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%I #4 Mar 30 2012 18:41:04
%S 0,0,2,6,22,62,189,444,1116,2594,6051,14141
%N Number of infinitary amicable pairs (i,j) with i<j and i<=10^n.
%C A divisor of n is called infinitary if it is a product of divisors of the form p^{y_a 2^a}, where p^y is a prime power dividing n and sum_a y_a 2^a is the binary representation of y.
%H Pedersen J. M., <a href="http://amicable.homepage.dk/knwnc2.htm">Known amicable pairs</a>.
%F Infinitary amicable pairs (m,n) satisfy isigma(m)=isigma(n)=m+n, with m<n
%e a(6)=62 because there are 62 infinitary amicable pairs (m,n) with m<n and m<=10^6
%t ExponentList[n_Integer, factors_List] := {#, IntegerExponent[n, # ]} & /@ factors; InfinitaryDivisors[1] := {1}; InfinitaryDivisors[n_Integer?Positive] := Module[ { factors = First /@ FactorInteger[n], d = Divisors[n] }, d[[Flatten[Position[ Transpose[ Thread[Function[{f, g}, BitOr[f, g] == g][ #, Last[ # ]]] & /@ Transpose[Last /@ ExponentList[ #, factors] & /@ d]], _?( And @@ # &), {1}]] ]] ] Null; properinfinitarydivisorsum[k_] := Plus @@ InfinitaryDivisors[k] - k; InfinitaryAmicableNumberQ[k_] := If[Nest[properinfinitarydivisorsum, k, 2] == k && ! properinfinitarydivisorsum[k] == k, True, False]; data1 = Select[ Range[10^6], InfinitaryAmicableNumberQ[ # ] &]; data2 = properinfinitarydivisorsum[ # ] & /@ data1; data3 = Table[{data1[[k]], data2[[k]]}, {k, 1, Length[data1]}]; data4 = Select[data3, First[ # ] < Last[ # ] &]; Table[Length[Select[data4, First[ # ] < 10^k &]], {k, 1, 6}]
%Y Cf. A126169, A049417, A126168, A037445, A126170.
%K hard,nonn
%O 1,3
%A _Ant King_, Dec 22 2006