A302786 Index of the smallest Fermi-Dirac factor of n, a(1) = 0 by convention: a(n) = A302778(A223490(n)).

0, 1, 2, 3, 4, 1, 5, 1, 6, 1, 7, 2, 8, 1, 2, 9, 10, 1, 11, 3, 2, 1, 12, 1, 13, 1, 2, 3, 14, 1, 15, 1, 2, 1, 4, 3, 16, 1, 2, 1, 17, 1, 18, 3, 4, 1, 19, 2, 20, 1, 2, 3, 21, 1, 4, 1, 2, 1, 22, 2, 23, 1, 5, 3, 4, 1, 24, 3, 2, 1, 25, 1, 26, 1, 2, 3, 5, 1, 27, 4, 28, 1, 29, 2, 4, 1, 2, 1, 30, 1, 5, 3, 2, 1, 4, 1, 31, 1, 6, 3, 32, 1, 33, 1, 2

Offset: 1

Author

Antti Karttunen, Apr 13 2018

Keywords

• nonn

Links

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

Crossrefs

A left inverse of A050376.

Cf. A001511, A052331, A223490, A302778, A302785, A302787, A302788 (ordinal transform), A302789.

Cf. also A055396.

Programs

Mathematica

nn = 105; t = {}; k = 1;
While[lim = nn^(1/k); lim > 2,
     t = Join[t, Prime[Range[PrimePi[lim]]]^k]; k = 2 k];
A050376 = Union[t];
A223490[n_] := Table[{p, e} = pe; p^(2^IntegerExponent[e, 2]), {pe, FactorInteger[n]}] // Min;
a[n_] := If[n == 1, 0, FirstPosition[A050376, A223490[n]][[1]]];
Array[a, nn] (* Jean-François Alcover, Jan 08 2022, after T. D. Noe in A050376 *)

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); };
A001511(n) = 1+valuation(n,2);
A302786(n) = if(1==n,0,A001511(A052331(n)));

Formula

a(n) = A302778(A223490(n)).

a(1) = 0; for n > 1, a(n) = A001511(A052331(n)).

For n >= 1, a(A050376(n)) = n.

For n > 1, A050376(a(n)) = A223490(n).