|
| |
|
|
A226587
|
|
Numbers n having at least two complementary pairs of divisors (q, p) and (p', q') such that n = p*q = p'*q' where the decimal digits of p' are the 9's complement of the decimal digits of p and the decimal digits of q' are the 9's complement of the decimal digits of q.
|
|
0
|
|
|
|
88, 154, 198, 220, 888, 1554, 1998, 2220, 8888, 9768, 15554, 17094, 19998, 21978, 22220, 24420, 88888, 89890, 97768, 105444, 112918, 120190, 127260, 134128, 140794, 147258, 153520, 155554, 159580, 165438, 171094, 176548, 181800, 186850, 191698, 196344, 199998, 200788, 205030
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
COMMENTS
|
The 9's complement of a number m equals 10^d - 1 - m where d is the number of digits in m. If u is a digit in m replace it with 9 - u.
A pair of integer (p, q) is complementary for multiplication when the product p*q is the same as the product p'*q' where the decimal digits of p' are the 9's complement of the decimal digits of p and the decimal digits of q' are the 9's complement of the decimal digits of q.
A double pair shows a complementary structure, for example: 77*2 = 7*22; 888*11 = 88*111; 8989*10 = 89*1010.
The sequence is infinite: let two integers x and y with the decimal representation x = ppp...p (i times) and y = (9-p)(9-p)...(9-p) (j times). The product x*y = p*(9-p)*R_i*R_j where R_k is a string of k 1's (or a Repunit number of the form (10^k - 1)/9). But x’ = (9-p)*R_i and y' = p*R_j => x*y => x'*y'.
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
198 is in the sequence because 66*3 = 6*33 = 198.
|
|
|
MAPLE
|
with(numtheory):for n from 1 to 210000 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('(9-m1[i])*10^(i-1)', 'i'=1..nn1): s2:=sum('(9-m2[i])*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:
# warning: there were missing terms, so the above Maple program may be wrong. - N. J. A. Sloane, Sep 17 2017
|
|
|
PROG
|
(PARI) compl(n) = my(dn = digits(n)); fromdigits(vector(#dn, k, 9 - dn[k]));
isok(n) = sumdiv(n, d, if ((d^2<n), compl(d)*compl(n/d) == n)) >= 2; \\ Michel Marcus, Sep 16 2017
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,base,less
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
Missing terms 88, 888, 8888, 88888 added by Michel Marcus, Sep 16 2017
|
|
|
STATUS
|
approved
|
| |
|
|
