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Length of runs of consecutive numbers in A367090, i.e., size of gaps in the set of sums of distinct powers of 3 and distinct powers of 4.
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%I #14 Dec 15 2023 16:30:38

%S 2,2,36,36,2,2,23,2,2,36,36,2,2,23,2,2,36,2,2,36,2,2,23,2,2,36,36,2,2,

%T 23,2,2,2,2,2,2,23,2,2,36,36,2,2,23,2,2,36,2,2,14,14,2,2,36,2,2,23,2,

%U 2,36,36,2,2,23,2,2,2,2,2,2,23,2,2,36,36,2,2,23,2,2,36,2,2,36

%N Length of runs of consecutive numbers in A367090, i.e., size of gaps in the set of sums of distinct powers of 3 and distinct powers of 4.

%C The numbers that occur in this sequence are, in order of first appearance: 2, 36, 23, 14, 1081, 20, ... It is not known which numbers will eventually appear and which numbers will never occur in this sequence.

%C The first 1's (which correspond to isolated numbers in A367090, or gaps that are a singleton) appear as a(131) = a(132) = 1.

%C This set exhibits an interesting self-similar, pseudo-symmetric structure. This is due to the Proposition given in A367090.

%H Hugo Pfoertner, <a href="/A367091/b367091.txt">Table of n, a(n) for n = 1..10000</a>

%e Sequence A367090 (= numbers that are not the sum of distinct powers of 3 or 4) starts (62, 63, 143, 144, 207, 208, 209, 210, ...), so the first two runs of consecutive terms are 2 = #{62, 63} and 2 = #{143, 144}, the next run is of length 36.

%o (PARI) D(v)=v[^1]-v[^-1] \\ first differences

%o A367091_upto(N, DA=D(A367090_upto(N)))= D([ k | k<-[0..#DA], !k|| DA[k]-1 ])

%Y Cf. A367090; A005836 and A000695 (sums of distinct powers of 3 resp. 4).

%K nonn

%O 1,1

%A _M. F. Hasler_, Nov 08 2023