A126171 Number of infinitary amicable pairs (i,j) with i<j and i<=10^n.

0, 0, 2, 6, 22, 62, 189, 444, 1116, 2594, 6051, 14141

Offset

1,3

Comments

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.

Links

Table of n, a(n) for n=1..12.

Pedersen J. M., Known amicable pairs.

Formula

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

Example

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

Mathematica

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}]

Crossrefs

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

Sequence in context: A324913 A027561 A321626 * A228396 A219766 A002839

Adjacent sequences: A126168 A126169 A126170 * A126172 A126173 A126174

Keyword

hard,nonn

Author

Ant King, Dec 22 2006

Status

approved