A124663 Number of reduced infinitary amicable pairs (i,j) with i

0, 0, 0, 1, 2, 3, 14, 25, 51, 112, 213

Offset: 1

Author

Ant King, Dec 23 2006

Keywords

• hard
• nonn

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.

Examples

			a(7)=14 because there are 14 reduced infinitary amicable pairs (m,n) with m<n and m<=10^7
		

Links

• Pedersen J. M., Known amicable pairs.

Crossrefs

Cf. A126169, A049417, A126168, A037445, A126170, A126171, A126172, A126173, A126174, A126175, A126176.

Programs

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; ReducedInfinitaryAmicableNumberQ[n_] := If[properinfinitarydivisorsum[properinfinitarydivisorsum[ n] - 1] == n + 1 && n > 1, True, False]; ReducedInfinitaryAmicablePairList[k_] := (anlist = Select[Range[k], ReducedInfinitaryAmicableNumberQ[ # ] &]; prlist = Table[Sort[{anlist[[n]], properinfinitarydivisorsum[anlist[[n]]] - 1}], {n, 1, Length[anlist]}]; amprlist = Union[prlist, prlist]); data1 = ReducedInfinitaryAmicablePairList[10^7]; Table[Length[Select[data1, First[ # ] < 10^k &]], {k, 1, 7}]

Formula

Reduced infinitary amicable pairs (m,n) satisfy isigma(m)=isigma(n)=m+n+1, with m

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