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%I #42 Oct 18 2023 09:23:07
%S 0,0,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,
%T 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,
%U 49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75
%N a(n) = the greatest non-divisor h of n (1 < h < n), or 0 if no such h exists.
%C From _Paul Curtz_, Feb 09 2015: (Start)
%C The nonnegative numbers with 0 instead of 1. See A254667(n), which is linked to the Bernoulli numbers A164555(n)/A027642(n), an autosequence of the second kind.
%C Offset 0 could be chosen.
%C An autosequence of the second kind is a sequence whose main diagonal is the first upper diagonal multiplied by 2. If the first upper diagonal is
%C s0, s1, s2, s3, s4, s5, ...,
%C the sequence is
%C Ssk(n) = 2*s0, s0, s0 + 2*s1, s0 +3*s1, s0 + 4*s1 + 2*s2, s1 + 5*s1 + 5*s2, etc.
%C The corresponding coefficients are A034807(n), a companion to A011973(n).
%C The binomial transform of Ssk(n) is (-1)^n*Ssk(n).
%C Difference table of a(n):
%C 0, 0, 2, 3, 4, 5, 6, 7, ...
%C 0, 2, 1, 1, 1, 1, 1, ...
%C 2, -1, 0, 0, 0, 0 ...
%C -3, 1, 0, 0, 0, ...
%C 4, -1, 0, 0, ...
%C -5, 1, 0, ...
%C 6, -1, ...
%C 7, ...
%C etc.
%C a(n) is an autosequence of the second kind. See A054977(n).
%C The corresponding autosequence of the first kind (a companion) is 0, 0 followed by the nonnegative numbers (A001477(n)). Not in the OEIS.
%C Ssk(n) = 2*Sfk(n+1) - Sfk(n) where Sfk(n) is the corresponding sequence of the first kind (see A254667(n)).
%C (End)
%C Number of binary sequences of length n-1 that contain exactly one 0 and at least one 1. - _Enrique Navarrete_, May 11 2021
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F a(n) = n-1 for n >= 3.
%F E.g.f.: 1-x^2/2+(x-1)*exp(x). - _Enrique Navarrete_, May 11 2021
%t Join[{0,0},Table[Max[Complement[Range[n],Divisors[n]]],{n,3,70}]] (* or *) Join[{0,0},Range[2,70]] (* _Harvey P. Dale_, May 31 2014 *)
%o (PARI) if(n>2,n-1,0) \\ _Charles R Greathouse IV_, Sep 02 2015
%Y Cf. A199968 (the smallest non-divisor h of n (1<h<n)), A199970. A001477, A011973, A034807, A054977, A254667.
%Y Cf. A007978.
%Y Essentially the same as A000027, A028310, A087156 etc.
%K nonn,easy,less
%O 1,3
%A _Jaroslav Krizek_, Nov 26 2011
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