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A126169
Smaller member of an infinitary amicable pair.
19
114, 594, 1140, 4320, 5940, 8640, 10744, 12285, 13500, 25728, 35712, 44772, 60858, 62100, 67095, 67158, 74784, 79296, 79650, 79750, 86400, 92960, 118500, 118944, 142310, 143808, 177750, 185368, 204512, 215712, 298188, 308220, 356408, 377784, 420640, 462330
OFFSET
1,1
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
Jan Munch Pedersen, Tables of Aliquot Cycles.
FORMULA
The values of m for which isigma(m)=isigma(n)=m+n and m<n.
EXAMPLE
a(5)=5940 because the fifth infinitary amicable pair is (5940,8460) and 5940 is its smallest member
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[First[data4[[k]]], {k, 1, Length[data4]}]
fun[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; infs[n_] := Times @@ (fun @@@ FactorInteger[n]) - n; s = {}; Do[k = infs[n]; If[k > n && infs[k] == n, AppendTo[s, n]], {n, 2, 10^5}]; s (* Amiram Eldar, Jan 22 2019 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Ant King, Dec 21 2006
EXTENSIONS
a(33)-a(36) from Amiram Eldar, Jan 22 2019
STATUS
approved