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A328142
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Elements of cycles for iterations of A329623: n -> |n - concat(sum of adjacent digits of n)|.
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5
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1, 2, 3, 4, 5, 6, 7, 8, 9, 182, 273, 364, 455, 546, 637, 728, 1728, 2637, 3546, 4455, 5364, 6273, 7182, 17182, 26273, 35364, 44455, 53546, 62637, 71728, 171728, 262637, 353546, 444455, 535364, 626273, 717182, 1717182, 2626273, 3535364, 4444455, 5353546, 6262637, 7171728
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OFFSET
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1,2
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COMMENTS
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Equivalently: range of A328865 \ {-1}.
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.
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.
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.
Sequence A328279 lists the smallest member of each cycle.
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LINKS
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EXAMPLE
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The single-digit numbers 1, ..., 9 and the numbers f(k) = 4*(10^k-1)/9 + 11, k >= 3, are fixed points of A329623.
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).
For a(10) = 182, we have A329623(182) = 728 and A329623(728) = 182, so these are members of the 2-cycle (182, 728).
For a(11) = 273, we have A329623(273) = 637 and A329623(637) = 273, so these are members of the 2-cycle (273, 637).
Similarly for all subsequent terms except the f(k) of the form 4...455.
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PROG
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(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.
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CROSSREFS
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See A328279 for the smallest representative of each cycle.
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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