The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A356241

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A356241

a(n) is the number of distinct Fermat numbers dividing n.
(history; published version)

#11 by Peter Luschny at Sun Jul 31 07:46:30 EDT 2022

STATUS

reviewed

approved

#10 by Michel Marcus at Sun Jul 31 02:59:31 EDT 2022

STATUS

proposed

reviewed

#9 by Amiram Eldar at Sun Jul 31 02:47:29 EDT 2022

STATUS

editing

proposed

#8 by Amiram Eldar at Sun Jul 31 02:17:01 EDT 2022

LINKS

Amiram Eldar, <a href="/A356241/b356241.txt">Table of n, a(n) for n = 1..10000</a>

STATUS

approved

editing

#7 by N. J. A. Sloane at Sat Jul 30 19:40:51 EDT 2022

STATUS

proposed

approved

#6 by Amiram Eldar at Sat Jul 30 04:48:08 EDT 2022

STATUS

editing

proposed

#5 by Amiram Eldar at Sat Jul 30 04:47:12 EDT 2022

COMMENTS

A051179(n) is the least number k with such that a(k) = n.

#4 by Amiram Eldar at Sat Jul 30 04:45:42 EDT 2022

FORMULA

a(A003593(n)) = A112753(n).

#3 by Amiram Eldar at Sat Jul 30 04:40:28 EDT 2022

NAME

allocated for Amiram Eldara(n) is the number of distinct Fermat numbers dividing n.

DATA

0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 2, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 1, 0, 0, 1, 2, 0, 1

OFFSET

1,15

COMMENTS

A051179(n) is the least number k with a(k) = n.

The asymptotic density of occurrences of 0 is 1/2.

The asymptotic density of occurrences of 1 is (1/2) * Sum_{k>=0} 1/2^(2^k) = (1/2) * A007404 = 0.4082107545... .

LINKS

Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FermatNumber.html">Fermat Number</a>.

Wikipedia, <a href="http://en.wikipedia.org/wiki/Fermat_number">Fermat number</a>.

FORMULA

a(A000215(n)) = 1.

a(A051179(n)) = n.

a(n) <= A356242(n).

a(A080307(n)) > 0 and a(A080308(n)) = 0.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=0} 1/(2^(2^k)+1) = 0.5960631721... (A051158).

MATHEMATICA

f = Table[(2^(2^n) + 1), {n, 0, 5}]; a[n_] := Count[f, _?(Divisible[n, #] &)]; Array[a, 100]

CROSSREFS

Cf. A000215, A007404, A051158, A051179, A356242.

Cf. A080307 (positions of nonzeros), A080308 (positions of 0's).

KEYWORD

allocated

nonn

AUTHOR

Amiram Eldar, Jul 30 2022

STATUS

approved

editing

#2 by Amiram Eldar at Sat Jul 30 04:37:37 EDT 2022

KEYWORD

allocating

allocated