A223491 Largest Fermi-Dirac factor of n.

1, 2, 3, 4, 5, 3, 7, 4, 9, 5, 11, 4, 13, 7, 5, 16, 17, 9, 19, 5, 7, 11, 23, 4, 25, 13, 9, 7, 29, 5, 31, 16, 11, 17, 7, 9, 37, 19, 13, 5, 41, 7, 43, 11, 9, 23, 47, 16, 49, 25, 17, 13, 53, 9, 11, 7, 19, 29, 59, 5, 61, 31, 9, 16, 13, 11, 67, 17, 23, 7, 71, 9

Offset: 1

Author

Reinhard Zumkeller, Mar 20 2013

Keywords

• nonn

Comments

Greatest Fermi-Dirac factor of n: Largest divisor of n of the form p^(2^k), for some prime p and k >= 0, with a(1) = 1. Thus for n > 1, the largest term of A050376 that divides n. - Antti Karttunen, Apr 13 2018

Links

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

Crossrefs

Cf. A223490, A050376, A034699, A000040 (subsequence), A302776, A302785, A302789 (ordinal transform).

Cf. also A006530, A034699.

Programs

Haskell

a223491 = last . a213925_row

Mathematica

f[p_, e_] := p^(2^Floor[Log2[e]]); a[n_] := Max @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 26 2020 *)

PARI

ispow2(n) = (n && !bitand(n,n-1));
A223491(n) = if(1==n,n,fordiv(n, d, if(ispow2(isprimepower(n/d)), return(n/d)))); \\ Antti Karttunen, Apr 13 2018

Formula

a(n) = A213925(n,A064547(n)).

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

From Antti Karttunen, Apr 13 2018: (Start)

a(1) = 1; for n > 1, a(n) = A050376(A302785(n)).

a(n) = n/A302776(n).

(End)