cp's OEIS Frontend

A302790 Number of runs of consecutive Fermi-Dirac factors of n (the runs are separated by gaps between indices of factors): a(n) = A069010(A052331(n)).

%I A302790 #11 Apr 16 2018 22:11:22
%S A302790 0,1,1,1,1,1,1,2,1,2,1,1,1,2,2,1,1,2,1,1,2,2,1,1,1,2,2,2,1,2,1,2,2,2,
%T A302790 1,2,1,2,2,2,1,2,1,2,2,2,1,2,1,2,2,2,1,2,2,3,2,2,1,1,1,2,1,2,2,2,1,2,
%U A302790 2,2,1,3,1,2,2,2,2,2,1,2,1,2,1,2,2,2,2,3,1,3,2,2,2,2,2,2,1,2,1,2,1,2,1,3,2
%N A302790 Number of runs of consecutive Fermi-Dirac factors of n (the runs are separated by gaps between indices of factors): a(n) = A069010(A052331(n)).
%H A302790 Antti Karttunen, <a href="/A302790/b302790.txt">Table of n, a(n) for n = 1..65537</a>
%F A302790 a(n) = A069010(A052331(n)).
%F A302790 a(n) = A069010(A302787(n)).
%F A302790 a(n) = A001221(A302024(n)).
%F A302790 For all n >= 1, a(n) <= A064547(n).
%e A302790 n = 84 has Fermi-Dirac factorization as A050376(2) * A050376(3) * A050376(5) = 3*4*7. Because there is a gap between A050376(3) and A050376(5), the factors occur in two separate runs (3*4 and 7), thus a(84) = 2.
%o A302790 (PARI)
%o A302790 up_to = 65537;
%o A302790 v050376 = vector(up_to);
%o A302790 ispow2(n) = (n && !bitand(n,n-1));
%o A302790 i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to,break));
%o A302790 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 A302790 A069010(n) = ((1 + (hammingweight(bitxor(n, n>>1)))) >> 1); \\ From A069010
%o A302790 A302790(n) = A069010(A052331(n));
%Y A302790 Cf. A052331, A069010, A302787.
%K A302790 nonn
%O A302790 1,8
%A A302790 _Antti Karttunen_, Apr 13 2018