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%I #22 Feb 07 2024 13:21:43
%S 1,2,3,4,5,6,7,8,9,10,10,1,13,14,15,16,17,18,19,20,21,20,10,2,25,26,
%T 27,28,29,30,31,32,30,21,1,3,37,38,39,40,41,42,43,40,31,20,13,4,49,50,
%U 51,52,53,54,50,41,32,10,14,60,5,62,63,64,65,60,51,42,30,70,2,15,6,74,75
%N a(n) = smallest k such that the commas sequence (cf. A121805) with initial term k contains n.
%C Every k >= 1 appears in this sequence exactly A330128(k) times. So there are 2137453 1's, 194697747222394 2's, 2 3's, 209534289952018960 6's, and so on.
%C a(n) is the most remote ancestor of n in the comma-successor graph.
%H Michael S. Branicky, <a href="/A367366/b367366.txt">Table of n, a(n) for n = 1..10000</a>
%H Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, <a href="http://arxiv.org/abs/2401.14346">arXiv:2401.14346</a>, <a href="https://www.youtube.com/watch?v=_EHAdf6izPI">Youtube</a>
%e All terms n in A121805 have a(n) = 1, all n in A139284 have a(n) = 2, all n in A366492 have a(n) = 4, and so on.
%o (Python)
%o def comma_predecessor(n): # A367614(n)
%o y = int(str(n)[0])
%o x = (n-y)%10
%o k = n - y - 10*x
%o kk = k + 10*x + y-1
%o return k if k > 0 and int(str(kk)[0]) != y-1 else -1
%o def a(n):
%o an = n
%o while (cp:=comma_predecessor(an)) > 0: an = cp
%o return an
%o print([a(n) for n in range(1, 76)]) # _Michael S. Branicky_, Dec 18 2023
%Y Cf. A121805, A139284, A330128, A366492, A367338, A367341, A367617.
%K nonn,base
%O 1,2
%A _N. J. A. Sloane_, Dec 05 2023
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