A285004 Numbers n that enter a cycle of greater length than that for any k < n in the iteration sequence s(0)=n, s(k+1) = s(k) + (-1)^k*d(s(k)), where d(n) is the number of divisors of n (A000005).

3, 93, 273, 684, 3993, 58695, 91355, 167055, 441519, 2337513, 20225991, 20625997, 148789675, 470944675, 494359089, 3252701700, 3472027000, 9096968436

Offset

1,1

Comments

Claudia Spiro conjectured that for every initial number n the iteration sequence is eventually periodic.

The corresponding cycle lengths are: 2, 4, 6, 10, 12, 14, 16, 20, 22, 26, 28, 32, 34, 36, 44, 46, 52, 54.

Links

Table of n, a(n) for n=1..18.

Claudia Spiro, An Iteration Problem Involving the Divisor Function, Acta Arithmetica, Vol. 46, No. 3 (1986), pp. 215-225.

Example

The sequence for 93 is 93, 93 + d(93) = 97, 97 - d(97) = 95, 95 + d(95) = 99, 99 - d(99) = 93, 93 + d(93) = 97, ... with a cycle of (93, 97, 95, 99) whose length is 4. All initial numbers < 93 end in a cycle whose length < 4, thus 93 is in the sequence.

Mathematica

lengths={}; Records={}; For[n=2, n<100, n++; c=1; v={}; m=0; s=1; a=n; i={}; While[m<10^6, AppendTo[v, a]; If[Length[v] > 3, i=LongestCommonSubsequencePositions[v[[1;; -3]], v[[-2;; -1]]], i = {}]; If[Length[i]==2 && Differences[i[[1]]][[1]]==1, c=Length[v]-i[[1]][[1]]-1; Break[]]; m++; a = a + s*DivisorSigma[0, a]; s = -s; ];  If[Length[lengths]==0 || c>lengths[[-1]], AppendTo[lengths, c]; AppendTo[Records, n]]]; Records

Crossrefs

Cf. A000005, A049820, A062249, A175304 (numbers with cycle of length 2), A288070.

Sequence in context: A213987 A297479 A116161 * A288070 A116292 A139543

Adjacent sequences: A285001 A285002 A285003 * A285005 A285006 A285007

Keyword

nonn,more

Author

Amiram Eldar, Apr 07 2017

Extensions

a(11)-a(18) from Giovanni Resta, Apr 07 2017

Wrong term 1 removed by Amiram Eldar, Jun 05 2017

Status

approved