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)).

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, 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, 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

Offset: 1

Author

Antti Karttunen, Apr 13 2018

Keywords

• nonn

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A052331, A069010, A302787.

Programs

PARI

up_to = 65537;
v050376 = vector(up_to);
ispow2(n) = (n && !bitand(n,n-1));
i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to,break));
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); };
A069010(n) = ((1 + (hammingweight(bitxor(n, n>>1)))) >> 1); \\ From A069010
A302790(n) = A069010(A052331(n));

Formula

a(n) = A069010(A052331(n)).

a(n) = A069010(A302787(n)).

a(n) = A001221(A302024(n)).

For all n >= 1, a(n) <= A064547(n).