A225547 - OEIS

%I #23 Aug 10 2022 13:09:06

%S 1,2,9,12,18,24,80,108,160,216,625,720,960,1250,1440,1792,1920,2025,

%T 3584,4050,5625,7500,8640,11250,15000,16128,17280,18225,21504,24300,

%U 32256,36450,43008,48600,50000,67500,100000,135000,143360,162000,193536,218700,286720,321489,324000,387072,437400,450000,600000

%N Fixed points of A225546.

%C Every number in this sequence is the product of a unique subset of A225548.

%C From _Peter Munn_, Feb 11 2020: (Start)

%C The terms are the numbers whose Fermi-Dirac factors (see A050376) occur symmetrically about the main diagonal of A329050.

%C 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 (End)

%H Paul Tek, <a href="/A225547/b225547.txt">Table of n, a(n) for n = 1..10000</a>

%e 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 (PARI) A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));

%o 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 pos(k, fs) = for (i=1, #fs, if (fs[i] == k, return(i)););

%o 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 isok(n) = my(fa=factor(n), fb=ff(fa)); normalize(fb) == fa; \\ _Michel Marcus_, Aug 05 2022

%Y Cf. A050376, A225546, A329050.

%Y Closed under A059895, A059896, A059897, A306697, A329329.

%Y Subsequences: A191554, A191555, A225548.

%Y Cf. fixed points of the comparable A122111 involution: A088902.

%K nonn

%O 1,2

%A _Paul Tek_, May 10 2013