A381962 -id:A381962 - cp's OEIS frontend

A078627 Write n in binary; repeatedly sum the "digits" until reaching 1; a(n) = 1 + number of steps required.

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

Offset: 1

Frank Schwellinger (nummer_eins(AT)web.de), Dec 12 2002

The terms a(n) are unbounded. The smallest n with a(n) = m, n_min(m), however may be exorbitantly large, even for small m. It can be calculated by the following recurrence: n_min(1) = 1; n_min(2) = 2; n_min(m) = 2^n_min(m-1) - 1 {if m > 2};

			a(13) = 4 because 13 = (1101) -> (1+1+0+1 = 11) -> (1+1 = 10) -> (1+0 = 1) = 1. (Three iterations were required to reach 1.)

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences related to binary expansion of n

Cf. A000120.

One more than A180094. Row lengths of A381962.

for n from 1 to 500 do h := n:a[n] := 1:while(h>1) do a[n] := a[n]+1: b := convert(h,base,2):h := sum(b[j],j=1..nops(b)):od:od:seq(a[j],j=1..500);

Table[Length[NestWhileList[Total[IntegerDigits[#,2]]&,n,#>1&]],{n,110}] (* Harvey P. Dale, Oct 10 2011 *)

A078627(n) = { my(k=1); while(n>1, n = hammingweight(n); k += 1); (k); }; \\ Antti Karttunen, Jul 09 2017

def a(n):
    c = 1 if n > 1 else 0
    while (n:=n.bit_count()) > 1:
        c += 1
    return c + 1
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Mar 12 2025

a(1) = 1; for n > 1, a(n) = 1 + a(A000120(n)), where A000120 gives the number of occurrences of digit 1 in binary representation of n.

a(n) = 1 + A180094(n). - Antti Karttunen, Jul 09 2017

Description corrected by Antti Karttunen, Jul 09 2017

A381963 Irregular triangle read by rows, where row n lists the iterates of f(x), starting at x = n until f(x) < 10, where f(x) is the digital sum of x (A007953).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 11, 2, 12, 3, 13, 4, 14, 5, 15, 6, 16, 7, 17, 8, 18, 9, 19, 10, 1, 20, 2, 21, 3, 22, 4, 23, 5, 24, 6, 25, 7, 26, 8, 27, 9, 28, 10, 1, 29, 11, 2, 30, 3, 31, 4, 32, 5, 33, 6, 34, 7, 35, 8, 36, 9, 37, 10, 1, 38, 11, 2, 39, 12, 3

Offset: 0

Paolo Xausa, Mar 11 2025

			Triangle begins:
  n\k|  0   1   2
  ---------------
   0 |  0;
   1 |  1;
   2 |  2;
   3 |  3;
   4 |  4;
   5 |  5;
   6 |  6;
   7 |  7;
   8 |  8;
   9 |  9;
  10 | 10,  1;
  11 | 11,  2;
  12 | 12,  3;
  13 | 13,  4;
  14 | 14,  5;
  15 | 15,  6;
  16 | 16,  7;
  17 | 17,  8;
  18 | 18,  9;
  19 | 19, 10,  1;
  20 | 20,  2;
  ...

Paolo Xausa, Table of n, a(n) for n = 0..11648 (rows 0..4000 of triangle, flattened).

Cf. A007953, A010888 (right border), A031286 (row lengths - 1), A381964 (row sums).

Cf. A381962, A381965.

A381963row[n_] := NestWhileList[DigitSum, n, # >= 10 &];
Array[A381963row, 40, 0]

T(n,0) = n and, for k = 1..A031286(n), T(n,k) = A007953(T(n,k-1)).

A381965 Irregular triangle read by rows, where row n lists the iterates of f(x), starting at x = n until f(x) < 10, where f(x) is the multiplicative digital root of x (A031347).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 11, 1, 12, 2, 13, 3, 14, 4, 15, 5, 16, 6, 17, 7, 18, 8, 19, 9, 20, 0, 21, 2, 22, 4, 23, 6, 24, 8, 25, 10, 0, 26, 12, 2, 27, 14, 4, 28, 16, 6, 29, 18, 8, 30, 0, 31, 3, 32, 6, 33, 9, 34, 12, 2, 35, 15, 5, 36, 18, 8, 37, 21, 2

Offset: 0

Paolo Xausa, Mar 11 2025

			Triangle begins:
  n\k|  0   1   2
  ---------------
   0 |  0;
   1 |  1;
   2 |  2;
   3 |  3;
   4 |  4;
   5 |  5;
   6 |  6;
   7 |  7;
   8 |  8;
   9 |  9;
  10 | 10,  0;
  11 | 11,  1;
  12 | 12,  2;
  13 | 13,  3;
  14 | 14,  4;
  15 | 15,  5;
  16 | 16,  6;
  17 | 17,  7;
  18 | 18,  8;
  19 | 19,  9;
  20 | 20,  0;
  21 | 21,  2;
  22 | 22,  4;
  23 | 23,  6;
  24 | 24,  8;
  25 | 25, 10,  0;
  ...

Paolo Xausa, Table of n, a(n) for n = 0..11256 (rows 0..3500 of triangle, flattened).

Cf. A031346 (row lengths - 1), A031347 (right border), A381966 (row sums).

Cf. A381962, A381963.

A381965row[n_] := NestWhileList[Times @@ IntegerDigits[#] &, n, # >= 10 &];
Array[A381965row, 50, 0]

T(n,0) = n and, for k = 1..A031346(n), T(n,k) = A031347(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

cp's OEIS Frontend

A078627 Write n in binary; repeatedly sum the "digits" until reaching 1; a(n) = 1 + number of steps required.

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A381963 Irregular triangle read by rows, where row n lists the iterates of f(x), starting at x = n until f(x) < 10, where f(x) is the digital sum of x (A007953).

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A381965 Irregular triangle read by rows, where row n lists the iterates of f(x), starting at x = n until f(x) < 10, where f(x) is the multiplicative digital root of x (A031347).

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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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Mathematica

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