A258059 - OEIS

%I #92 Nov 17 2015 18:59:51

%S 1,0,0,0,2,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,3,0,0,0,1,0,0,0,1,0,0,0,1,0,

%T 0,0,2,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,2,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,

%U 2,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,4,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,2,0,0,0,1

%N Let n = Sum_{i=0..k} d_i*4^i be the base-4 expansion of n, with 0 <= d_i < 4. Then a(n) = minimal i such that d_i is not 1, or k+1 if there is no such i.

%C This is the "General Ruler Sequence Base 4 Focused at 1" of Webster (2015).

%H N. J. A. Sloane, <a href="/A258059/b258059.txt">Table of n, a(n) for n = 1..10000</a>

%H Richard C. Webster, <a href="https://onedrive.live.com/redir?resid=820CC6AC8786ECC8!26212&amp;authkey=!AJKZJSXac5BuoDc&amp;ithint=file%2cpdf">One Sequence to Rule Them All: The Ruler Sequence and Its Relation to Odd Perfect Numbers and Multiplicative Order</a>, MS Thesis, California State Polytechnic University, Pomona, CA, 2015.

%F Recurrence: a(1)=1; thereafter a(4*n+1) = a(n)+1, a(4*n+j) = 0 for j = 0,2,3. G.f. g(x) = Sum_{k>=0} k * x^((4^k-1)/3) * (1 + x^(2*4^k) + x^(3*4^k))/(1 - x^(4*4^k)) satisfies g(x) = x*g(x^4) + x/(1-x^4). - _Robert Israel_, Jun 08 2015

%e 1 = 0*4+1, so a(1)=1.

%e 7 = 1*4+3, so a(7)=0.

%e 21 = 0*4^3+1*4^2+1*4+1, so a(21)=3.

%e 523 base 10 is 20023 in base 4, so a(523)=0.

%e 1365 base 10 is 111111 in base 4, so a(1365)=6.

%p f:= proc(n)

%p   if n mod 4 = 1 then procname((n-1)/4) + 1 else 0 fi

%p end proc:

%p map(f, [$1..1000]); # _Robert Israel_, Jun 08 2015

%o (PARI) a(n) = {v = Vecrev(digits(n, 4)); for (i=1, #v, if (v[i] != 1, return (i-1));); return(#v);}

%o (Haskell)

%o a258059 = f 0 . a030386_row where

%o    f i [] = i

%o    f i (t:ts) = if t == 1 then f (i + 1) ts else i

%o -- _Reinhard Zumkeller_, Nov 08 2015

%Y The nonzero terms give A263845.

%Y This sequence and A263845 are analogs of the pair of ruler sequences A007814 and A001511.

%Y Cf. A007090, A235127.

%Y Cf. A030386.

%K nonn,easy,base

%O 1,5

%A _Richard C. Webster_, May 17 2015

%E Edited by _N. J. A. Sloane_, Oct 31 2015 and Nov 06 2015.