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%I #24 Aug 26 2020 06:09:22
%S 1,9,42,64,75,255,312,378,525,648,738,1111,1278,2224,2448,2784,2817,
%T 3504,3864,3875,4977,5238,5495,5888,8992,9712,10368,11358,11817,12348,
%U 12875,13136,13584,13775,13832,13944,15351,15384,15744,15900,16912,17768,18095,19344,20448
%N Colombian numbers that are also Bogotá numbers.
%C 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).
%C 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).
%C A336983, A336985, A336986 and this sequence form a partition of the set of positive integers N*.
%H David A. Corneth, <a href="/A336984/b336984.txt">Table of n, a(n) for n = 1..10000</a>
%H Giovanni Resta, <a href="http://www.numbersaplenty.com/set/self_number/">Self or Colombian number</a>, Numbers Aplenty.
%H Puzzling Stackexchange, <a href="https://puzzling.stackexchange.com/questions/98998/pairs-of-bogot%c3%a1-numbers?noredirect=1#comment281441_98998">Pairs of Bogotá Numbers</a>.
%H <a href="https://oeis.org/wiki/Index_to_OEIS:_Section_Coi#Colombian">Index to sequences related to Colombian numbers</a>.
%e 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.
%e 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.
%e 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.
%t 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 *)
%o (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
%Y Intersection of A003052 and A336826.
%Y Cf. A336983 (Bogotá and not Colombian), A336985 (Colombian not Bogotá), A336986 (not Colombian and not Bogotá).
%Y Cf. A336944, A336983, A337208.
%K nonn,base
%O 1,2
%A _Bernard Schott_, Aug 22 2020
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