A277071 Numbers n for which A277070(n) does not equal A237442(n).

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

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) = 1 at {41, 43, 59, 86, 88, 91, 113, 118, ...}

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

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

References

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

Links

Michael De Vlieger, Table of n, a(n) for n = 1..3000

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

Cf. A003586, A237442, A276380, A277070.

Sequence in context: A007643 A259552 A290365 * A186401 A180548 A157194

Adjacent sequences: A277068 A277069 A277070 * A277072 A277073 A277074

Keyword

nonn

Author

Michael De Vlieger, Sep 27 2016

Status

approved