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.

A112754 Total number of prime factors of n-th number of the form 3^i*5^j.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 3, 3, 4, 3, 4, 4, 5, 4, 5, 4, 5, 6, 5, 6, 5, 6, 7, 5, 6, 7, 6, 7, 8, 6, 7, 8, 6, 7, 8, 9, 7, 8, 9, 7, 8, 9, 10, 7, 8, 9, 10, 8, 9, 10, 11, 8, 9, 10, 11, 8, 9, 10, 11, 12, 9, 10, 11, 12, 9, 10, 11, 12, 13, 9, 10, 11, 12, 13, 10, 11, 12, 13, 14, 10, 11, 12, 13, 14, 10, 11, 12
Offset: 1

Views

Author

Reinhard Zumkeller, Sep 18 2005

Keywords

Links

Crossrefs

Cf. A001222, A003593, A022336, A022337, A069352, A112753, A112759.

Programs

  • Mathematica
    s = {}; m = 12; Do[n = 5^k; While[n <= 5^m, AppendTo[s, n]; n *= 3], {k, 0, m}]; PrimeOmega[Union[s]] (* Amiram Eldar, Feb 06 2020 *)

Formula

a(n) = A001222(A003593(n)) = A022336(n) + A022337(n).

A112758 Number of distinct prime factors of n-th 5-smooth number.

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, Sep 18 2005

Keywords

Links

Crossrefs

Cf. A001221, A051037, A086412, A112753, A112759, A112760, A112761, A112762.

Programs

  • Mathematica
    aa = {}; Do[If[8 n - EulerPhi[30 n] == 0, AppendTo[aa, n]], {n, 1, 100}]; PrimeNu[aa]  (* G. C. Greubel, May 07 2017 *)
PrimeNu[#]&/@Select[Range[2000],Max[FactorInteger[#][[All,1]]]<6&] (* Harvey P. Dale, Apr 12 2020 *)

Formula

a(n) = A001221(A051037(n)).
a(n) = 3 - 0^A112760(n) - 0^A112761(n) - 0^A112762(n).
a(n) <= 3.

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).
Showing 1-3 of 3 results.

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