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%I #13 Feb 11 2014 19:05:24
%S 36,62,168,326,9936,14056,16198,19862,45304,51910,82662,90152,337688,
%T 388102,472902,479672,1970586,2353756,2969288,3769942,6319544,8454886,
%U 12276056,13125574,16783976,17948854,18818780,20825882,21738114,22479040,25960468,31470614
%N f-amicable numbers where f(n) = n+1.
%C 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).
%C 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
%H J. L. Pe, <a href="http://numeratus.net/fperfect/fperfect.html">On a Generalization of Perfect Numbers</a>, J. Rec. Math., 31(3) (2002-2003), 168-172.
%e 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.
%t 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}]
%o (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
%Y Cf. A066229, A066218.
%K nonn
%O 1,1
%A _Joseph L. Pe_, Jan 04 2002
%E More terms from _John W. Layman_, Nov 11 2002
%E More terms from _R. J. Mathar_, Sep 07 2006
%E a(17)-a(32) from _Donovan Johnson_, Jun 23 2012
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