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A298268 a(1) = 1, and for any n > 1, if n is the k-th number with greatest prime factor p, then a(n) is the k-th number with least prime factor p.

%I A298268 #37 Feb 02 2018 03:25:57
%S A298268 1,2,3,4,5,9,7,6,15,25,11,21,13,49,35,8,17,27,19,55,77,121,23,33,65,
%T A298268 169,39,91,29,85,31,10,143,289,119,45,37,361,221,95,41,133,43,187,115,
%U A298268 529,47,51,161,125,323,247,53,57,209,203,437,841,59,145,61,961
%N A298268 a(1) = 1, and for any n > 1, if n is the k-th number with greatest prime factor p, then a(n) is the k-th number with least prime factor p.
%C A298268 This sequence is a permutation of the natural numbers, with inverse A298882.
%C A298268 For any prime p and k > 0:
%C A298268 - if s_p(k) is the k-th p-smooth number and r_p(k) is the k-th p-rough number,
%C A298268 - then a(p * s_p(k)) = p * r_p(k),
%C A298268 - for example: a(11 * A051038(k)) = 11 * A008364(k).
%H A298268 Rémy Sigrist, <a href="/A298268/b298268.txt">Table of n, a(n) for n = 1..10000</a>
%H A298268 Rémy Sigrist, <a href="/A298268/a298268.png">Colored logarithmic scatterplot of the first 100000 terms</a> (with prime and semiprime values highlighted)
%H A298268 Rémy Sigrist, <a href="/A298268/a298268_1.png">Colored logarithmic scatterplot of the first 100000 terms</a> (where the color is function of A006530(n))
%H A298268 Rémy Sigrist, <a href="/A298268/a298268_2.png">Colored logarithmic scatterplot of the first 100000 terms</a> (where the color is function of A176506(k) when a(n) is the k-th semiprime)
%H A298268 Rémy Sigrist, <a href="/A298268/a298268.gp.txt">PARI program for A298268</a>
%H A298268 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A298268 a(1) = 1.
%F A298268 a(A125624(n, k)) = A083140(n, k) for any n > 0 and k > 0.
%F A298268 a(n) = A083140(A061395(n), A078899(n)) for any n > 1.
%F A298268 Empirically:
%F A298268 - a(n) = n iff n belongs to A046022,
%F A298268 - a(2^k) = 2 * k for any k > 0,
%F A298268 - a(2 * p) = p^2 for any prime p,
%F A298268 - a(3 * p) = p * A151800(p) for any odd prime p.
%e A298268 The first terms, alongside A006530(n), are:
%e A298268   n     a(n)   gpf(n)
%e A298268   --    ----   ------
%e A298268    1      1      1
%e A298268    2      2      2
%e A298268    3      3      3
%e A298268    4      4      2
%e A298268    5      5      5
%e A298268    6      9      3
%e A298268    7      7      7
%e A298268    8      6      2
%e A298268    9     15      3
%e A298268   10     25      5
%e A298268   11     11     11
%e A298268   12     21      3
%e A298268   13     13     13
%e A298268   14     49      7
%e A298268   15     35      5
%e A298268   16      8      2
%e A298268   17     17     17
%e A298268   18     27      3
%e A298268   19     19     19
%e A298268   20     55      5
%o A298268 (PARI) See Links section.
%Y A298268 Cf. A006530, A008364, A046022, A051038, A061395, A078899, A083140, A125624, A151800, A176506, A298268, A298882 (inverse).
%K A298268 nonn
%O A298268 1,2
%A A298268 _Rémy Sigrist_, Jan 27 2018