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 1-3 of 3 results.

A080307 Multiples of the Fermat numbers 2^(2^n)+1.

Original entry on oeis.org

3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 21, 24, 25, 27, 30, 33, 34, 35, 36, 39, 40, 42, 45, 48, 50, 51, 54, 55, 57, 60, 63, 65, 66, 68, 69, 70, 72, 75, 78, 80, 81, 84, 85, 87, 90, 93, 95, 96, 99, 100, 102, 105, 108, 110, 111, 114, 115, 117, 119, 120, 123, 125, 126, 129, 130, 132
Offset: 1

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Author

Matthew Vandermast, Feb 16 2003

Keywords

Comments

Since all the Fermat numbers are relatively prime to each other (see link), the probability that a given integer is not a multiple of the first k Fermat numbers is 2^((2^k)-1) / 2^(2^k)-1, the limit of which is 0.5 as k increases infinitely; therefore the probability that an integer is a Fermat multiple, as well as the probability that it is not, is 0.5.

Links

Crossrefs

Cf. A000215, A080308, A080309.

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

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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 1-3 of 3 results.

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