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 A361511 a(1) = 1. Thereafter if a(n-1) is a novel term, a(n) = d(a(n-1)); otherwise, if a(n-1) is the t-th non-novel term, a(n) = a(n-1) + d(a(t)), where d is the divisor function A000005. 8
 1, 1, 2, 2, 3, 2, 4, 3, 5, 2, 4, 6, 4, 7, 2, 4, 6, 8, 4, 7, 11, 2, 5, 7, 9, 3, 6, 10, 4, 8, 11, 13, 2, 4, 6, 8, 10, 13, 15, 4, 8, 12, 6, 9, 13, 15, 17, 2, 4, 7, 11, 15, 19, 2, 4, 8, 11, 15, 21, 4, 8, 11, 13, 17, 19, 21, 24, 8, 10, 12, 16, 5, 7, 9, 12, 16, 18, 6, 10, 14, 4, 7, 11, 13, 15, 17, 19, 23 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Inspired by A360179, but uses a simpler rule for non-novel terms. It is an obvious conjecture that every number eventually appears, but is there a proof? LINKS Michael De Vlieger, Table of n, a(n) for n = 1..40000 Michael De Vlieger, Scatterplot of a(n), n = 1..2^16, showing records in red, smallest missing numbers in blue (small until they enter sequence, then large), terms deriving from novel predecessors in gold, otherwise green. Michael De Vlieger, Plot that shows the increasing subsequences that form the rows when the sequence is regarded as an irregular triangle EXAMPLE The initial terms (in the third column, N = novel term, D = non-novel term): .n.a(n).....t .1,..1,.N, .2,..1,.D,..1 .3,..2,.N, .4,..2,.D,..2 .5,..3,.N, .6,..2,.D,..3 .7,..4,.N, .8,..3,.D,..4 .9,..5,.N, 10,..2,.D,..5 11,..4,.D,..6 12,..6,.N, 13,..4,.D,..7 14,..7,.N, 15,..2,.D,..8 16,..4,.D,..9 17,..6,.D,.10 18,..8,.N, 19,..4,.D,.11 20,..7,.D,.12 21,.11,.N, 22,..2,.D,.13 ... If n=8, for example, a(8) = 3 is a non-novel term, the 4th such, so a(9) = a(8) + d(a(4)) = 3 + d(2) = 5. Comment from Michael De Vlieger, Apr 08 2023 (Start) Can be read as an irregular triangle of increasing subsequences: 1; 1, 2; 2, 3; 2, 4; 3, 5; 2, 4, 6; 4, 7; 2, 4, 6, 8; 4, 7, 11; 2, 5, 7, 9; 3, 6, 10; 4, 8, 11, 13; 2, 4, 6, 8, 10, 13, 15; 4, 8, 12; 6, 9, 13, 15, 17; 2, 4, 7, 11, 15, 19; etc. (End) The rows end with the novel terms - see A361512, A361513 - and their lengths are given by A361514. MATHEMATICA nn = 120; c[_] = False; f[n_] := DivisorSigma[0, n]; a[1] = m = 1; Do[(If[c[#], a[n] = # + f[a[m]] ; m++, a[n] = f[#] ]; c[#] = True) &[a[n - 1]], {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Apr 08 2023 *) CROSSREFS Cf. A000005, A360179, A361512-A361516, A362095. Sequence in context: A246029 A245564 A360179 * A345147 A214126 A205378 Adjacent sequences: A361508 A361509 A361510 * A361512 A361513 A361514 KEYWORD nonn,tabf AUTHOR N. J. A. Sloane, Apr 08 2023 STATUS approved

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Last modified May 29 03:48 EDT 2024. Contains 372921 sequences. (Running on oeis4.)