



41, 43, 59, 86, 88, 91, 113, 118, 123, 135, 155, 172, 176, 177, 182, 185, 209, 215, 226, 236, 239, 248, 261, 267, 270, 273, 275, 279, 307, 310, 311, 337, 339, 344, 347, 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
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OFFSET

1,1


COMMENTS

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


REFERENCES

V. Dimitrov, G. Jullien, R. Muscedere, Multiple Number Base System Theory and Applications, 2nd ed., CRC Press, 2012, pp. 3539.


LINKS



EXAMPLE

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}.
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}.


MATHEMATICA

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 *)


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



