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.

64, 95, 130, 242, 325, 326, 392, 396, 435, 504, 544, 552, 572, 585, 632, 644, 664, 693, 740, 748, 756, 762, 770, 784, 806, 868, 952, 968, 973, 986, 990, 995, 1008

Offset: 1

Author

Francesco Di Matteo, Sep 29 2015

Keywords

• nonn
• base

Comments

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.

The name recalls the "genetic" metaphor of such numbers, that even if genes/digits are recombined, they remain with the same "destiny".

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.

Examples

			64  = 1*64 = 4*16;
95  = 1*95 = 5*19;
130 = 2*65 = 5*26;
242 = 2*121 = 11*22, etc.
		

References

• Francesco Di Matteo, Sequenze ludiche, Game Edizioni (2015), pages 28-37.

Links

• Francesco Di Matteo, Table of n, a(n) for n = 1..2815
• F. Di Matteo and A. Marchini, All the first 2815 terms calculated.

Crossrefs

Cf. A245386, A262873.

Programs

Mathematica

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 *)