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A109299 Primal codes of canonical finite permutations on positive integers.

%I A109299 #14 Aug 14 2025 04:08:23
%S A109299 1,2,12,18,360,540,600,1350,1500,2250,75600,105840,113400,126000,
%T A109299 158760,246960,283500,294000,315000,411600,472500,555660,735000,
%U A109299 864360,992250,1296540,1389150,1440600,1653750,2572500,3241350,3601500,3858750
%N A109299 Primal codes of canonical finite permutations on positive integers.
%C A109299 A canonical finite permutation on positive integers is a bijective mapping of [n] = {1, ..., n} to itself, counting the empty mapping as a permutation of the empty set.
%C A109299 From _Rémy Sigrist_, Sep 18 2021: (Start)
%C A109299 As usual with lists, the terms of the sequence are given in ascending order.
%C A109299 Equivalently, these are the numbers m such that A001221(m) = A051903(m) = A061395(m) = A071625(m).
%C A109299 This sequence has connections with A175061; here the prime factorizations, there the run-lengths in binary expansions, encode finite permutations.
%C A109299 There are m! terms with m distinct prime factors, the least one being A006939(m) and the greatest one being A076954(m); these m! terms are not necessarily contiguous. (End)
%D A109299 Suggested by Franklin T. Adams-Watters
%H A109299 J. Awbrey, <a href="https://oeis.org/wiki/Riffs_and_Rotes">Riffs and Rotes</a>
%H A109299 Rémy Sigrist, <a href="/A109299/a109299.gp.txt">PARI program for A109299</a>
%e A109299 Writing (prime(i))^j as i:j, we have this table:
%e A109299 Primal Codes of Canonical Finite Permutations
%e A109299         1 = { }
%e A109299         2 = 1:1
%e A109299        12 = 1:2 2:1
%e A109299        18 = 1:1 2:2
%e A109299       360 = 1:3 2:2 3:1
%e A109299       540 = 1:2 2:3 3:1
%e A109299       600 = 1:3 2:1 3:2
%e A109299      1350 = 1:1 2:3 3:2
%e A109299      1500 = 1:2 2:1 3:3
%e A109299      2250 = 1:1 2:2 3:3
%e A109299     75600 = 1:4 2:3 3:2 4:1
%e A109299    105840 = 1:4 2:3 3:1 4:2
%e A109299    113400 = 1:3 2:4 3:2 4:1
%e A109299    126000 = 1:4 2:2 3:3 4:1
%e A109299    158760 = 1:3 2:4 3:1 4:2
%e A109299    246960 = 1:4 2:2 3:1 4:3
%e A109299    283500 = 1:2 2:4 3:3 4:1
%e A109299    294000 = 1:4 2:1 3:3 4:2
%e A109299    315000 = 1:3 2:2 3:4 4:1
%e A109299    411600 = 1:4 2:1 3:2 4:3
%e A109299    472500 = 1:2 2:3 3:4 4:1
%e A109299    555660 = 1:2 2:4 3:1 4:3
%e A109299    735000 = 1:3 2:1 3:4 4:2
%e A109299    864360 = 1:3 2:2 3:1 4:4
%e A109299    992250 = 1:1 2:4 3:3 4:2
%e A109299   1296540 = 1:2 2:3 3:1 4:4
%e A109299   1389150 = 1:1 2:4 3:2 4:3
%e A109299   1440600 = 1:3 2:1 3:2 4:4
%e A109299   1653750 = 1:1 2:3 3:4 4:2
%e A109299   2572500 = 1:2 2:1 3:4 4:3
%e A109299   3241350 = 1:1 2:3 3:2 4:4
%e A109299   3601500 = 1:2 2:1 3:3 4:4
%e A109299   3858750 = 1:1 2:2 3:4 4:3
%e A109299   5402250 = 1:1 2:2 3:3 4:4
%o A109299 (PARI) \\ See Links section.
%o A109299 (PARI) is(n) = { my (f=factor(n), p=f[,1]~, e=f[,2]~); Set(e)==[1..#e] && (#p==0 || p[#p]==prime(#p)) } \\ _Rémy Sigrist_, Sep 18 2021
%Y A109299 Cf. A001221, A006939, A051903, A061395, A071625, A076954, A106177, A108352, A108371, A109297, A109298, A109301, A175061, A347758.
%K A109299 nonn
%O A109299 1,2
%A A109299 _Jon Awbrey_, Jul 09 2005
%E A109299 Offset changed to 1 and data corrected by _Rémy Sigrist_, Sep 18 2021