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A086342
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Smallest number of 1's in binary expansion of any positive multiple of n.
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6
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0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 2, 2, 2, 3, 4, 1, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 3, 2, 4, 5, 1, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 2, 2, 4, 3, 3, 2, 3, 2, 4, 2, 2, 2, 3, 3, 2, 2, 2, 4, 2, 5, 6, 1, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 4, 2, 3, 3, 3, 2, 2, 2, 2, 3, 4, 2, 3, 2, 4, 4, 3, 3, 5, 3, 3, 2, 2, 3, 2, 2, 2, 4, 3, 2
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OFFSET
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0,4
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COMMENTS
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If n is a power of 2 then a(n)=1. All other positive n have a(n)>1. a(n)=2 precisely in cases where some multiple of n is a factor of 2^q+1 for some q.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..10000
Trevor Clokie et al., Computational Aspects of Sturdy and Flimsy Numbers, arxiv preprint arXiv:2002.02731 [cs.DS], February 7 2020.
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FORMULA
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a(2^k-1) = k. - Thomas Dybdahl Ahle, May 01 2013
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EXAMPLE
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a(n)=2 for n=53, 59, 61, 67, 81, 97 and 101 because n divides 2^k+1 for k=26, 29, 30, 33, 27, 24 and 50, respectively. - T. D. Noe, Jul 22 2008
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PROG
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(PARI) a(n)=if(!n, return(0)); n>>=valuation(n, 2); my(o=znorder(Mod(2, n)), v1=Set(powers(Mod(2, n), o)), v=v1, s=1); while(!setsearch(v, Mod(0, n)), v=setbinop((x, y)->x+y, v, v1); s++); s \\ Charles R Greathouse IV, Dec 07 2016
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CROSSREFS
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Cf. A005360 (flimsy numbers), A125121 (sturdy numbers), A143069 (least multiple).
Sequence in context: A305301 A106140 A211097 * A274036 A194449 A261923
Adjacent sequences: A086339 A086340 A086341 * A086343 A086344 A086345
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KEYWORD
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base,nonn
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AUTHOR
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Sean A. Irvine, Sep 02 2003
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EXTENSIONS
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More terms from Robert G. Wilson v, Feb 21 2005
Corrected by T. D. Noe, Jul 22 2008
An incorrect Mathematica program was deleted Aug 01 2008
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STATUS
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approved
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