A126173 Larger element of a reduced infinitary amicable pair.

2295, 75495, 817479, 1902215, 1341495, 1348935, 2226014, 2421704, 3123735, 3010215, 5644415, 4282215, 7509159, 10106504, 12900734, 24519159, 31356314, 41950359, 43321095, 80870615, 42125144, 85141719, 87689415, 87802407, 86477895, 105993657, 168669879, 129081735

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..278

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)=817479 because 817479 is the largest member of the third reduced infinitary amicable pair, (573560,817479)

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; ReducedInfinitaryAmicableNumberQ[n_] := If[properinfinitarydivisorsum[properinfinitarydivisorsum[ n] - 1] == n + 1 && n > 1, True, False]; ReducedInfinitaryAmicablePairList[k_] := (anlist = Select[Range[k], ReducedInfinitaryAmicableNumberQ[ # ] &]; prlist = Table[Sort[{anlist[[n]], properinfinitarydivisorsum[anlist[[n]]] - 1}], {n, 1, Length[anlist]}]; amprlist = Union[prlist, prlist]); data1 = ReducedInfinitaryAmicablePairList[10^7]; Table[Last[data1[[k]]], {k, 1, Length[data1]}]
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, 10^5}]; s (* Amiram Eldar, Jan 22 2019 *)

Crossrefs

Cf. A126169, A049417, A126168, A037445, A126170, A126171, A126172, A126174, A126175, A126176.

Sequence in context: A274359 A306871 A123911 * A083572 A270854 A260292

Adjacent sequences: A126170 A126171 A126172 * A126174 A126175 A126176

Keyword

nonn

Author

Ant King, Dec 23 2006

Extensions

a(15)-a(28) from Amiram Eldar, Jan 22 2019

Status

approved