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 1-5 of 5 results.

A384048 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is squarefree.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 7, 8, 10, 11, 9, 13, 14, 15, 15, 17, 16, 19, 15, 21, 22, 23, 21, 24, 26, 26, 21, 29, 30, 31, 31, 33, 34, 35, 24, 37, 38, 39, 35, 41, 42, 43, 33, 40, 46, 47, 45, 48, 48, 51, 39, 53, 52, 55, 49, 57, 58, 59, 45, 61, 62, 56, 63, 65, 66, 67, 51
Offset: 1

Views

Author

Amiram Eldar, May 18 2025

Keywords

Links

Crossrefs

Unitary analog of A063659.
Cf. A000010, A005117, A055231, A057521, A065466, A384046, A384047.
The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), this sequence (squarefree), A384049 (cubefree), A384050 (powerful), A384051 (cubefull), A384052 (square), A384053 (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).

Programs

  • Mathematica
    f[p_, e_] := If[e == 1, p, p^e-1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - if(f[i,2] == 1, 0, 1));}

Formula

Multiplicative with a(p) = p and a(p^e) = p^e - 1 if e >= 2.
a(n) = n * A047994(n) / A384050(n).
a(n) = A047994(A057521(n)) * A055231(n) = A000010(A055231(n)) * A057521(n).
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^s - 1/p^(2*s) + 1/p^(3*s-1)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/(p^3*(p+1))) = 0.947733... (A065466).

A058035 Largest 4th-power-free number dividing n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 8, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 8, 65, 66, 67, 68, 69, 70, 71, 72
Offset: 1

Views

Author

Henry Bottomley, Nov 16 2000

Keywords

Examples

			a(96) = 24 since the factors of 96 are {1,2,3,4,6,8,12,16,24,32,48,96} but 32, 48 and 96 all contain a 4th power factor (16).

Links

Crossrefs

Cf. A007947, A007948, A008835, A046100, A053165, A065466.
Cf. A027748, A124010.

Programs

  • Haskell
    a058035 n = product $
   zipWith (^) (a027748_row n) (map (min 3) $ a124010_row n)
-- Reinhard Zumkeller, Jan 06 2012
  • Mathematica
    f[p_, e_] := p^Min[e, 3]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 09 2022 *)
  • PARI
    a(n) = my(f=factor(n)); for(k=1,#f~,f[k,2]=min(3,f[k,2])); factorback(f); \\ Michel Marcus, Sep 13 2017

Formula

Multiplicative with a(p^e) = p ^ min(e,3), p prime, e > 0. - Reinhard Zumkeller, Jan 06 2012
Sum_{k=1..n} a(k) ~ (1/2) * c * n^2, where c = Product_{p prime} (1 - 1/(p^3*(p+1))) = 0.947733... (A065466). - Amiram Eldar, Oct 13 2022

A116393 Decimal expansion of Product(1 - 1/(p+1)^3), p prime >= 2.

Original entry on oeis.org

9, 4, 0, 3, 0, 0, 4, 1, 4, 5, 7, 0, 3, 4, 5, 0, 7, 0, 6, 1, 9, 4, 6, 5, 3, 8, 5, 3, 7, 3, 2, 3, 9, 4, 4, 9, 9, 9, 8, 5, 7, 4, 0, 1, 4, 3, 6, 9, 1, 3, 0, 0, 7, 2, 5, 7, 0, 3, 1, 3, 3, 8, 3, 0, 5, 9, 4, 7, 4, 6, 2, 1, 0, 4, 5, 5, 4, 5, 7, 8, 0, 1, 2, 6, 7, 0, 4
Offset: 0

Views

Author

N. J. A. Sloane, Apr 15 2006

Keywords

Comments

Calculated from A065465 * A065467^3 * X / A065466^2 where X = 1.03108637675008536... = product[ 1+ (4p^13 +13p^12 +10p^11 -3p^10 -9p^9 -12p^8 -9p^7 +6p^5 +4p^4 +2p^3 +p^2 -p -1) /(p^5+p^4-1)^3 / (p^3+p^2-1) / (p+1)] over prime p >=2. - R. J. Mathar, Sep 10 2007

Examples

			0.9403004...

Crossrefs

Cf. A065472.

Programs

  • PARI
    prodeulerrat(1 - 1/(p+1)^3) \\ Amiram Eldar, Nov 29 2020

Extensions

More terms from John W. Layman and Zak Seidov, Apr 20 2006
More terms from R. J. Mathar, Sep 10 2007
More terms from Amiram Eldar, Nov 29 2020

A375031 Numbers whose prime factorization has at least one exponent that equals 2 and no higher even exponent.

Original entry on oeis.org

4, 9, 12, 18, 20, 25, 28, 36, 44, 45, 49, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 98, 99, 100, 108, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 164, 169, 171, 172, 175, 180, 188, 196, 198, 200, 204, 207, 212, 220, 225, 228, 234, 236, 242, 244, 245
Offset: 1

Views

Author

Amiram Eldar, Jul 28 2024

Keywords

Comments

Subsequence of A304365 and differs from it by not having the terms 1, 144, 216, 324, 400, ... .
Subsequence of A038109 and differs from it by not having the terms 144, 324, 400, 576, 720, ... .
Numbers whose largest unitary divisor that is a square (A350388) is a square of squarefree number (A062503) that is larger than 1.
Each term is a product of two coprime numbers: an exponentially odd number (A268335) and a square of a squarefree number (A062503) that is larger than 1.
The asymptotic density of this sequence is Product_{p prime} (1 - 1/(p^3*(p+1))) - Product_{p prime}(1 - 1/(p*(p+1))) = A065466 - A065463 = 0.2432910611445097832029... .

Examples

			4 = 2^2 is a term because it has the exponent 2 in its prime factorization, and no higher even exponent.
144 = 2^4 * 3^2 is not a term because it has the exponent 4 in its prime factorization which is even and larger than 2.

Links

Crossrefs

Subsequence of A013929, A038109 and A304365.
A062503 \ {1} is a subsequence.
Cf. A065463, A065466, A268335, A335275, A350388, A375033.

Programs

  • Mathematica
    q[n_] := Max[Select[FactorInteger[n][[;; , 2]], EvenQ]] == 2; Select[Range[250], q]
  • PARI
    is(k) = {my(e = select(x -> !(x % 2), factor(k)[,2])); #e > 0 && vecmax(e) == 2;}

Formula

A375033(a(n)) = 2.

A078085 Continued fraction expansion of Product_{p prime} (1 - 1/(p^3*(p+1))).

Original entry on oeis.org

0, 1, 18, 7, 1, 1, 5, 1, 3, 6, 1, 2, 1, 1, 1, 38, 1, 3, 1, 1, 1, 1, 7, 2, 2, 2, 1, 24, 8, 8, 1, 3, 1, 1, 2, 35, 2, 1, 4, 1, 15, 2, 3, 6, 64, 3, 2, 2, 1, 2, 1, 30, 2, 1, 1, 8, 1, 2, 2, 1, 4, 24, 3, 7, 2, 1, 2, 5, 2, 2, 4, 9, 2, 1, 1, 1, 3, 1, 4, 5, 1, 993, 1, 1, 2, 2, 3, 1, 1, 1, 1, 3, 14, 5, 1
Offset: 0

Views

Author

Benoit Cloitre, Dec 02 2002

Keywords

Crossrefs

Cf. A065466.

Programs

  • PARI
    contfrac(prodeulerrat(1 - 1/(p^3*(p+1)))) \\ Amiram Eldar, Mar 13 2021

Extensions

Offset changed by Andrew Howroyd, Jul 02 2024
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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