1,1

Let t(k) = psi(k) - k = A001615(k) - k be the sum of aliquot divisors d of k, such that k/d is squarefree. Penney & Pomerance proposed a problem to show that the aliquot-like sequence related to this function, i.e., the trajectory of an integer k under the repeated application of the map k -> t(k), can be unbounded. Since t(m^j * k) = m^j * t(k) if m|k, then if in the sequence a_0 = k, a_1 = t(k), a_2 = t(t(k)), ... there is a term a_{i1} = m^j * a_0 such that m|k and j > 0 then a_{i+i1} = m^j * a_i for all i and thus the sequence is unbounded.

Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, page 71, entry 318.

Kevin Brown and Charles Vanden Eynden, Pseudo-aliquot Sequences, Solution to Problem 10323, The American Mathematical Monthly, Volume 103, No. 8 (1996), pp. 697-698.

David E. Penney and Carl Pomerance, Problem 10323, The American Mathematical Monthly, Volume 100, No. 7 (1993), p. 688.

Eric Weisstein's World of Mathematics, Aliquot Sequence.

Wikipedia, Aliquot sequence.

318 is in the sequence since t(318) = psi(318) - 318 = 330, t(330) = 534, etc., and this repeated mapping yields an unbounded sequence.

t[1]=0; t[n_] := (Times@@(1+1/Transpose[FactorInteger[n]][[1]])-1)*n; rt[n_] := Module[{f=FactorInteger[n]}, e=GCD@@f[[;; , 2]]; Surd[n, e]]; divrootQ[n_, m_] := Divisible[n, rt[m]]; divQ[s_, n_] := If[n==0, 0, If[MemberQ[s, n], 1, If[ Length[Select[s, Divisible[n, #] && divrootQ[#, n/#] &]] > 0, 2, 3]]]; seqQ[n_] := Module[{n1=n}, s={}; While[divQ[s, n1] ==3, AppendTo[s, n1]; n1=t[n1]]; divQ[s, n1]] == 2; Select[Range[10000], seqQ]

Cf. A001615, A098007, A187778.

Sequence in context: A252246 A252254 A252247 * A323328 A045272 A359012

Adjacent sequences: A323324 A323325 A323326 * A323328 A323329 A323330

Amiram Eldar, Jan 11 2019

approved