cp's OEIS Frontend

A225547 Fixed points of A225546.

%I A225547 #23 Aug 10 2022 13:09:06
%S A225547 1,2,9,12,18,24,80,108,160,216,625,720,960,1250,1440,1792,1920,2025,
%T A225547 3584,4050,5625,7500,8640,11250,15000,16128,17280,18225,21504,24300,
%U A225547 32256,36450,43008,48600,50000,67500,100000,135000,143360,162000,193536,218700,286720,321489,324000,387072,437400,450000,600000
%N A225547 Fixed points of A225546.
%C A225547 Every number in this sequence is the product of a unique subset of A225548.
%C A225547 From _Peter Munn_, Feb 11 2020: (Start)
%C A225547 The terms are the numbers whose Fermi-Dirac factors (see A050376) occur symmetrically about the main diagonal of A329050.
%C A225547 Closed under the commutative binary operation A059897(.,.). As numbers are self-inverse under A059897, the sequence thereby forms a subgroup of the positive integers under A059897.
%C A225547 (End)
%H A225547 Paul Tek, <a href="/A225547/b225547.txt">Table of n, a(n) for n = 1..10000</a>
%e A225547 The Fermi-Dirac factorization of 160 is 2 * 5 * 16. The factors 2, 5 and 16 are A329050(0,0), A329050(2,0) and A329050(0,2), having symmetry about the main diagonal of A329050. So 160 is in the sequence.
%o A225547 (PARI) A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));
%o A225547 ff(fa) = {for (i=1, #fa~, my(p=fa[i, 1]); fa[i, 1] = A019565(fa[i, 2]); fa[i, 2] = 2^(primepi(p)-1); ); fa; } \\ A225546
%o A225547 pos(k, fs) = for (i=1, #fs, if (fs[i] == k, return(i)););
%o A225547 normalize(f) = {my(list = List()); for (k=1, #f~, my(fk = factor(f[k,1])); for (j=1, #fk~, listput(list, fk[j,1]));); my(fs = Set(list)); my(m = matrix(#fs, 2)); for (i=1, #m~, m[i,1] = fs[i]; for (k=1, #f~, m[i,2] += valuation(f[k,1], fs[i])*f[k,2];);); m;}
%o A225547 isok(n) = my(fa=factor(n), fb=ff(fa)); normalize(fb) == fa; \\ _Michel Marcus_, Aug 05 2022
%Y A225547 Cf. A050376, A225546, A329050.
%Y A225547 Closed under A059895, A059896, A059897, A306697, A329329.
%Y A225547 Subsequences: A191554, A191555, A225548.
%Y A225547 Cf. fixed points of the comparable A122111 involution: A088902.
%K A225547 nonn
%O A225547 1,2
%A A225547 _Paul Tek_, May 10 2013