OFFSET
1,1
COMMENTS
f-amicable pairs are defined similarly to f-perfect numbers in A066218. That is, a, b is a f-amicable pair if f(a) = D(b) and f(b) = D(a), where D(n) = sum_{k divides n, k<n} f(d).
Pairs are (36,62), (14056,16198), (9936,19862), (45304,51910), (82662,90152) (337688,388102) and (472902,479672). The sequence shows them unbundled, then elements sorted according to size. - R. J. Mathar, Sep 07 2006, Dec 07 2006
LINKS
J. L. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.
EXAMPLE
Proper divisors of 36 = {1, 2, 3, 4, 6, 9, 12, 18}. f applied to these divisors = {2, 3, 4, 5, 7, 10, 13, 19}; their sum = 63. So D(36) = f(62). proper divisors of 62 = {1, 2, 31}. f applied to these divisors = {2, 3, 32}; their sum = 37. So D(62) = f(36). Therefore 36, 62 is an f-amicable pair.
MATHEMATICA
f[x_] := x + 1; d[x_] := Apply[ Plus, Map[ f, Divisors[ x] ] ] - f[ x]; m = Table[{x, y}, {x, 1, 1000}, {y, 1, 1000}]; Do[a = m[[i, j]]; If[ (a[[1]] < a[[2]]) && (f[a[[1]]] == d[a[[2]]]) && (f[a[[2]]] == d[a[[1]]]), Print[{i, j}]], {i, 1, 1000}, {j, 1, 1000}]
PROG
(C++) #include <limits.h> #include <iostream> #include <vector> using namespace std ; inline int f(const int n) { return n+1 ; } int D(const int n) { int resul=0 ; for(int k=1 ; k < n ; k++) if ( n %k == 0) resul += k+1 ; return resul ; } int main(int argc, char *argv[]) { vector<int> fvec ; vector<int> Dvec ; fvec.push_back(1) ; Dvec.push_back(1) ; for(int a=1 ; a < INT_MAX ; a++) { fvec.push_back( f(a)) ; Dvec.push_back( D(a)) ; for(int b=1 ; b< a ; b++) { if ( fvec[a]==Dvec[b] && fvec[b] == Dvec[a]) cout << b << ", " << a << ", " ; } } return 0 ; } - R. J. Mathar, Sep 07 2006
CROSSREFS
KEYWORD
nonn
AUTHOR
Joseph L. Pe, Jan 04 2002
EXTENSIONS
More terms from John W. Layman, Nov 11 2002
More terms from R. J. Mathar, Sep 07 2006
a(17)-a(32) from Donovan Johnson, Jun 23 2012
STATUS
approved