cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A080308 Non-multiples of Fermat numbers 2^(2^n)+1.

Original entry on oeis.org

1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 22, 23, 26, 28, 29, 31, 32, 37, 38, 41, 43, 44, 46, 47, 49, 52, 53, 56, 58, 59, 61, 62, 64, 67, 71, 73, 74, 76, 77, 79, 82, 83, 86, 88, 89, 91, 92, 94, 97, 98, 101, 103, 104, 106, 107, 109, 112, 113, 116, 118, 121, 122, 124, 127, 128, 131
Offset: 1

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Author

Matthew Vandermast, Feb 16 2003

Keywords

Comments

Complement of A080307. A080307 and A080308 each comprise one-half of the integers; see A080307.
It appears that the first 128 terms of this sequence constitute all of the primitive elements of GF(256) if each term is the exponent of the minimum primitive element for the irreducible polynomial splitting GF(2). For example, when GF(2) is split by F(x) = x^8 + x^4 + x^3 + x + 1, the minimum primitive element is a = x + 1. Then the primitive elements of the finite field are a^1, a^2, a^4, a^7, ... - Cody Planteen, Jul 27 2019

Links

Crossrefs

Cf. A000215, A080307.

A281746 Nonnegative numbers k such that k == 0 (mod 3) or k == 0 (mod 5).

Original entry on oeis.org

0, 3, 5, 6, 9, 10, 12, 15, 18, 20, 21, 24, 25, 27, 30, 33, 35, 36, 39, 40, 42, 45, 48, 50, 51, 54, 55, 57, 60, 63, 65, 66, 69, 70, 72, 75, 78, 80, 81, 84, 85, 87, 90, 93, 95, 96, 99, 100, 102, 105, 108, 110, 111, 114, 115, 117, 120, 123, 125, 126, 129, 130, 132, 135
Offset: 1

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Author

Seiichi Manyama, Jan 29 2017

Keywords

Comments

In the game "FizzBuzz", players replace any number divisible by three with the word "Fizz", and any number divisible by five with the word "Buzz". But multiples of both three and five are replaced by "FizzBuzz". For example, 1, 2, Fizz, 4, Buzz, Fizz, 7, 8, Fizz, Buzz, 11, Fizz, 13, 14, FizzBuzz, 16, 17, Fizz, 19, Buzz, Fizz, 22, 23, Fizz, Buzz, 26, Fizz, 28, 29, FizzBuzz, ...
The asymptotic density of this sequence is 7/15. - Amiram Eldar, Mar 25 2021
For a neat way to supplement the set to achieve equal density with its complement, see A080307. - Peter Munn, Oct 12 2023

Links

Crossrefs

Complement of A229829.
Union of A008585 and A008587.
Subsequence of {0} U A080307.
Cf. A281787.

Programs

  • Mathematica
    Select[Range[0, 200], MemberQ[Mod[#, {3, 5}], 0]&] (* or *) LinearRecurrence[{1, 0, 0, 0, 0, 0, 1, -1}, {0, 3, 5, 6, 9, 10, 12, 15}, 80] (* Harvey P. Dale, Apr 01 2018 *)
Union[3Range[0, 33], 5Range[20]] (* Alonso del Arte, Sep 03 2018 *)
CoefficientList[Series[-(3*x^7 + 2*x^6 + x^5 + 3*x^4 + x^3 + 2*x^2 + 3*x) / (-x^8 + x^7 + x - 1) , {x, 0, 80}], x] (* Stefano Spezia, Sep 16 2018 *)
  • PARI
    concat(0, Vec(x^2*(3 + 2*x + x^2 + 3*x^3 + x^4 + 2*x^5 + 3*x^6) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)) + O(x^100))) \\ Colin Barker, Feb 07 2017

Formula

G.f.: -(3*x^8 + 2*x^7 + x^6 + 3*x^5 + x^4 + 2*x^3 + 3*x^2) / (-x^8 + x^7 + x - 1).
a(n) = a(n-1) + a(n-7) - a(n-8) for n > 8. - Colin Barker, Feb 07 2017
a(n) = 15n/7 + O(1). - Charles R Greathouse IV, Jan 13 2025

A080309 n-th even number equals n-th multiple of a Fermat number.

Original entry on oeis.org

3, 5, 6, 9, 10, 12, 15, 17, 24, 25, 27, 30, 33, 34, 39, 40, 42, 45, 48, 50, 51, 60, 63, 65, 66, 68, 69, 70, 72, 75, 78, 80, 81, 95, 96, 111, 119, 120, 123, 125, 126, 129, 130, 132, 135, 136, 144, 159, 160, 174, 175, 177, 180, 183, 185, 186, 187, 189, 190, 192, 195, 204
Offset: 1

Views

Author

Matthew Vandermast, Feb 16 2003

Keywords

Comments

Sequence includes only multiples of Fermat numbers (sequence A080307), but not all of them. It is not certain that A080309 is infinite, but it seems likely given that exactly one-half of all integers are multiples of Fermat numbers (see A080307).

Examples

			The first 3 even numbers are 2, 4 and 6; the first 3 multiples of Fermat numbers (the numbers of the form 2^(2^n)+1) are 3, 5 and 6. The third even number is also the third Fermat multiple; thus 3 is in the sequence.

Crossrefs

Cf. A000215 (the Fermat numbers), A080307.

A339344 Lexicographically earliest sequence of odd primes such that the asymptotic density of the numbers which are divisible by at least one of these primes is 1/2.

Original entry on oeis.org

3, 5, 17, 257, 65537, 4294967311, 1229782942255939601, 88962710886098567818446141338419231, 255302062200114858892457591448999891874349780170241684791167583265041
Offset: 1

Views

Author

Amiram Eldar, Nov 30 2020

Keywords

Comments

Given a set of prime numbers P, finite or infinite, the set of numbers which are divisible by at least one of the primes in P has an asymptotic density Product_{p in P} (1 - 1/p). If P is finite, then this density is equal to 1/2 only when P = {2}. Otherwise, the density is 1/2 for infinitely many sets P. This sequence is the lexicographically earliest infinite sequence of such primes.
The first 5 terms are the Fermat primes (A019434).
a(10) = 7.455916... * 10^135 is too large to be included in the data section.

Links

Crossrefs

Cf. A000215, A019434, A080307, A125045, A262228, A339345.

Programs

  • Mathematica
    s = {}; r = 1; p = 3; Do[AppendTo[s, p]; r *= 1 - 1/p; p = NextPrime[r/(r - 1/2)], {9}]; s

Formula

a(1) = 3, a(n) = nextprime(r(n-1)/(r(n-1) - 1/2)), where r(n) = Product_{k=1..n-1} 1 - 1/a(n).
Product_{n=>1} (1 - 1/a(n)) = 1/2.

A356241 a(n) is the number of distinct Fermat numbers dividing n.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Jul 30 2022

Keywords

Comments

A051179(n) is the least number k such that 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

Crossrefs

Cf. A000215, A007404, A051158, A051179, A356242.
Cf. A080307 (positions of nonzeros), A080308 (positions of 0's).

Programs

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

Formula

a(A000215(n)) = 1.
a(A051179(n)) = n.
a(A003593(n)) = A112753(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).

A356242 a(n) is the number of Fermat numbers dividing n, counted with multiplicity.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 0, 0, 2, 0, 1, 2, 0, 1, 1, 0, 0, 1, 2, 0, 3, 0, 0, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 0, 1, 0, 0, 3, 0, 0, 1, 0, 2, 2, 0, 0, 3, 1, 0, 1, 0, 0, 2, 0, 0, 2, 0, 1, 1, 0, 1, 1, 1, 0, 2, 0, 0, 3, 0, 0, 1, 0, 1, 4, 0, 0, 1, 2, 0, 1
Offset: 1

Views

Author

Amiram Eldar, Jul 30 2022

Keywords

Comments

The multiplicity of a divisor d (not necessarily a prime) of n is defined in A169594 (see also the first formula).
A000244(n) is the least number k such that 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) = (1/2) * A051158 = 0.2980315860... .

Links

Crossrefs

Cf. A000244, A000215, A007404, A051158, A051179, A169594, A356241.
Cf. A080307 (positions of nonzeros), A080308 (positions of 0's).

Programs

  • Mathematica
    f = Table[(2^(2^n) + 1), {n, 0, 5}]; a[n_] := Total[IntegerExponent[n, f]]; Array[a, 100]

Formula

a(n) = Sum_{k>=1} v(A000215(k), n), where v(m, n) is the exponent of the largest power of m that divides n.
a(A000215(n)) = 1.
a(A000244(n)) = a(A000351(n)) = a(A001026(n)) = n.
a(A003593(n)) = A112754(n).
a(n) >= A356241(n).
a(A051179(n)) = 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)) = 0.8164215090... (A007404).
Showing 1-6 of 6 results.

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