A262873 -id:A262873 - cp's OEIS frontend

A262743 Predestined numbers: every n number is generated by at least one pair of products, such as n = ab = cd, 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

Francesco Di Matteo, Sep 29 2015

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.

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

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

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

Cf. A245386, A262873.

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 *)

A228164 Numbers n having at least two distinct symmetrical pairs of divisors (a, b) and (b', a') such that n = ab = b'a' with a' = reverse(a) and b' = reverse(b).

Original entry on oeis.org

504, 756, 806, 1008, 1148, 1209, 1472, 1512, 2016, 2208, 2418, 2772, 2924, 3024, 4416, 4433, 5544, 6314, 8096, 8316, 8415, 8866, 10736, 11088, 12628, 13277, 13299, 14300, 16038, 16082, 16192, 16632, 17732, 20405, 21384, 22176, 24288, 24948, 25452, 26598, 26730

Offset: 1

Michel Lagneau, Aug 17 2013

A pair of integers (a, b) is symmetrical for multiplication when the product a*b is the same as the product b'*a' where a' = reverse(a) and b' = reverse(b). A double pair shows a symmetrical structure, for example:
23*64 = 46*32;
42*36 = 63*24;
21*36 = 63*12;
21*48 = 84*12;
31*26 = 62*13.
Because it is possible to obtain a number of double pairs equal to 1, 2, 3, ... we introduce the notion of "symmetrical order" denoted So(n) for each number n of the sequence corresponding to the number of double pairs.
The numbers of the sequence n = 50904, 55944, 76356, 81406, 83916, ... generate two double pairs of the form (a, b) and (b', a'), (c, d) and (d', c') such that n = a*b = b'*a' with a' = reverse(a) and b' = reverse(b) and n = c*d = d'*c' with c' = reverse(c) and d' = reverse(d). Hence So(50904) = 2, So(55944) = 2, ...
The number n = 101808 implies So(n) = 3 because this number generates 3 double couples (see the example below).
The sequence shows primitive and nonprimitive values: for example n = 504, 756, 806, ... are primitive values, but n = 1008 = 2*504, 1512 = 2*756, 2016 = 4*504, ... are not primitive values. A primitive number contains a couple of divisors (a, b) where a (and/or) b has decimal digits less than 5.

			504 is in the sequence because the two pairs of divisors (42, 12) and (21, 24) have the property 42*12 = 21*24 = 504 with 42 = reverse(24) and 12 = reverse(21).
50904 is in the sequence because we obtain two double pairs of divisors: (12, 4242) and (2424, 21), (42, 1212) and (2121, 24);
101808 is in the sequence because we obtain three double pairs of divisors: (12, 8484) and (4848, 21), (24, 4242) and (2424, 42), (48, 2121) and (1212, 84).
From _Michael De Vlieger_, Sep 15 2017: (Start)
First positions of numbers k of symmetrical pairs that appear for a(n) <= 10^7.
k     n     a(n)
----------------
2     1      504
3     4     1008
4    17     5544
6    98   101808
8   274   559944
(End)

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, 2nd Ed. (1997), p. 142.

Michael De Vlieger, Table of n, a(n) for n = 1..1000
Michael De Vlieger, Symmetrical divisor pairs for numbers m in A228164 with 1 <= m <= 10^7.

Cf. A262873 (a subsequence of predestined numbers A262743).

with(numtheory):for n from 2 to 50000 do:x:=divisors(n):n1:=nops(x):ii:=0:for a from 2 to n1-1 while(ii=0) do:m:=n/x[a]:m1:=convert(m,base,10):nn1:=nops(m1): m2:=convert(x[a],base,10):nn2:=nops(m2): s1:=sum('m1[nn1-i+1]*10^(i-1)', 'i'=1..nn1): s2:=sum('m2[nn2-i+1]*10^(i-1)', 'i'=1..nn2):for b from a+1 to n1-1 while(ii=0) do:q:=n/x[b]:if s1=q and s2=x[b] and m<>x[b] then ii:=1:printf(`%d, `,n):else fi:od:od:od:

Select[Range[10^7], Function[n, Count[Rest@ Select[Divisors@ n, # <= Sqrt@ n &], ?(And[IntegerReverse@ # != #, IntegerReverse@ # IntegerReverse[n/#] == n] &)] > 1]] (* _Michael De Vlieger, Oct 09 2015, updated Sep 15 2017 *)

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

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A228164 Numbers n having at least two distinct symmetrical pairs of divisors (a, b) and (b', a') such that n = ab = b'a' with a' = reverse(a) and b' = reverse(b).

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cp's OEIS Frontend

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.

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A228164 Numbers n having at least two distinct symmetrical pairs of divisors (a, b) and (b', a') such that n = a*b = b'*a' with a' = reverse(a) and b' = reverse(b).

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Maple

Mathematica

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

A228164 Numbers n having at least two distinct symmetrical pairs of divisors (a, b) and (b', a') such that n = ab = b'a' with a' = reverse(a) and b' = reverse(b).