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A078627
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Write n in binary; repeatedly sum the "digits" until reaching 1; a(n) = 1 + number of steps required.
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3
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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
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1,2
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COMMENTS
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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};
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.)
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MAPLE
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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);
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MATHEMATICA
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Table[Length[NestWhileList[Total[IntegerDigits[#, 2]]&, n, #>1&]], {n, 110}] (* Harvey P. Dale, Oct 10 2011 *)
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PROG
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(PARI) A078627(n) = { my(k=1); while(n>1, n = hammingweight(n); k += 1); (k); }; \\ Antti Karttunen, Jul 09 2017
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CROSSREFS
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KEYWORD
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base,easy,nonn
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AUTHOR
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Frank Schwellinger (nummer_eins(AT)web.de), Dec 12 2002
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EXTENSIONS
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STATUS
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approved
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