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A350833 Run lengths of even terms in A350877 (half if even, add next prime if odd).

%I A350833 #16 Jan 24 2022 16:15:00
%S A350833 0,1,3,3,2,4,1,2,1,2,1,1,1,1,1,5,1,2,1,2,1,1,2,1,5,1,2,1,1,2,1,2,6,1,
%T A350833 2,1,2,2,5,2,5,2,1,3,2,1,2,2,4,3,3,4,1,2,5,1,1,1,1,1,3,1,3,2,1,3,2,2,
%U A350833 3,1,2,1,3,1,2,1,4,1,2,2,4,3,3,1,1,1,1,1,2,4,1,1,1,1,1,1,3,1,1,2,2,1,2,2,2,1,2
%N A350833 Run lengths of even terms in A350877 (half if even, add next prime if odd).
%C A350833 Equals first differences of indices of odd terms (A350616) minus one.
%C A350833 After the initial 0, also equals the 2-valuation (A007814) of A350618, the terms following odd terms in A350877.
%C A350833 Record values are a(1) = 0, a(2) = 1, a(3) = 3, a(6) = 4, a(16) = 5, a(33) = 6, a(146) = 7, a(243) = 11, a(1596) = 12, a(2092) = 13, ... The first occurrences of the other values are: a(5) = 2, a(8) = 510, a(9) = 667, a(10) = 1526. No others up to n = 33000. - _M. F. Hasler_, Jan 24 2022
%H A350833 M. F. Hasler, <a href="/A350833/b350833.txt">Table of n, a(n) for n = 1..10000</a>
%F A350833 a(n) = A350616(n+1) - A350616(n) - 1 = A007814(A350618(n-1)) for n > 1.
%e A350833 Between the first odd term, A350877(1) = 1, and second odd term, A350877(2) = 3, there are no even terms, therefore a(1) = 0.
%e A350833 Between the second and third odd term, A350877(4) = 3, there is one even term, A350877(3) = 6, therefore a(2) = 1.
%o A350833 (PARI) A350833_upto(N)={N=A350616_upto(N);[N[k]-N[k-1]-1|k<-[2..#N]]}
%Y A350833 Cf. A007814, A350877, A350616, A350618.
%K A350833 nonn
%O A350833 1,3
%A A350833 _M. F. Hasler_, Jan 23 2022