A126174 Smaller member of an augmented infinitary amicable pair.

1252216, 1754536, 2166136, 2362360, 6224890, 7626136, 7851256, 9581320, 12480160, 12494856, 13324311, 15218560, 15422536, 19028296, 29180466, 36716680, 37542190, 40682824, 45131416, 45495352, 56523810, 67195305, 71570296, 80524665, 89740456, 93182440, 101304490

Offset: 1

Author

Ant King, Dec 23 2006

Keywords

• hard
• 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(3)=2166136 because 2166136 is the smaller element of the third augmented infinitary amicable pair, (2166136,2580105).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..276
• Jan Munch Pedersen, Tables of Aliquot Cycles.

Crossrefs

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

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; AugmentedInfinitaryAmicableNumberQ[n_] := If[properinfinitarydivisorsum[properinfinitarydivisorsum[ n] + 1] == n - 1 && ! properinfinitarydivisorsum[n] + 1 == n, True, False]; AugmentedInfinitaryAmicablePairList[k_] := (anlist = Select[Range[k], AugmentedInfinitaryAmicableNumberQ[ # ] &]; prlist = Table[ Sort[{anlist[[n]], properinfinitarydivisorsum[anlist[[n]]] + 1}], {n, 1, Length[anlist]}]; amprlist = Union[prlist, prlist]); data = AugmentedInfinitaryAmicablePairList[10^7]; Table[First[data[[k]]], {k, 1, Length[data]}]
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] + 1; If[k > n && infs[k] == n - 1, AppendTo[s, n]], {n, 2, 10^9}]; s (* Amiram Eldar, Jan 20 2019 *)

Formula

The values of m for which isigma(m)=isigma(n)=m+n-1, where mA049417(n).

Extensions

a(9)-a(27) from Amiram Eldar, Jan 20 2019