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Numbers n for which A277070(n) does not equal A237442(n).
4

%I #9 Oct 01 2016 12:17:15

%S 41,43,59,86,88,91,113,118,123,135,155,172,176,177,182,185,209,215,

%T 226,236,239,248,261,267,270,273,275,279,307,310,311,337,339,344,347,

%U 352,354,364,365,367,369,370,371,377,383,405,407,418,425,427,430,452,455,465,472,473,475,478,479,496,499

%N Numbers n for which A277070(n) does not equal A237442(n).

%C These are numbers n for which the greedy algorithm A276380(n) produces a partition of n with more than A237442(n) terms that are all unique and in A003586.

%C A276380(n) = A237442(n) if n is in A003586. There may be more than one partition of n that has terms that are unique and in A003586. The first n in a(n) with that quality is n = 88.

%C A277070(n)-A237442(n) = 1 at {41, 43, 59, 86, 88, 91, 113, 118, ...}

%C A277070(n)-A237442(n) = 2 at {279, 371, 558, 837, 1116, 1240, 1267, ...}

%C A277070(n)-A237442(n) = 3 at {2777, 5554, ...}

%D V. Dimitrov, G. Jullien, R. Muscedere, Multiple Number Base System Theory and Applications, 2nd ed., CRC Press, 2012, pp. 35-39.

%H Michael De Vlieger, <a href="/A277071/b277071.txt">Table of n, a(n) for n = 1..3000</a>

%e 41 is in the sequence because A276380(41) = {1,4,36}, thus A277070(41) = 3, but A237442(41) = 2. The partition of 41 with unique terms that are all in A003586 is {9,32}.

%e 88 is in the sequence because A276380(88) = {1,6,81}, thus A277070(88) = 3, but A237442(41) = 2. There are 2 partitions of 88 with unique terms that are all in A003586: {16,72} and {24,64}.

%t f[n_] := Length@ DeleteCases[Append[Abs@ Differences@ #, Last@ #], k_ /; k == 0] &@ NestWhileList[# - SelectFirst[# - Range[0, # - 1], Module[{a = #, b = 6}, While[And[a != 1, ! CoprimeQ[a, b]], b = GCD[a, b]; a = a/b]; a == 1] &] &, n, # > 1 &]; g[n_] := Block[{p = Select[Range@ n, FactorInteger[#][[-1, 1]] < 4 &], k = 1}, While[{} == Quiet@ IntegerPartitions[n, {k}, p, 1], k++]; k]; Select[Range@ 500, f@ # != g@ # &] (* function g after _Giovanni Resta_ at A237442 *)

%Y Cf. A003586, A237442, A276380, A277070.

%K nonn

%O 1,1

%A _Michael De Vlieger_, Sep 27 2016