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A277070 Row length of A276380(n).

%I A277070 #19 Sep 15 2022 02:18:03
%S A277070 1,1,1,1,2,1,2,1,1,2,2,1,2,2,2,1,2,1,2,2,2,2,3,1,2,2,1,2,2,2,2,1,2,2,
%T A277070 2,1,2,2,2,2,3,2,3,2,2,3,3,1,2,2,2,2,3,1,2,2,2,2,3,2,3,2,2,1,2,2,2,2,
%U A277070 3,2,3,1,2,2,2,2,3,2,3,2,1,2,2,2,2,3,2,3,2,2,3,3,2,3,3,1,2,2,2,2
%N A277070 Row length of A276380(n).
%C A277070 a(n) represents the partition size generated by greedy algorithm at A276380(n) such that all parts k are unique and in A003586.
%C A277070 See A276380 for further comments about the greedy algorithm.
%C A277070 Row n = 1 for n that are in A003586.
%C A277070 A237442(n) represents the smallest possible partition size such that all k are distinct and in A003586. The reference defines the "canonic" representation of n in the "dual-base number system", i.e., base(2,3), essentially as those which have length A237442(n).
%C A277070 a(n) differs from A237442(n) at n = 41, 43, 59, 86, 88, 91, 113, 118, 123, 135, 155, 172, 176, 177, 182, 185, 209, 215, 226, 236, 239, 248, ... (i.e., A277071).
%D A277070 V. Dimitrov, G. Jullien, and R. Muscedere, Multiple Number Base System Theory and Applications, 2nd ed., CRC Press, 2012, pp. 35-39.
%H A277070 Michael De Vlieger, <a href="/A277070/b277070.txt">Table of n, a(n) for n = 1..10000</a>
%e A277070 a(n) Terms k in row n of A276380:
%e A277070 1    1
%e A277070 1    2
%e A277070 1    3
%e A277070 1    4
%e A277070 2    1,4
%e A277070 1    6
%e A277070 2    1,6
%e A277070 1    8
%e A277070 1    9
%e A277070 2    1,9
%e A277070 2    2,9
%e A277070 1    12
%e A277070 2    1,12
%e A277070 2    2,12
%e A277070 2    3,12
%e A277070 1    16
%e A277070 2    1,16
%e A277070 1    18
%e A277070 2    1,18
%e A277070 2    2,18
%e A277070 2    3,18
%e A277070 2    4,18
%e A277070 3    1,4,18
%e A277070 ...
%e A277070 a(41) = 3 since A276380(41) = {1,4,36}, but {9,32} is the shortest possible partition of 41 such that all terms are distinct and in A003586.
%e A277070 a(88) = 3 since A276380(88) = {1,6,81}, but {16,72} and {24,64} are shorter and have A237442(88) = 2 terms.
%t A277070 Table[Length@ DeleteCases[Append[Abs@ Differences@ #, Last@ #], k_ /; k == 0] &@ NestWhileList[# - SelectFirst[# - Range[0, # - 1], Block[{m = #, n = 6}, While[And[m != 1, ! CoprimeQ[m, n]], n = GCD[m, n]; m = m/n]; m == 1] &] &, n, # > 1 &], {n, 100}]
%o A277070 (Python)
%o A277070 from itertools import count, takewhile
%o A277070 N = 100
%o A277070 def B(p): return list(takewhile(lambda x: x<=N, (p**i for i in count(0))))
%o A277070 B23set = set(b*t for b in B(2) for t in B(3) if b*t <= N)
%o A277070 B23lst = sorted(B23set, reverse=True)
%o A277070 def a(n):
%o A277070     if n in B23set: return 1
%o A277070     big = next(t for t in B23lst if t <= n)
%o A277070     return a(n - big) + 1
%o A277070 print([a(n) for n in range(1, N+1)]) # _Michael S. Branicky_, Sep 14 2022
%Y A277070 Cf. A003586, A237442, A276380, A277071.
%K A277070 nonn
%O A277070 1,5
%A A277070 _Michael De Vlieger_, Sep 27 2016