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.

A374324 The maximal exponent in the prime factorization of the numbers whose maximal exponent in their prime factorization is even.

Original entry on oeis.org

0, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 6, 2, 2, 2, 4, 4, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 4, 2, 2, 6, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 8, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2
Offset: 1

Views

Author

Amiram Eldar, Jul 04 2024

Keywords

Links

Crossrefs

Cf. A051903, A368714.
Similar sequences: A374325, A374326, A374327, A374328.

Programs

  • Mathematica
    f[n_] := Module[{e = If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]}, If[EvenQ[e], e, Nothing]]; Array[f, 350]
  • PARI
    lista(kmax) = {my(e); print1(0, ", "); for(k = 2, kmax, e = vecmax(factor(k)[, 2]); if(!(e % 2), print1(e, ", ")));}

Formula

a(n) = A051903(A368714(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} (2*k * (1/zeta(2*k+1) - 1/zeta(2*k))) / Sum_{k>=2} (-1)^k * (1 - 1/zeta(k)) = 2.48584683692026915946... .

A374325 The maximal exponent in the prime factorization of the numbers whose maximal exponent in their prime factorization is a square.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Jul 04 2024

Keywords

Comments

First differs from A369935 at n = 95.
The first occurrence of k^2, for k = 0, 1, ..., is at 1, 2, 12, 335, 42563, ..., which is the position of 2^(k^2) at A369937.

Links

Crossrefs

Cf. A051903, A369935, A369937.
Similar sequences: A374324, A374326, A374327, A374328.

Programs

  • Mathematica
    f[n_] := Module[{e = If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]}, If[IntegerQ@ Sqrt[e], e, Nothing]]; Array[f, 200]
  • PARI
    lista(kmax) = {my(e); print1(0, ", "); for(k = 2, kmax, e = vecmax(factor(k)[, 2]); if(issquare(e), print1(e, ", ")));}

Formula

a(n) = A374326(n)^2.
a(n) = A051903(A369937(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} k^2 * d(k) / Sum_{k>=1} d(k) = 1.19949058780036416105..., where d(k) = 1/zeta(k^2+1) - 1/zeta(k^2) for k>=2, and d(1) = 1/zeta(2).

A374326 The square root of the maximal exponent in the prime factorization of the numbers whose maximal exponent in their prime factorization is a square.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Jul 04 2024

Keywords

Comments

First differs from A369936 at n = 95.
The first occurrence of k = 0, 1, ... is at 1, 2, 12, 335, 42563, ..., which is the position of 2^(k^2) at A369937.

Links

Crossrefs

Cf. A051903, A369936, A369937.
Similar sequences: A374324, A374325, A374327, A374328.

Programs

  • Mathematica
    f[n_] := Module[{s = Sqrt[If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]]}, If[IntegerQ[s], s, Nothing]]; Array[f, 200]
  • PARI
    lista(kmax) = {my(e); print1(0, ", "); for(k = 2, kmax, e = vecmax(factor(k)[, 2]); if(issquare(e), print1(sqrtint(e), ", ")));}

Formula

a(n) = sqrt(A374325(n)).
a(n) = sqrt(A051903(A369937(n))).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} k * d(k) / Sum_{k>=1} d(k) = 1.06543556163434367736..., where d(k) = 1/zeta(k^2+1) - 1/zeta(k^2) for k>=2, and d(1) = 1/zeta(2).

A374328 The base-2 logarithm of the maximal exponent in the prime factorization of the numbers whose maximal exponent in their prime factorization is a power of 2.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Jul 04 2024

Keywords

Comments

First differs from {A369934(n+1), n>=1} at n = 378.
The first occurrence of k = 0, 1, ... is at 1, 3, 14, 224, 57307, ..., which is the position of 2^(2^k) at A369938.

Links

Crossrefs

Cf. A051903, A369934, A369938.
Similar sequences: A374324, A374325, A374326, A374327.

Programs

  • Mathematica
    f[n_] := Module[{e = If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]}, If[e == 2^IntegerExponent[e, 2], IntegerExponent[e, 2], Nothing]]; Array[f, 150]
  • PARI
    lista(kmax) = {my(e); for(k = 2, kmax, e = vecmax(factor(k)[, 2]); if(e >> valuation(e, 2) == 1, print1(valuation(e, 2), ", ")));}

Formula

a(n) = log_2(A374327(n)).
a(n) = log_2(A051903(A369938(n))).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=0} k * d(k) / Sum_{k>=0} d(k) = 0.35575339438419889972..., where d(k) = 1/zeta(2^k+1) - 1/zeta(2^k) for k>=1, and d(0) = 1/zeta(2).
Showing 1-4 of 4 results.

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