|
%I #35 May 21 2022 14:03:53
%S 504,756,806,1008,1148,1209,1472,1512,2016,2208,2418,2772,2924,3024,
%T 4416,4433,5544,6314,8096,8316,8415,8866,10736,11088,12628,13277,
%U 13299,14300,16038,16082,16192,16632,17732,20405,21384,22176,24288,24948,25452,26598,26730
%N 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).
%C 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:
%C 23*64 = 46*32;
%C 42*36 = 63*24;
%C 21*36 = 63*12;
%C 21*48 = 84*12;
%C 31*26 = 62*13.
%C 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.
%C 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, ...
%C The number n = 101808 implies So(n) = 3 because this number generates 3 double couples (see the example below).
%C 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.
%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers, 2nd Ed. (1997), p. 142.
%H Michael De Vlieger, <a href="/A228164/b228164.txt">Table of n, a(n) for n = 1..1000</a>
%H Michael De Vlieger, <a href="/A228164/a228164.txt">Symmetrical divisor pairs for numbers m in A228164 with 1 <= m <= 10^7.</a>
%e 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).
%e 50904 is in the sequence because we obtain two double pairs of divisors: (12, 4242) and (2424, 21), (42, 1212) and (2121, 24);
%e 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).
%e From _Michael De Vlieger_, Sep 15 2017: (Start)
%e First positions of numbers k of symmetrical pairs that appear for a(n) <= 10^7.
%e k n a(n)
%e ----------------
%e 2 1 504
%e 3 4 1008
%e 4 17 5544
%e 6 98 101808
%e 8 274 559944
%e (End)
%p 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:
%t 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 *)
%Y Cf. A262873 (a subsequence of predestined numbers A262743).
%K nonn,base
%O 1,1
%A _Michel Lagneau_, Aug 17 2013
|
