1,1

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.

Pedersen J. M., Known amicable pairs.

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

A(3)=2166136 because 2166136 is the smaller element of the third augmented infinitary amicable pair, (2166136,2580105)

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]}]

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

Sequence in context: A144694 A156113 A118900 this_sequence A072892 A090615 A081430

Adjacent sequences: A126171 A126172 A126173 this_sequence A126175 A126176 A126177

hard,nonn

Ant King (mathstutoring(AT)ntlworld.com), Dec 23 2006