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%I #43 Sep 23 2021 08:03:29
%S 0,1,2,3,4,5,6,7,8,9,10,11,12,16,15,13,14,17,18,21,19,20,22,23,24,36,
%T 55,25,26,27,28,29,30,42,54,66,78,90,144,259,45,33,31,32,34,35,37,38,
%U 39,40,50,43,41,44,46,47,48,76,64,63,49,51,52,53,56,57,58,59,60,108,172
%N a(0) = 0; a(1) = 1; for n > 1, a(n) = A001065(a(n-1)) = sigma(a(n-1)) - a(n-1) (the sum of aliquot parts of a(n-1)) if this is not yet in the sequence; otherwise a(n) is the smallest number missing from the sequence.
%C This sequence is a permutation of the nonnegative integers iff Catalan's aliquot sequence conjecture (also called Catalan-Dickson conjecture) is true.
%C a(563) = 276 is the smallest number whose aliquot sequence has not yet been fully determined.
%C As long as the aliquot sequence of 276 is not known to be finite or eventually periodic, a(563+k) = A008892(k).
%H Eric Chen, <a href="/A347769/b347769.txt">Table of n, a(n) for n = 0..2703</a> (all currently known terms, after the b-file of A008892)
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AliquotSequence.html">Aliquot sequence</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CatalansAliquotSequenceConjecture.html">Catalan's Aliquot Sequence Conjecture</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Aliquot_sequence">Aliquot sequence</a>
%e a(0) = 0, a(1) = 1;
%e since A001065(a(1)) = 0 has already appeared in this sequence, a(2) = 2;
%e since A001065(a(2)) = 1 has already appeared in this sequence, a(3) = 3;
%e ...
%e a(11) = 11;
%e since A001065(a(11)) = 1 has already appeared in this sequence, a(12) = 12;
%e since A001065(a(12)) = 16 has not yet appeared in this sequence, a(13) = A001065(a(12)) = 16;
%e since A001065(a(13)) = 15 has not yet appeared in this sequence, a(14) = A001065(a(13)) = 15;
%e since A001065(a(14)) = 9 has already appeared in this sequence, a(15) = 13;
%e ...
%o (PARI) A347769_list(N)=print1(0, ", "); if(N>0, print1(1, ", ")); v=[0, 1]; b=1; for(n=2, N, if(setsearch(Set(v), sigma(b)-b), k=1; while(k<=n, if(!setsearch(Set(v), k), b=k; k=n+1, k++)), b=sigma(b)-b); print1(b, ", "); v=concat(v, b))
%Y Cf. A032451.
%Y Cf. A001065 (sum of aliquot parts).
%Y Cf. A003023, A044050, A098007, A098008: ("length" of aliquot sequences, four versions).
%Y Cf. A007906.
%Y Cf. A115060 (maximum term of aliquot sequences).
%Y Cf. A115350 (termination of the aliquot sequences).
%Y Cf. A098009, A098010 (records of "length" of aliquot sequences).
%Y Cf. A290141, A290142 (records of maximum term of aliquot sequences).
%Y Cf. A000396, A063990, A122726, A206708, A347770.
%Y Cf. A080907, A063769, A121507, A121508, A131884, A126016.
%Y Aliquot sequences starting at various numbers: A000004 (0), A000007 (1), A033322 (2), A010722 (6), A143090 (12), A143645 (24), A010867 (28), A008885 (30), A143721 (38), A008886 (42), A143722 (48), A143723 (52), A008887 (60), A143733 (62), A143737 (68), A143741 (72), A143754 (75), A143755 (80), A143756 (81), A143757 (82), A143758 (84), A143759 (86), A143767 (87), A143846 (88), A143847 (96), A143919 (100), A008888 (138), A008889 (150), A008890 (168), A008891 (180), A203777 (220), A008892 (276), A014360 (552), A014361 (564), A074907 (570), A014362 (660), A269542 (702), A045477 (840), A014363 (966), A014364 (1074), A014365 (1134), A074906 (1521), A143930 (3630), A072891 (12496), A072890 (14316), A171103 (46758), A072892 (1264460).
%K nonn
%O 0,3
%A _Eric Chen_, Sep 13 2021
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