A328279 Smallest elements of the cycles of (iterations of) A329623: n -> |concat(sum of adjacent digits of n) - n|.

1, 2, 3, 4, 5, 6, 7, 8, 9, 182, 273, 364, 455, 1728, 2637, 3546, 4455, 17182, 26273, 35364, 44455, 171728, 262637, 353546, 444455, 1717182, 2626273, 3535364, 4444455, 17171728, 26262637, 35353546, 44444455, 171717182, 262626273, 353535364, 444444455, 1717171728, 2626262637, 3535353546

Offset: 1

Author

M. F. Hasler, Dec 02 2019

Keywords

• nonn
• base

Comments

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.

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

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.

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.

Examples

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

Crossrefs

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

Programs

PARI

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

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.

Formula

Conjectures from Colin Barker, Dec 05 2019: (Start)

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)).

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

(End)