A328142 Elements of cycles for iterations of A329623: n -> |n - concat(sum of adjacent digits of n)|.

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

Offset: 1

Author

M. F. Hasler, Dec 02 2019

Keywords

• nonn
• base

Comments

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.

Examples

			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.
		

Crossrefs

Cf. A329623, A053392 (concatenate sums of adjacent digits of n), A328865, A329624.

See A328279 for the smallest representative of each cycle.

Programs

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.