A336984 Colombian numbers that are also Bogotá numbers.

1, 9, 42, 64, 75, 255, 312, 378, 525, 648, 738, 1111, 1278, 2224, 2448, 2784, 2817, 3504, 3864, 3875, 4977, 5238, 5495, 5888, 8992, 9712, 10368, 11358, 11817, 12348, 12875, 13136, 13584, 13775, 13832, 13944, 15351, 15384, 15744, 15900, 16912, 17768, 18095, 19344, 20448

Offset

1,2

Comments

Equivalently, numbers m that are not of the form k + sum of digits of k for any k (A003052), but are of the form q * product of digits of q for some q (A336826).

Repunits are trivially Bogotá numbers but the indices m of the repunits R_m that are Colombian numbers are in A337208. No known prime belongs to this sequence (see A004023).

A336983, A336985, A336986 and this sequence form a partition of the set of positive integers N*.

Links

David A. Corneth, Table of n, a(n) for n = 1..10000

Giovanni Resta, Self or Colombian number, Numbers Aplenty.

Puzzling Stackexchange, Pairs of Bogotá Numbers.

Index to sequences related to Colombian numbers.

Example

42 = 21 * (2*1) is a Bogotá number and there does not exist m < 42 such that 42 = m + sum of digits of m, hence 42 is a Colombian number and 42 is a term.
56 = 14 * (1*4) is a Bogotá number but as 56 = 46 + (4+6), 56 is not a Colombian number, hence 56 is not a term.
648 = 36 * (3*6) = 81 * (8*1) but there does not exist m < 648 such that 648 = m + sum of digits of m, hence 648 is a Colombian number, so 648 is a term that has two different representations as the product of a number and of its decimal digits.

Mathematica

m = 21000; Intersection[Complement[Range[m], Select[Union[Table[n + Plus @@ IntegerDigits[n], {n, 1, m}]], # <= m &]], Select[Union[Table[n * Times @@ IntegerDigits[n], {n, 1, m}]], # <= m &]] (* Amiram Eldar, Aug 22 2020 *)

Prog

(PARI) lista(nn) = Vec(setintersect(setminus([1..nn], Set(vector(nn, k, k+sumdigits(k)))), Set(vector(nn, k, k*vecprod(digits(k)))))); \\ Michel Marcus, Aug 23 2020

Crossrefs

Intersection of A003052 and A336826.

Cf. A336983 (Bogotá and not Colombian), A336985 (Colombian not Bogotá), A336986 (not Colombian and not Bogotá).

Cf. A336944, A336983, A337208.

Sequence in context: A065792 A118546 A334147 * A075233 A062783 A172464

Adjacent sequences: A336981 A336982 A336983 * A336985 A336986 A336987

Keyword

nonn,base

Author

Bernard Schott, Aug 22 2020

Status

approved