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Search: id:A098860
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| A098860 |
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Primes that are conjectured to lead to a one-cycle for every natural number x in the following (nontrivial) generalization of the (3x+1) problem. |
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1
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| 3, 5, 7, 29, 41, 79, 89, 107, 109, 127, 149, 157, 179, 191, 199, 211
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Start with a number x and construct a successor by the following iterative procedure: first remove all factors 2, 3, 5, ..., p(k) from x, where p(k) is the k-th prime number. When no further such factors remain then take the number ([p(k+1)*x]+1)/2 as the successor. The (3x+1) problem is the special case k=1 in the sequence that lists the p(k+1) leading to a one-cycle.
For other primes there is at least 1 supplementary cycle: e.g. when p(k+1)=11 there is also a cycle starting with 17; when p(k+1)=19 there is also a cycle starting with 46063; when p(k+1)=61 there are 3 supplementary cycles starting resp. with 97, 199, 26833; etc.
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MATHEMATICA
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v[n_, k_]:=Block[{m=n}, Do[While[Mod[m, Prime[i]]==0, m=m/Prime[i]], {i, k}]; If[m!=1, Prepend[v[m*Prime[k+1]+1, k], m], v[m, k]={1}]] b[r_, s_, t_]:=Table[v[n, r], {n, s, t}]
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CROSSREFS
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Sequence in context: A046931 A154551 A058047 this_sequence A106920 A060273 A124077
Adjacent sequences: A098857 A098858 A098859 this_sequence A098861 A098862 A098863
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KEYWORD
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nonn,more
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AUTHOR
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Herman Roelants (herman.roelants(AT)hiw.kuleuven.ac.be), Oct 11 2004
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