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 1-4 of 4 results.

A375032 The maximum odd exponent in the prime factorization of n, or 0 if no such exponent exists.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, 3, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 3, 0, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 0, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 0, 1, 1, 1, 3, 1
Offset: 1

Views

Author

Amiram Eldar, Jul 28 2024

Keywords

Comments

The asymptotic density of the occurrences of 0's is 0 (the asymptotic density of squares).
The asymptotic density of the occurrences of 1's is d(0) = Product_{p prime} (1 - 1/(p^2*(p+1))) = 0.881513... (A065465, asymptotic density of A335275).
The asymptotic density of the occurrences of 2*k+1, for k = 1, 2, ..., is d(k) = Product_{p prime} (1 - 1/(p^(2*k+2)*(p+1))) - Product_{p prime} (1 - 1/(p^(2*k)*(p+1))).

Links

Crossrefs

Cf. A000290, A051903, A065465, A162642, A335275, A350389, A375033.

Programs

  • Mathematica
    a[n_] := Max[0, Max[Select[FactorInteger[n][[;; , 2]], OddQ]]]; a[1] = 0; Array[a, 100]
  • PARI
    a(n) = {my(e = select(x -> (x % 2), factor(n)[,2])); if(#e == 0, 0, vecmax(e));}

Formula

max(a(n), A375033(n)) = A051903(n).
a(n) = 0 if and only if n is a square (A000290).
a(n) = 1 if and only if n is in A335275 \ A000290.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=0} (2*k+1) * d(k) = 1.30000522546018852138..., where d(k) is defined in the Comments section above.
a(n) = A051903(A350389(n)). - Amiram Eldar, Aug 17 2024

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.

A375850 The maximum even exponent in the prime factorization of n!, or 0 if no such exponent exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 4, 2, 4, 8, 8, 10, 10, 2, 6, 6, 6, 16, 16, 18, 18, 4, 4, 22, 22, 10, 6, 6, 6, 26, 26, 14, 4, 32, 32, 34, 34, 8, 18, 38, 38, 6, 6, 6, 10, 42, 42, 46, 46, 22, 12, 12, 12, 50, 50, 26, 4, 54, 54, 56, 56, 28, 30, 30, 30, 64, 64, 66, 66, 32, 32, 70
Offset: 0

Views

Author

Amiram Eldar, Aug 31 2024

Keywords

Comments

The sequence of indices of record values, 0, 6, 10, 12, 18, 20, 24, 30, 34, 36, 40, ..., are the evil numbers (A001969) multiplied by 2 (A125592).

Links

Crossrefs

Cf. A000142, A001969, A011371, A125592, A375033, A375849.

Programs

  • Mathematica
    a[n_] := Max[0, Max[Select[FactorInteger[n!][[;; , 2]], EvenQ]]]; Array[a, 100, 0]
  • PARI
    a(n) = {my(e = select(x -> !(x % 2), factor(n!)[, 2])); if(#e == 0, 0, vecmax(e));}
  • Python
    from collections import Counter
from sympy import factorint
def A375850(n): return max(filter(lambda x: x&1^1,sum((Counter(factorint(i)) for i in range(2,n+1)),start=Counter()).values()),default=0) # Chai Wah Wu, Aug 31 2024

Formula

a(n) = A375033(n!).
max(a(n), A375849(n)) = A011371(n).

A375034 The difference between the maximum odd exponent and the maximum even exponent in the prime factorization of n, where 0 is assigned to each maximum exponent if no such exponent exists.

Original entry on oeis.org

0, 1, 1, -2, 1, 1, 1, 3, -2, 1, 1, -1, 1, 1, 1, -4, 1, -1, 1, -1, 1, 1, 1, 3, -2, 1, 3, -1, 1, 1, 1, 5, 1, 1, 1, -2, 1, 1, 1, 3, 1, 1, 1, -1, -1, 1, 1, -3, -2, -1, 1, -1, 1, 3, 1, 3, 1, 1, 1, -1, 1, 1, -1, -6, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, -1, -1, 1, 1, 1, -3
Offset: 1

Views

Author

Amiram Eldar, Jul 28 2024

Keywords

Comments

The indices of high value records are 1, 2, 8, 32, 128, 512, ... (A081294 with offset 1), and the indices of low value records are 1, 4, 16, 64, 256, 1024, ... (A000302 with offset 1).

Links

Crossrefs

Cf. A000302, A081294, A368714, A375032, A375033.

Programs

  • Mathematica
    a[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Max[0, Max[Select[e, OddQ]]] - Max[0, Max[Select[e, EvenQ]]]]; a[1] = 0; Array[a, 100]
  • PARI
    a(n) = {my(e = factor(n)[,2], e1 = select(x -> (x % 2), e), e2 = select(x -> !(x % 2), e)); if(#e1 == 0, 0, vecmax(e1)) - if(#e2 == 0, 0, vecmax(e2));}

Formula

a(n) = A375032(n) - A375033(n).
a(n) = 0 if and only if n = 1.
a(n) <= 0 if and only if n is in A368714.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} (-1)^(k+1)*k*d(k) = 0.5741591604302832339078..., where d(k) = Product_{p prime} (1 - 1/(p^(k+1)*(p+1))) - Product_{p prime} (1 - 1/(p^(k-1)*(p+1))) for k >= 2, and d(1) = Product_{p prime} (1 - 1/(p^2*(p+1))).
Showing 1-4 of 4 results.

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