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

Author

Michael De Vlieger, Sep 27 2016

Keywords

• nonn

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, ...}

Examples

			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}.
		

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

Crossrefs

Cf. A003586, A237442, A276380, A277070.

Programs

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 *)