

A078627


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


3



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
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OFFSET

1,2


COMMENTS

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(m1)  1 {if m > 2};


LINKS

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


FORMULA

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


EXAMPLE

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.)


MAPLE

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);


MATHEMATICA

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


PROG

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


CROSSREFS

Cf. A000120.
One more than A180094.
Sequence in context: A129759 A137467 A261461 * A106370 A128630 A273040
Adjacent sequences: A078624 A078625 A078626 * A078628 A078629 A078630


KEYWORD

base,easy,nonn


AUTHOR

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


EXTENSIONS

Description corrected by Antti Karttunen, Jul 09 2017


STATUS

approved



