A126170 Larger member of an infinitary amicable pair.

126, 846, 1260, 7920, 8460, 11760, 10856, 14595, 17700, 43632, 45888, 49308, 83142, 62700, 71145, 73962, 96576, 83904, 107550, 88730, 178800, 112672, 131100, 125856, 168730, 149952, 196650, 203432, 206752, 224928, 306612, 365700, 399592, 419256, 460640, 548550

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

Amiram Eldar, Table of n, a(n) for n = 1..7916

Jan Munch Pedersen, Tables of Aliquot Cycles.

Formula

The values of n for which isigma(m)=isigma(n)=m+n and n>m.

Example

a(5)=8460 because the fifth infinitary amicable pair is (5940,8460) and 8460 is its largest 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[Last[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, k]], {n, 2, 10^5}]; s (* Amiram Eldar, Jan 22 2019 *)

Crossrefs

Cf. A126169, A049417, A126168, A037445.

Sequence in context: A181254 A004008 A292981 * A324709 A342307 A151989

Adjacent sequences: A126167 A126168 A126169 * A126171 A126172 A126173

Keyword

nonn

Author

Ant King, Dec 21 2006

Extensions

a(33)-a(36) from Amiram Eldar, Jan 22 2019

Status

approved