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A109297 Primal codes of finite permutations on positive integers.

%I A109297 #30 Aug 07 2022 07:53:09
%S A109297 1,2,9,12,18,40,112,125,250,352,360,540,600,675,832,1008,1125,1350,
%T A109297 1500,2176,2250,2268,2352,2401,3168,3969,4802,4864,7488,7938,10692,
%U A109297 11616,11776,14000,19584,21609,27440,28812,29403,29696,32448,35000,37908,43218,43776
%N A109297 Primal codes of finite permutations on positive integers.
%C A109297 A finite permutation is a bijective mapping from a finite set to itself, counting the empty mapping as a permutation of the empty set.
%C A109297 Also Heinz numbers of integer partitions where the set of distinct parts is equal to the set of distinct multiplicities. These partitions are counted by A114640. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - _Gus Wiseman_, Apr 02 2019
%H A109297 Amiram Eldar, <a href="/A109297/b109297.txt">Table of n, a(n) for n = 1..300</a> (terms 1..100 from Alois P. Heinz)
%H A109297 Jon Awbrey, <a href="https://oeis.org/wiki/Riffs_and_Rotes">Riffs and Rotes</a>.
%e A109297 Writing (prime(i))^j as i:j, we have the following table:
%e A109297 Primal Codes of Finite Permutations on Positive Integers
%e A109297 ` ` ` 1 = { }
%e A109297 ` ` ` 2 = 1:1
%e A109297 ` ` ` 9 = 2:2
%e A109297 ` ` `12 = 1:2 2:1
%e A109297 ` ` `18 = 1:1 2:2
%e A109297 ` ` `40 = 1:3 3:1
%e A109297 ` ` 112 = 1:4 4:1
%e A109297 ` ` 125 = 3:3
%e A109297 ` ` 250 = 1:1 3:3
%e A109297 ` ` 352 = 1:5 5:1
%e A109297 ` ` 360 = 1:3 2:2 3:1
%e A109297 ` ` 540 = 1:2 2:3 3:1
%e A109297 ` ` 600 = 1:3 2:1 3:2
%e A109297 ` ` 675 = 2:3 3:2
%e A109297 ` ` 832 = 1:6 6:1
%e A109297 ` `1008 = 1:4 2:2 4:1
%e A109297 ` `1125 = 2:2 3:3
%e A109297 ` `1350 = 1:1 2:3 3:2
%e A109297 ` `1500 = 1:2 2:1 3:3
%e A109297 ` `2176 = 1:7 7:1
%e A109297 ` `2250 = 1:1 2:2 3:3
%p A109297 a:= proc(n) option remember; local k; for k from 1+`if`(n=1, 0,
%p A109297       a(n-1)) while (l-> sort(map(i-> i[2], l)) <> sort(map(
%p A109297       i-> numtheory[pi](i[1]), l)))(ifactors(k)[2]) do od; k
%p A109297     end:
%p A109297 seq(a(n), n=1..45);  # _Alois P. Heinz_, Mar 08 2019
%t A109297 Select[Range[1000],#==1||Union[PrimePi/@First/@FactorInteger[#]]==Union[Last/@FactorInteger[#]]&] (* _Gus Wiseman_, Apr 02 2019 *)
%o A109297 (PARI) is(n) = {my(f = factor(n), p = f[,1], e = vecsort(f[,2])); for(i=1, #p, if(primepi(p[i]) != e[i], return(0))); 1}; \\ _Amiram Eldar_, Jul 30 2022
%Y A109297 Subsequence of A130091.
%Y A109297 Cf. A076954, A106177, A108352, A108371, A109298, A109301, A056239, A112798, A114640, A118914.
%Y A109297 Cf. A324524, A324525, A324571, A325127, A325128, A325130, A325131.
%K A109297 nonn
%O A109297 1,2
%A A109297 _Jon Awbrey_, Jul 08 2005
%E A109297 More terms from _Franklin T. Adams-Watters_, Dec 19 2005
%E A109297 Offset set to 1 by _Alois P. Heinz_, Mar 08 2019