1,1

Based on empirical evidence, approximately 98.9 % of the infinitary aliquot sequences generated by the odd integers are monotonically decreasing. This sequence represents the 1.1 % of odd integers that are the exceptions to this.

Cohen, Graeme L.; On an Integer's Infinitary Divisors, Mathematics of Computation, Vol. 54, No. 189. (1990), pp. 395-411.

Pedersen, Jan Munch, Tables of Aliquot Cycles.

a(5)=2823 because 2823 is the fifth odd integer whose infinitary aliquot sequence is not monotonically decreasing.

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; g[n_] := If[n > 0, properinfinitarydivisorsum[n], 0]; iTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]]; u[n_]:=Table[n[[k+1]]<n[[k]], {k, 1, Length[n]-1}]; v[n_]:=If[ !MemberQ[u[n], False], True, False]; data=iTrajectory/@Range[1, 10^4, 2]; First/@Select[data, !v[ # ] &]

Cf. A126168, A127661, A127666.

Sequence in context: A125013 A005231 A006038 this_sequence A109729 A127666 A133818

Adjacent sequences: A127664 A127665 A127666 this_sequence A127668 A127669 A127670

nonn

Ant King (mathstutoring(AT)ntlworld.com), Jan 26 2007