, Mar 11 2025

			Triangle begins:
  n\k|  0  1  2  3
  ----------------
   0 |  0;
   1 |  1;
   2 |  2, 1;
   3 |  3, 2, 1;
   4 |  4, 1;
   5 |  5, 2, 1;
   6 |  6, 2, 1;
   7 |  7, 3, 2, 1;
   8 |  8, 1;
   9 |  9, 2, 1;
  10 | 10, 2, 1;
  ...

Paolo Xausa, Table of n, a(n) for n = 0..11724 (rows 0..3000 of triangle, flattened).

Cf. A000120, A078627 (row lengths), A078677 (row sums), A180094.

Cf. A381963, A381965.

A381962row[n_] := NestWhileList[DigitSum[#, 2] &, n, # > 1 &];
Array[A381962row, 30, 0]

def row(n):
    out = [n] if n > 1 else []
    while (n:=n.bit_count()) > 1:
        out += [n]
    return out + [n]
print([e for n in range(27) for e in row(n)]) # Michael S. Branicky, Mar 12 2025

T(n,0) = n and, for k = 1..A180094(n), T(n,k) = A000120(T(n,k-1)).

A078677 Write n in binary; repeatedly sum the "digits" until reaching 1; a(n) = sum of these sums (including '1' and n itself).

Original entry on oeis.org

1, 3, 6, 5, 8, 9, 13, 9, 12, 13, 17, 15, 19, 20, 20, 17, 20, 21, 25, 23, 27, 28, 28, 27, 31, 32, 32, 34, 34, 35, 39, 33, 36, 37, 41, 39, 43, 44, 44, 43, 47, 48, 48, 50, 50, 51, 55, 51, 55, 56, 56, 58, 58, 59, 63, 62, 62, 63, 67, 65, 69, 70, 72, 65, 68, 69, 73, 71, 75, 76, 76, 75

Offset: 1

Frank Schwellinger (nummer_eins(AT)web.de), Dec 17 2002

			a(13) = 19 because 13 = (1101) -> (1+1+0+1 = 11) -> (1+1 = 10) -> (1+0 = 1) = 1 and 1101+11+10+1 (binary) = 19 (decimal).

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A000120, A078627, A180094.

Row sums of A381962.

A078677[n_] := Total[NestWhileList[DigitSum[#, 2] &, n, # > 1 &]];
Array[A078677, 100] (* Paolo Xausa, Mar 11 2025 *)

def a(n):
    s = n if n > 1 else 0
    while (n:=n.bit_count()) > 1:
        s += n
    return s + 1
print([a(n) for n in range(1, 73)]) # Michael S. Branicky, Mar 12 2025

a(1) = 1; for n > 1, a(n) = n + a(A000120(n)).

a(n) = Sum_{k = 0..A180094(n)} A381962(n,k). - Paolo Xausa, Mar 12 2025

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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.

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A381962 Irregular triangle read by rows, where row n lists the iterates of f(x), starting at x = n until f(x) <= 1, where f(x) is the Hamming weight of x (A000120).

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A078677 Write n in binary; repeatedly sum the "digits" until reaching 1; a(n) = sum of these sums (including '1' and n itself).

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