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A292876 Irregular table whose n-th row lists all k such that A039654(k) = prime(n).

%I A292876 #15 Sep 27 2017 09:22:00
%S A292876 2,3,5,7,4,6,11,13,10,17,19,8,14,15,23,29,21,31,37,20,26,41,43,22,33,
%T A292876 35,47,34,53,18,24,38,59,61,67,9,12,16,25,27,28,30,39,46,51,55,71,73,
%U A292876 57,79,44,65,83,40,58,89
%N A292876 Irregular table whose n-th row lists all k such that A039654(k) = prime(n).
%C A292876 This can also be considered as a list of all orbits of A039653, ordered by their maximal element p = A039654(k) for any k of this orbit.
%C A292876 Indeed, A039653(x) >= x with equality iff x is prime, and all orbits of A039653 are conjectured to end in such a fixed point prime p = A039654(k) for any k in this orbit.
%C A292876 Row lengths are given by A177343.
%C A292876 This sequence is also a permutation of all integers > 1, where each prime p(k) is immediately preceded by A177343(k)-1 composite numbers less than p(k). It follows that each composite is preceded either by a smaller composite or by a larger prime, and followed by a larger composite or prime. Thus, the primes appear in their natural order, but the composites do not.
%C A292876 The first element of each row (i.e., the first column of this table) is given by A292874.
%C A292876 We see (cf. a-file) that powers of 2 are often the first element (or at least part) of relatively long orbits: A177343(A000720(A039654(2^k))) = (1, 3, 4, 12, 25, 5, 10, 35, 61, 143, 143, 220, 365, ...)
%H A292876 M. F. Hasler, <a href="/A292876/a292876_1.txt">Table rows n = 1..1229</a>
%e A292876 The table starts:
%e A292876     n  p(n)  { k | A039654(k) = p(n) }
%e A292876     1    2   { 2 }
%e A292876     2    3   { 3 }
%e A292876     3    5   { 5 }
%e A292876     4    7   { 7 }
%e A292876     5   11   { 4, 6, 11 }
%e A292876     6   13   { 13 }
%e A292876     7   17   { 10, 17 }
%e A292876     8   19   { 19 }
%e A292876     9   23   { 8, 14, 15, 23 }
%o A292876 (PARI) A292876(n,p=prime(n))=select(k->A039654(k)==p,[2..p]) \\ Not optimized nor efficient; mainly for illustrational purpose. - _M. F. Hasler_, Sep 25 2017
%Y A292876 Cf. A039654, A039653, A177343, A292874, A292112, A292113.
%K A292876 nonn,tabf
%O A292876 1,1
%A A292876 _M. F. Hasler_, Sep 25 2017