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 1-6 of 6 results.

A372505 a(n) = log_2(A368473(n)).

Original entry on oeis.org

0, 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, 2, 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
Offset: 1

Views

Author

Amiram Eldar, May 04 2024

Keywords

Comments

The first position of k, for k = 0, 1, ..., is 1, 4, 15, 126, 1134, ..., which is the position of A085629(2^k) in A138302.

Links

Crossrefs

Cf. A005361, A085629, A138302, A368473.
Cf. A369934, A370078, A372467, A372469.

Programs

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

Formula

a(n) = log_2(A005361(A138302(n))).

A368472 Product of exponents of prime factorization of the exponentially odd numbers.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Dec 26 2023

Keywords

Comments

The odd terms of A005361.
The first position of 2*k-1, for k = 1, 2, ..., is 1, 7, 24, 91, 154, 1444, 5777, 610, 92349, ..., which is the position of A085629(2*k-1) in A268335.

Links

Crossrefs

Cf. A005361, A013661, A065463, A085629, A098198, A268335.
Cf. A366438, A366439.
Similar sequences: A322327, A368473, A368474.

Programs

  • Mathematica
    f[n_] := Module[{p = Times @@ FactorInteger[n][[;; , 2]]}, If[OddQ[p], p, Nothing]]; Array[f, 150]
  • PARI
    lista(kmax) = {my(p); for(k = 1, kmax, p = vecprod(factor(k)[, 2]); if(p%2, print1(p, ", ")));}

Formula

a(n) = A005361(A268335(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (zeta(2)^2/d) * Product_{p prime} (1 - 3/p^2 + 3/p^3 - 1/p^5) = 1.38446562720473484463..., where d = A065463 is the asymptotic density of the exponentially odd numbers.

A368474 Product of exponents of prime factorization of the numbers whose exponents in their prime power factorization are squares (A197680).

Original entry on oeis.org

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, 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, Dec 26 2023

Keywords

Comments

All the terms are squares (A000290).
The first position of k^2, for k = 1, 2, ..., is 1, 12, 331, 834, 21512290, 26588, ..., which is the position of A085629(k^2) in A197680.

Links

Crossrefs

Cf. A000290, A005361, A085629, A197680, A357016.
Similar sequences: A322327, A368472, A368473.

Programs

  • Mathematica
    f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, IntegerQ[Sqrt[#]] &], Times @@ e, Nothing]]; Array[f, 150]
  • PARI
    lista(kmax) = {my(e, ok); for(k = 1, kmax, e = factor(k)[, 2]; ok = 1; for(i = 1, #e, if(!issquare(e[i]), ok = 0; break)); if(ok, print1(vecprod(e), ", ")));}

Formula

a(n) = A005361(A197680(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1/d) * Product_{p prime} (1 + Sum_{k>=1} k^2/p^(k^2)) = 1.16776748073813763932..., where d = A357016 is the asymptotic density of A197680.

A369933 The maximal exponent in the prime factorization of the exponentially 2^n numbers (A138302).

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2
Offset: 1

Views

Author

Amiram Eldar, Feb 06 2024

Keywords

Comments

Differs from A368473 at n = 1, 32, 89, 126, 159, ... .

Links

Crossrefs

Cf. A013661, A051903, A271727, A138302, A368473, A369934.

Programs

  • Mathematica
    pow2Q[n_] := n == 2^IntegerExponent[n, 2]; f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, pow2Q], Max @@ e, Nothing]]; f[1] = 0; Array[f, 150]
  • PARI
    ispow2(n) = n >> valuation(n, 2) == 1;
lista(kmax) = {my(e); print1(0, ", "); for(k = 2, kmax, e = factor(k)[, 2]; if(ispow2(vecprod(e)), print1(vecmax(e), ", "))); }

Formula

a(n) = A051903(A138302(n)).
a(n) = 2^A369934(n), for n >= 2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1/zeta(2) + Sum_{k>=1} (2^k * (d(k) - d(k-1)))) / A271727 = 1.40540547368932408503..., where d(k) = Product_{p prime} (1 - 1/p^3 + Sum_{i=2..k} (1/p^(2^i)-1/p^(2^i+1))) for k >= 1, and d(0) = 1/zeta(2).

A372504 Multiplicative with a(p^e) = e if e is a power of 2, and 0 otherwise.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 0, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 0, 2, 1, 0, 2, 1, 1, 1, 0, 1, 1, 1, 4, 1, 1, 1, 0, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 0, 1, 0, 1, 1, 1, 2, 1, 1, 2, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, May 04 2024

Keywords

Links

Crossrefs

Cf. A005361, A048298, A138302, A209229, A355823, A368473, A372470.

Programs

  • Mathematica
    f[p_, e_] := If[e == 2^IntegerExponent[e, 2], e, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = vecprod(apply(x -> if(x == 1 << valuation(x, 2), x, 0), factor(n)[, 2]));

Formula

Multiplicative with a(p^e) = A048298(e) = A209229(e) * e.
a(n) = A355823(n) * A005361(n).
a(A138302(n)) = A005361(A138302(n)) = A368473(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.31285540951965780409..., where f(x) = (1-x) * (1 + Sum_{k>=0} 2^k * x^(2^k)).

A382787 The product of exponents in the prime factorization of the numbers whose prime factorization contains exponents that are either 1 or even.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Apr 05 2025

Keywords

Comments

First differs from A368473 at n = 57.

Links

Crossrefs

Cf. A005361, A013661, A065465, A335275, A368473.

Programs

  • Mathematica
    f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, # == 1 || EvenQ[#] &], Times @@ e, Nothing]]; Array[f, 150]
  • PARI
    list(lim) = {my(e, ok); for(k = 1, lim, e = factor(k)[, 2]; ok = 1; for(i = 1, #e, if(e[i] > 1 && e[i]%2, ok = 0; break)); if(ok, print1(vecprod(e), ", ")));}

Formula

a(n) = A005361(A335275(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (zeta(2)^2 / A065465) * Product_{p prime} (1 - 1/p^2 - 2/p^3 + 3/p^4 - 1/p^6) = 1.568148713987289233406... .
Showing 1-6 of 6 results.

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

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