%I #18 Jan 22 2019 08:21:56

%S 1252216,1754536,2166136,2362360,6224890,7626136,7851256,9581320,

%T 12480160,12494856,13324311,15218560,15422536,19028296,29180466,

%U 36716680,37542190,40682824,45131416,45495352,56523810,67195305,71570296,80524665,89740456,93182440,101304490

%N Smaller member of an augmented 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="/A126174/b126174.txt">Table of n, a(n) for n = 1..276</a>

%H Jan Munch Pedersen, <a href="http://62.198.248.44/aliquot/tables.htm">Tables of Aliquot Cycles</a>.

%F The values of m for which isigma(m)=isigma(n)=m+n-1, where m<n and isigma(n) is given by A049417(n).

%e a(3)=2166136 because 2166136 is the smaller element of the third augmented infinitary amicable pair, (2166136,2580105).

%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; AugmentedInfinitaryAmicableNumberQ[n_] := If[properinfinitarydivisorsum[properinfinitarydivisorsum[ n] + 1] == n - 1 && ! properinfinitarydivisorsum[n] + 1 == n, True, False]; AugmentedInfinitaryAmicablePairList[k_] := (anlist = Select[Range[k], AugmentedInfinitaryAmicableNumberQ[ # ] &]; prlist = Table[ Sort[{anlist[[n]], properinfinitarydivisorsum[anlist[[n]]] + 1}], {n, 1, Length[anlist]}]; amprlist = Union[prlist, prlist]); data = AugmentedInfinitaryAmicablePairList[10^7]; Table[First[data[[k]]], {k, 1, Length[data]}]

%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] + 1; If[k > n && infs[k] == n - 1, AppendTo[s, n]], {n, 2, 10^9}]; s (* _Amiram Eldar_, Jan 20 2019 *)

%Y Cf. A126169, A049417, A126168, A037445, A126170, A126171, A126173, A126175, A126176.

%K hard,nonn

%O 1,1

%A _Ant King_, Dec 23 2006

%E a(9)-a(27) from _Amiram Eldar_, Jan 20 2019