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%I #12 Jan 24 2019 03:42:11
%S 126,846,1260,7920,8460,11760,10856,14595,17700,43632,45888,49308,
%T 83142,62700,71145,73962,96576,83904,107550,88730,178800,112672,
%U 131100,125856,168730,149952,196650,203432,206752,224928,306612,365700,399592,419256,460640,548550
%N Larger member of an infinitary amicable pair.
%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 Amiram Eldar, <a href="/A126170/b126170.txt">Table of n, a(n) for n = 1..7916</a>
%H Jan Munch Pedersen, <a href="http://62.198.248.44/aliquot/tables.htm">Tables of Aliquot Cycles</a>.
%F The values of n for which isigma(m)=isigma(n)=m+n and n>m.
%e a(5)=8460 because the fifth infinitary amicable pair is (5940,8460) and 8460 is its largest member.
%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[Last[data4[[k]]], {k, 1, Length[data4]}]
%t fun[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; infs[n_] := Times @@ (fun @@@ FactorInteger[n]) - n; s = {}; Do[k = infs[n]; If[k > n && infs[k] == n, AppendTo[s, k]], {n, 2, 10^5}]; s (* _Amiram Eldar_, Jan 22 2019 *)
%Y Cf. A126169, A049417, A126168, A037445.
%K nonn
%O 1,1
%A _Ant King_, Dec 21 2006
%E a(33)-a(36) from _Amiram Eldar_, Jan 22 2019