1,1

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 (mathar(AT)strw.leidenuniv.nl), Sep 07 2006, Dec 07 2006

J. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.

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.

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}]

(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 (mathar(AT)strw.leidenuniv.nl), Sep 07 2006

Cf. A066229, A066218.

Sequence in context: A124941 A116321 A080469 this_sequence A039419 A043242 A044022

Adjacent sequences: A066502 A066503 A066504 this_sequence A066506 A066507 A066508

more,nonn

Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Jan 04 2002

More terms from John W. Layman (layman(AT)math.vt.edu), Nov 11 2002

More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 07 2006