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Smallest Fermi-Dirac factor of n.
10

%I #14 Nov 26 2020 03:18:50

%S 1,2,3,4,5,2,7,2,9,2,11,3,13,2,3,16,17,2,19,4,3,2,23,2,25,2,3,4,29,2,

%T 31,2,3,2,5,4,37,2,3,2,41,2,43,4,5,2,47,3,49,2,3,4,53,2,5,2,3,2,59,3,

%U 61,2,7,4,5,2,67,4,3,2,71,2,73,2,3,4,7,2,79

%N Smallest Fermi-Dirac factor of n.

%C Note that this is not equal to the smallest Fermi-Dirac prime (A050376) dividing n, which is always A020639(n). - _Antti Karttunen_, Apr 15 2018

%H Reinhard Zumkeller, <a href="/A223490/b223490.txt">Table of n, a(n) for n = 1..10000</a>

%H OEIS Wiki, <a href="/wiki/%22Fermi-Dirac_representation%22_of_n">"Fermi-Dirac representation" of n</a>

%F a(n) = A213925(n,1).

%F A209229(A100995(a(n))) = 1; A010055(a(n)) = 1.

%F From _Antti Karttunen_, Apr 15 2018: (Start)

%F a(1) = 1; and for n > 1, a(n) = A050376(A302786(n)).

%F a(n) = n / A302792(n).

%F a(n) = A302023(A020639(A302024(n))).

%F (End)

%t f[p_, e_] := p^(2^IntegerExponent[e, 2]); a[n_] := Min @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 26 2020 *)

%o (Haskell)

%o a223490 = head . a213925_row

%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)))); \\ _Antti Karttunen_, Apr 15 2018

%Y Cf. A223491, A050376, A028233, A000040 (subsequence).

%Y Cf. A302023, A302024, A302786, A302792.

%Y Cf. also A020639.

%K nonn,look

%O 1,2

%A _Reinhard Zumkeller_, Mar 20 2013