A126175 Larger member of an augmented infinitary amicable pair.

1483785, 2479065, 2580105, 4895241, 7336455, 9100905, 10350345, 16367481, 17307105, 24829945, 15706090, 27866241, 15439545, 23872185, 53763535, 63075321, 41337555, 60923577, 51394665, 56802249, 110691295, 73809496, 89870985, 82771336, 92586585, 150672921, 108212055

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..276

Jan Munch Pedersen, Tables of Aliquot Cycles.

Formula

The values of n for which isigma(m)=isigma(n)=m+n-1, where n>m and isigma(n) is given by A049417(n).

Example

a(3)=2580105 because 2580105 is the larger member of the third augmented infinitary amicable pair, (2166136,2580105).

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[Last[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, k]], {n, 2, 10^9}]; s (* Amiram Eldar, Jan 20 2019 *)

Crossrefs

Cf. A126169, A049417, A126168, A037445, A126170, A126171, A126173, A126174, A126176.

Sequence in context: A204801 A234837 A233391 * A128837 A297881 A335289

Adjacent sequences: A126172 A126173 A126174 * A126176 A126177 A126178

Keyword

hard,nonn

Author

Ant King, Dec 23 2006

Extensions

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

Status

approved