%I #20 Nov 19 2022 04:46:22
%S 1,1,1,1,1,3,1,4,1,5,1,4,1,7,5,1,1,9,1,5,7,11,1,12,1,13,9,7,1,15,1,16,
%T 11,17,7,9,1,19,13,20,1,21,1,11,9,23,1,16,1,25,17,13,1,27,11,28,19,29,
%U 1,20,1,31,9,16,13,33,1,17,23,35,1,36,1,37,25,19,11,39,1,16,1,41,1,28,17,43,29,44,1,45,13,23,31,47,19,48,1
%N a(1) = 1; for n>1, a(n) = n/(smallest Fermi-Dirac factor of n).
%C The positive integers that are absent from this sequence are A036554, integers that have 2 as a Fermi-Dirac factor. - _Peter Munn_, Apr 23 2018
%C a(n) is the largest aliquot infinitary divisor of n, for n > 1 (cf. A077609). - _Amiram Eldar_, Nov 19 2022
%H Antti Karttunen, <a href="/A302792/b302792.txt">Table of n, a(n) for n = 1..65537</a>
%F a(n) = n / A223490(n).
%F a(n) = A300841(A302023(A032742(A302024(n)))).
%t f[p_, e_] := p^(2^IntegerExponent[e, 2]); a[n_] := n / Min @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 27 2020 *)
%o (PARI)
%o up_to = 65537;
%o v050376 = vector(up_to);
%o A050376(n) = v050376[n];
%o ispow2(n) = (n && !bitand(n,n-1));
%o i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to,break));
%o A052331(n) = { my(s=0,e); while(n > 1, fordiv(n, d, if(((n/d)>1)&&ispow2(isprimepower(n/d)), e = vecsearch(v050376, n/d); if(!e, print("v050376 too short!"); return(1/0)); s += 2^(e-1); n = d; break))); (s); };
%o A001511(n) = 1+valuation(n,2);
%o A223490(n) = if(1==n,n,A050376(A001511(A052331(n))));
%o A302792(n) = (n/A223490(n));
%o (PARI) a(n) = {if(n==1, 1, my(f = factor(n)); for(i=1, #f~, f[i,1] = f[i,1]^(1<<valuation(f[i,2], 2))); n/vecmin(f[,1]));} \\ _Amiram Eldar_, Nov 19 2022
%Y Cf. A001511, A036554, A050376, A052331, A077609, A223490, A302023, A302024, A302776, A300841.
%Y Cf. A084400 (gives the positions of 1's).
%Y Cf. also A032742.
%K nonn
%O 1,6
%A _Antti Karttunen_, Apr 13 2018