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
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.
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).
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)
MAPLE
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:
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
Michel Lagneau, Aug 17 2013
STATUS
approved