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

Author

Ant King, Dec 21 2006

Keywords

• 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(5)=8460 because the fifth infinitary amicable pair is (5940,8460) and 8460 is its largest member.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..7916
• Jan Munch Pedersen, Tables of Aliquot Cycles.

Crossrefs

Cf. A126169, A049417, A126168, A037445.

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; 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 *)

Formula

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

Extensions

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