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Search: id:A126171
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| A126171 |
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Number of infinitary amicable pairs (i,j) with i<j and i<=10^n. |
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+0
9
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| 0, 0, 2, 6, 22, 62, 189, 444, 1116, 2594, 6051, 14141
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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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.
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LINKS
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Pedersen J. M., Known amicable pairs.
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FORMULA
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Infinitary amicable pairs (m,n) satisfy isigma(m)=isigma(n)=m+n, with m<n
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EXAMPLE
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a(6)=62 because there are 62 infinitary amicable pairs (m,n) with m<n and m<=10^6
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MATHEMATICA
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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}]
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CROSSREFS
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Cf. A126169, A049417, A126168, A037445, A126170.
Sequence in context: A062085 A147800 A027561 this_sequence A002839 A109194 A014334
Adjacent sequences: A126168 A126169 A126170 this_sequence A126172 A126173 A126174
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KEYWORD
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hard,nonn
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AUTHOR
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Ant King (mathstutoring(AT)ntlworld.com), Dec 22 2006
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