A127667 Odd integers that do not generate monotonically decreasing infinitary aliquot sequences.

945, 1743, 2175, 2655, 2823, 2865, 3105, 3375, 3537, 3585, 3729, 4209, 4665, 5775, 6559, 6681, 6969, 7257, 7263, 7785, 8457, 8583, 9657, 10017, 10047, 10113, 10395, 10599, 10743, 12285, 13815, 14055, 14145, 15015, 15597, 16065, 17955, 18529, 18777, 19305, 19635

Offset

1,1

Comments

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.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp., 54 (1990), 395-411.

J. O. M. Pedersen, Tables of Aliquot Cycles [Broken link]

J. O. M. Pedersen, Tables of Aliquot Cycles [Via Internet Archive Wayback-Machine]

J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]

Example

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

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; 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[ # ] &]

Crossrefs

Cf. A126168, A127661, A127666.

Sequence in context: A188439 A275472 A275066 * A252184 A188263 A188342

Adjacent sequences: A127664 A127665 A127666 * A127668 A127669 A127670

Keyword

nonn

Author

Ant King, Jan 26 2007

Extensions

More terms from Amiram Eldar, Sep 16 2019

Status

approved