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Smallest elements of the cycles of (iterations of) A329623: n -> |concat(sum of adjacent digits of n) - n|.
2

%I #46 Jan 14 2021 21:17:55

%S 1,2,3,4,5,6,7,8,9,182,273,364,455,1728,2637,3546,4455,17182,26273,

%T 35364,44455,171728,262637,353546,444455,1717182,2626273,3535364,

%U 4444455,17171728,26262637,35353546,44444455,171717182,262626273,353535364,444444455,1717171728,2626262637,3535353546

%N Smallest elements of the cycles of (iterations of) A329623: n -> |concat(sum of adjacent digits of n) - n|.

%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.

%C Differs from A328142 = range of A328865 \ {-1} in that only the smallest element of a cycle is listed here.

%C Infinite subsequences include: 1{71}*82, 2{62}*73, 3{53}*64, 4+55, {17}+28, {26}+37 and {35}+46, where "*" (resp. "+") means 0 (resp. 1) or more occurrences. As long as there are no other terms, we have a simple formula for a(n), cf. PARI code.

%C So far the known fixed points of A329623 are the single-digit numbers and numbers of the form 4...455: {4*(10^k-1)/9 + 11; k >= 3}. All other known terms of this sequence and A328142 are part of 2-cycles, i.e., A329623(A329623(a(n))) = a(n) for all n. No other cycles are known so far.

%F Conjectures from _Colin Barker_, Dec 05 2019: (Start)

%F G.f.: x*(1 + x + x^2 + x^3 - 8*x^4 - 8*x^5 - 8*x^6 - 8*x^7 - 18*x^8 + 154*x^9 + 72*x^10 + 72*x^11 + 72*x^12 - 294*x^13 + 80*x^14 + 80*x^15 + 80*x^16 - 460*x^17) / ((1 - x)*(1 + x^4)*(1 - 10*x^4)).

%F a(n) = a(n-1) + 9*a(n-4) - 9*a(n-5) + 10*a(n-8) - 10*a(n-9) for n>17.

%F (End)

%e The single-digit numbers 1, ..., 9 as well as numbers f(k) = 4*(10^k-1)/9 + 11, k >= 3, are fixed points of A329623.

%e Indeed, A053392(f(k)) = A053392(4...455) = 8...8910 = 8*(10^k-1)/9 + 22 = 2*f(k), and therefore A329623(f(k)) = A053392(f(k)) - f(k) = f(k).

%e For a(10) = 182, we have A329623(182) = 728 and A329623(728) = 182, so this is the smallest member of the 2-cycle (182, 728).

%e For a(11) = 273, we have A329623(273) = 637 and A329623(637) = 273, so this is the smallest member of the 2-cycle (273, 637).

%e Similarly for all subsequent terms except the f(k) of the form 4...455.

%o (PARI) is_A328279(n)={n==vecmin(vector(9,i,n=A329623(n)))}

%o (PARI) apply( A328279(n)={if(n<10,n, bittest((n=divrem(n-10,4)+[1,2]~)[1],0), (n[2]*9-1)*10^(n[1]-1)\99*1000+n[2]*91, (-1+n[2]*=9)*10^n[1]\99*100+10+n[2])}, [1..40]) \\ Valid as long as there is no other term > 9 than those of the 7 infinite subfamilies mentioned in the comment.

%Y Cf. A328142, A329623, A053392 (concatenate sums of adjacent digits of n), A328865.

%K nonn,base

%O 1,2

%A _M. F. Hasler_, Dec 02 2019