Search: id:A126171
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%I A126171
%S A126171 0,0,2,6,22,62,189,444,1116,2594,6051,14141
%N A126171 Number of infinitary amicable pairs (i,j) with i<j and i<=10^n.
%C A126171 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 A126171 Pedersen J. M., <a href="http://amicable.homepage.dk/knwnc2.htm">Known 
               amicable pairs</a>.
%F A126171 Infinitary amicable pairs (m,n) satisfy isigma(m)=isigma(n)=m+n, with 
               m<n
%e A126171 a(6)=62 because there are 62 infinitary amicable pairs (m,n) with m<n 
               and m<=10^6
%t A126171 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 A126171 Cf. A126169, A049417, A126168, A037445, A126170.
%Y A126171 Sequence in context: A062085 A147800 A027561 this_sequence A002839 A109194 
               A014334
%Y A126171 Adjacent sequences: A126168 A126169 A126170 this_sequence A126172 A126173 
               A126174
%K A126171 hard,nonn
%O A126171 1,3
%A A126171 Ant King (mathstutoring(AT)ntlworld.com), Dec 22 2006

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