1,3

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.

Infinitary amicable pairs (m,n) satisfy isigma(m)=isigma(n)=m+n, with m<n

a(6)=62 because there are 62 infinitary amicable pairs (m,n) with m<n and m<=10^6

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}]

Cf. A126169, A049417, A126168, A037445, A126170.

Sequence in context: A324913 A027561 A321626 * A228396 A381385 A219766

Adjacent sequences: A126168 A126169 A126170 * A126172 A126173 A126174

hard,nonn

Ant King, Dec 22 2006

approved