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A294658
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Number of steps required to reach either 5 or 13, starting with n, when iterating the map A125256: x -> smallest odd prime divisor of n^2+1; or a(n) = -1 in case 5 is never reached.
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3
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1, 1, 2, 0, 2, 1, 1, 4, 3, 5, 1, 0, 2, 2, 2, 1, 1, 4, 3, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 3, 3, 3, 1, 1, 1, 2, 3, 1, 1, 2, 4, 3, 1, 1, 2, 2, 2, 1, 1, 4, 1, 4, 1, 1, 2, 2, 2, 1, 1, 5, 1, 3, 1, 1, 2, 3, 2, 1, 1, 4, 2, 2, 1, 1, 2, 2, 1, 1
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OFFSET
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2,3
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COMMENTS
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The orbit or trajectory under A125256 appears to end in the cycle 5 -> 13 -> 5 -> etc. for any initial value n.
Sequence A294656 gives the size of the complete orbit of n under the map A125256, including the two elements 5 and 13 of the terminating cycle. Thus a(n) is 2 less than A294656(n) for all n. This is confirmed by careful examination of special cases - assuming, of course, that all trajectories end in the cycle (5, 13).
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LINKS
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Ray Chandler, Table of n, a(n) for n = 2..20001
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FORMULA
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a(n) = A294656(n) - 2.
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EXAMPLE
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For n = 1 the map A125256 is not defined.
a(2) = 1 because under A125256, 2 -> 2^2+1 = 5 (= its smallest odd prime factor), so 5 is reached after just a(2) = 1 iteration of this map.
a(3) = 1 because A125256(3) = 5, least odd prime factor of 3^2+1 = 10 = 2*5, so here again 5 is reached after just a(2) = 1 iteration of A125256.
a(4) = 2 because A125256(4) = 4^2 + 1 = 17, and A125256(17) = 5 = least odd prime factor of 17^2 + 1 = 289 + 1 = 2*5*29, so 5 is reached after a(4) = 2 iterations of A125256.
a(5) = a(13) = 0 because for these initial values 5 and 13, no iteration is needed until either 5 or 13 is reached.
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PROG
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(PARI) A294658(n)=for(k=0, oo, bittest(8224, n)&&return(k); n=A125256(n)) \\ 8224 = 2^5 + 2^13. One could add 2^0 + 2^1 = 3 to avoid an error message for initial values 0 and 1, for which A125256 is not defined.
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CROSSREFS
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Cf. A125256, A294656, A294657.
Sequence in context: A112467 A112466 A166348 * A127543 A280830 A068907
Adjacent sequences: A294655 A294656 A294657 * A294659 A294660 A294661
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KEYWORD
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nonn
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AUTHOR
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M. F. Hasler, Nov 06 2017
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STATUS
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approved
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