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A126174 Smaller member of an augmented infinitary amicable pair. 8
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 (list; graph; refs; listen; history; text; internal format)
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 m for which isigma(m)=isigma(n)=m+n-1, where m<n and isigma(n) is given by A049417(n).

EXAMPLE

a(3)=2166136 because 2166136 is the smaller element 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[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 *)

CROSSREFS

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

Sequence in context: A251156 A210132 A251118 * A186089 A208489 A257268

Adjacent sequences:  A126171 A126172 A126173 * A126175 A126176 A126177

KEYWORD

hard,nonn

AUTHOR

Ant King, Dec 23 2006

EXTENSIONS

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

STATUS

approved

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Last modified May 25 19:19 EDT 2022. Contains 354071 sequences. (Running on oeis4.)