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%I #52 Jan 14 2021 21:17:49
%S 1,2,3,4,5,6,7,8,9,182,273,364,455,546,637,728,1728,2637,3546,4455,
%T 5364,6273,7182,17182,26273,35364,44455,53546,62637,71728,171728,
%U 262637,353546,444455,535364,626273,717182,1717182,2626273,3535364,4444455,5353546,6262637,7171728
%N Elements of cycles for iterations of A329623: n -> |n - concat(sum of adjacent digits of n)|.
%C Equivalently: range of A328865 \ {-1}.
%C By a k-cycle (or cycle of length k) of A329623 we mean a vector (x_1, ..., x_k) such that A329623(x_i) = x_{i+1} for i < k, and A329623(x_k) = x_1. We include here the cycles of length k = 1 which are the fixed points of A329623, viz A329623(x_1) = x_1. No cycle with k > 2 is known.
%C There are 7 infinite subsequences: for initial digit 1 <= d <= 7, alternate digit d and 8-d to form an undulating (A033619) number of arbitrary length L >= 3, then add 11.
%C The terms with initial digit d > 4 are the larger member of a 2-cycle having a term with d < 4 as smaller member. The terms with d = 4 (and those <= 9) are fixed points. So far no other fixed points or other cycles are known. As far as this remains valid, the terms of this sequence are characterized by A329623(A329623(x)) = x.
%C Sequence A328279 lists the smallest member of each cycle.
%e The single-digit numbers 1, ..., 9 and the numbers f(k) = 4*(10^k-1)/9 + 11, k >= 3, are fixed points of A329623.
%e Indeed, for f(k) = 4...455 we have A053392(f(k)) = 8...910 = 2*f(k), so A329623(f(k)) = 2*f(k) - f(k) = f(k).
%e For a(10) = 182, we have A329623(182) = 728 and A329623(728) = 182, so these are members of the 2-cycle (182, 728).
%e For a(11) = 273, we have A329623(273) = 637 and A329623(637) = 273, so these are members of the 2-cycle (273, 637).
%e Similarly for all subsequent terms except the f(k) of the form 4...455.
%o (PARI) apply( {A328142(n)=if(n>9,fromdigits(vector((n+8)\/7,i,n=if(i>1, 8-n,(n+4)%7+1)))+11,n)}, [1..40]) \\ As far as there are no other terms than those described in COMMENTS. - _M. F. Hasler_, Dec 06 2019, replacing earlier code.
%Y Cf. A329623, A053392 (concatenate sums of adjacent digits of n), A328865, A329624.
%Y See A328279 for the smallest representative of each cycle.
%K nonn,base
%O 1,2
%A _M. F. Hasler_, Dec 02 2019
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