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A124663 Number of reduced infinitary amicable pairs (i,j) with i<j and i<=10^n. 0
0, 0, 0, 1, 2, 3, 14, 25, 51, 112, 213 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

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..11.

Pedersen J. M., Known amicable pairs.

FORMULA

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

EXAMPLE

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

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}]

CROSSREFS

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

Sequence in context: A089876 A025092 A112636 * A101005 A029998 A117461

Adjacent sequences:  A124660 A124661 A124662 * A124664 A124665 A124666

KEYWORD

hard,nonn

AUTHOR

Ant King, Dec 23 2006

STATUS

approved

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Last modified October 21 23:02 EDT 2018. Contains 316431 sequences. (Running on oeis4.)