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).

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

Author

Michel Lagneau, Aug 17 2013

Keywords

• nonn
• base

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.

Examples

			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)
		

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.

Crossrefs

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

Programs

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 *)