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A262743 Predestined numbers: every n number is generated by at least one pair of products, such as n = a*b = c*d, and the multiset of the digits of a and b coincides with the multiset of the digits of c and d.

%I A262743 #36 Oct 09 2015 17:53:56
%S A262743 64,95,130,242,325,326,392,396,435,504,544,552,572,585,632,644,664,
%T A262743 693,740,748,756,762,770,784,806,868,952,968,973,986,990,995,1008
%N A262743 Predestined numbers: every n number is generated by at least one pair of products, such as n = a*b = c*d, and the multiset of the digits of a and b coincides with the multiset of the digits of c and d.
%C A262743 For each pair of products, no more than 1 of the 4 numbers involved may have a trailing zero. E.g., 20 = 1*20 = 2*10 is trivial. Even 3920 = 49*80 = 4*980 is not valid (but note that 392 = 4*98 = 8*49 is a valid term). This rule is similar to that of the vampire numbers (A014575), and prevents trivial proliferations.
%C A262743 The name recalls the "genetic" metaphor of such numbers, that even if genes/digits are recombined, they remain with the same "destiny".
%C A262743 This sequence looks similar to A048936 (a subset of A014575, vampire numbers) but it's not the same because the factors' digits are exclusively the same as those of n, and also see, e.g., 11930170 = 1301*9170 = 1310*9107, which is not a valid predestined number.
%D A262743 Francesco Di Matteo, Sequenze ludiche, Game Edizioni (2015), pages 28-37.
%H A262743 Francesco Di Matteo, <a href="/A262743/b262743.txt">Table of n, a(n) for n = 1..2815</a>
%H A262743 F. Di Matteo and A. Marchini, <a href="/A262743/a262743.txt">All the first 2815 terms calculated</a>.
%e A262743 64  = 1*64 = 4*16;
%e A262743 95  = 1*95 = 5*19;
%e A262743 130 = 2*65 = 5*26;
%e A262743 242 = 2*121 = 11*22, etc.
%t A262743 good[w_] := Block[{L = {}}, Do[If[Length[Select[Join[w[[i]], w[[j]]], Mod[#, 10] == 0 &]] <= 1, AppendTo[L, {w[[i]], w[[j]]}]], {i, Length@w}, {j, i - 1}]; L]; prQ[n_] := Block[{t, d = Select[Divisors@n, #^2 <= n &]}, t = (Last /@ #) & /@ Select[SplitBy[ Sort@ Table[{ Sort@ Join[ IntegerDigits@ e, IntegerDigits [n/e]], {e, n/e}}, {e, d}], First], Length[#] > 1 &]; g = Select[good /@ t, # != {} &]; g != {}]; (* then *) Select[Range[1000], prQ] (* or *) Do[If[prQ@ n, Print[n," ", Flatten[g, 1]]], {n, 10^5}] (* _Giovanni Resta_, Oct 07 2015 *)
%Y A262743 Cf. A245386, A262873.
%K A262743 nonn,base
%O A262743 1,1
%A A262743 _Francesco Di Matteo_, Sep 29 2015