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A066505
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f-amicable numbers where f(n) = n+1.
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0
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36, 62, 168, 326, 9936, 14056, 16198, 19862, 45304, 51910, 82662, 90152, 337688, 388102, 472902, 479672, 1970586, 2353756, 2969288, 3769942, 6319544, 8454886, 12276056, 13125574, 16783976, 17948854, 18818780, 20825882, 21738114, 22479040, 25960468, 31470614
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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}]
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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