A223490 Smallest Fermi-Dirac factor of n.

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

Offset: 1

Author

Reinhard Zumkeller, Mar 20 2013

Keywords

• nonn
• look

Comments

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

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• OEIS Wiki, "Fermi-Dirac representation" of n

Crossrefs

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

Cf. A302023, A302024, A302786, A302792.

Cf. also A020639.

Programs

Haskell

a223490 = head . a213925_row

Mathematica

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

PARI

up_to = 65537;
v050376 = vector(up_to);
A050376(n) = v050376[n];
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);
A223490(n) = if(1==n,n,A050376(A001511(A052331(n)))); \\ Antti Karttunen, Apr 15 2018

Formula

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

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

From Antti Karttunen, Apr 15 2018: (Start)

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

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

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

(End)