

A323327


Numbers that start an unbounded aliquotlike sequence based on Dedekind psi function (A001615).


4



318, 330, 498, 510, 534, 546, 636, 660, 786, 798, 942, 954, 978, 990, 996, 1020, 1068, 1092, 1110, 1122, 1254, 1272, 1320, 1398, 1410, 1470, 1494, 1506, 1518, 1530, 1572, 1596, 1602, 1614, 1626, 1638, 1734, 1884, 1908, 1938, 1950, 1956, 1980, 1992, 2040, 2046
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OFFSET

1,1


COMMENTS

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 aliquotlike 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 mk, 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 mk and j > 0 then a_{i+i1} = m^j * a_i for all i and thus the sequence is unbounded.


REFERENCES

JeanMarie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, page 71, entry 318.


LINKS

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


EXAMPLE

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


MATHEMATICA

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]


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



