

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).


2



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
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OFFSET

1,1


COMMENTS

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.


REFERENCES

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


LINKS



EXAMPLE

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).
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)


MAPLE

with(numtheory):for n from 2 to 50000 do:x:=divisors(n):n1:=nops(x):ii:=0:for a from 2 to n11 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[nn1i+1]*10^(i1)', 'i'=1..nn1): s2:=sum('m2[nn2i+1]*10^(i1)', 'i'=1..nn2):for b from a+1 to n11 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:


MATHEMATICA

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 *)


CROSSREFS



KEYWORD

nonn,base


AUTHOR



STATUS

approved



