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.

A368711 The maximal exponent in the prime factorization of the exponentially odd numbers (A268335).

Original entry on oeis.org

0, 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, Jan 04 2024

Keywords

Comments

Differs from A368472 at n = 1, 154, 610, 707, 762, ... .

Links

Crossrefs

Cf. A033150, A051903, A268335, A368472.
Similar sequences: A368710, A368712, A368713.

Programs

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

Formula

a(n) = A051903(A268335(n)).
a(n) is odd for n >= 2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + 2 * Sum_{k>=1} (1 - Product_{p prime} (1 - 1/(p^(2*k-1)*(p^2+p-1)))) = 1.34877064483679975726... .

A368473 Product of exponents of prime factorization of the exponentially 2^n-numbers (A138302).

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, 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, Dec 26 2023

Keywords

Comments

The terms of A005361 that are powers of 2 (A000079).
The first position of 2^k, for k = 0, 1, ..., is 1, 4, 15, 126, 1134, ..., which is the position of A085629(2^k) in A138302.

Links

Crossrefs

Cf. A000079, A005361, A085629, A138302, A271727.
Cf. A366538, A367169.
Similar sequences: A322327, A368472, A368474.

Programs

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

Formula

a(n) = A005361(A138302(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1/d) * Product_{p prime} (1 + Sum_{k>=0} 2^k/p^(2^k)) = 1.504710204899266020302..., where d = A271727 is the asymptotic density of the exponentially 2^n-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.

A370079 The product of the odd exponents of the prime factorization of n.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Feb 09 2024

Keywords

Comments

First differs from A363329 at n = 32.

Links

Crossrefs

Cf. A005361, A013661, A268335, A335275, A350387, A350389, A363329, A368472, A370080.

Programs

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

Formula

a(n) = A005361(A350389(n)).
Multiplicative with a(p^e) = e if e is odd, and 1 if e is even.
a(n) = A005361(n)/A370080(n).
a(n) >= 1, with equality if and only if n is in A335275.
a(n) <= A005361(n), with equality if and only if n is an exponentially odd number (A268335).
Dirichlet g.f.: zeta(2*s)^2 * Product_{p prime} (1 + 1/p^s - 1/p^(2*s) + 1/p^(3*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2)^2 * Product_{p prime} (1 - 2/p^2 + 2/p^3 - 1/p^4) = 1.32800597172596287374... .
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 2/((p^s - 1)*(p^s + 1)^2)). - Vaclav Kotesovec, Feb 11 2024
Showing 1-4 of 4 results.

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

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