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 1-3 of 3 results.

A375933 The second-largest exponent in the prime factorization of n, or 0 if it does not exist.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Sep 03 2024

Keywords

Comments

First differs from A363127 at n = 60, and from A363131 at n = 72.
The position of the first occurrence of k = 1, 2, ..., is A167747(k+1) = 2*6^k.

Examples

			12 = 2^2 * 3^1 has 2 exponents in its prime factorization: 1 and 2. 2 is the largest and 1 is the second-largest. Therefore a(12) = 1.

Links

Crossrefs

Cf. A051903, A054861, A072774, A167747, A375932, A375934.
Cf. A363127, A363131.

Programs

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

Formula

a(n) = A051903(A375932(n)).
a(n) = 0 if and only if n is a power of a squarefree number (A072774).
a(n) = 1 if and only if n is in A375934.
a(n) <= A051903(n), with equality if and only if n = 1.
a(n!) = A054861(n) for n != 3.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{i >= 1} i * d(i) = 0.42745228287872473252..., where d(i) = Sum_{j >= i+1} d_2(i, j) and d_2(i, j) = Product_{p prime} (1 - 1/p^(i+1) + 1/p^j - 1/p^(j+1)) - Product_{p prime} (1 - 1/p^(i+1)) + [i > 1] * (Product_{p prime} (1 - 1/p^i) - Product_{p prime} (1 - 1/p^i + 1/p^j - 1/p^(j+1))), and [] is the Iverson bracket.

A375931 The product of the prime powers in the prime factorization of n that have an exponent that is equal to the maximum exponent in this factorization.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 4, 13, 14, 15, 16, 17, 9, 19, 4, 21, 22, 23, 8, 25, 26, 27, 4, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 8, 41, 42, 43, 4, 9, 46, 47, 16, 49, 25, 51, 4, 53, 27, 55, 8, 57, 58, 59, 4, 61, 62, 9, 64, 65, 66, 67, 4, 69, 70, 71
Offset: 1

Views

Author

Amiram Eldar, Sep 03 2024

Keywords

Comments

Differs from A327526 at n = 12, 20, 28, 40, 44, 45, ... .
Each positive number appears in this sequence either once or infinitely many times:
1. If m is squarefree then the only solution to a(x) = m is x = m.
2. If m = s^k is a power of a squarefree number s with k >= 2, then x = m * i is a solution to a(x) = m for all numbers i that are k-free numbers (i.e., having exponents in their prime factorizations that are all less than k) that are coprime to m.

Examples

			180 = 2^2 * 3^2 * 5, and the maximum exponent in the prime factorization of 180 is 2, which is the exponent of its prime factors 2 and 3. Therefore a(180) = 2^2 * 3^2 = (2*3)^2 = 36.

Links

Crossrefs

Cf. A001221 (omega), A001222 (Omega), A005117, A051903, A060818, A072774, A104141, A261969, A327526, A356862, A362611, A375932.

Programs

  • Mathematica
    a[n_] := Module[{f = FactorInteger[n], p, e, i, m}, p = f[[;; , 1]]; e = f[[;; , 2]]; m = Max[e]; i = Position[e, m] // Flatten; (Times @@ p[[i]])^m]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n), p = f[,1], e = f[,2], m); if(n == 1, 1, m = vecmax(e); prod(i = 1, #p, if(e[i] == m, p[i], 1))^m);}

Formula

If n = Product_{i} p_i^e_i (where p_i are distinct primes) then a(n) = Product_{i} p_i^(e_i * [e_i = max_{j} e_j]), where [] is the Iverson bracket.
a(n) = A261969(n)^A051903(n).
a(n) = n / A375932(n).
a(n) = n if and only if n is a power of a squarefree number (A072774).
A051903(a(n)) = A051903(n).
omega(a(n)) = A362611(n).
omega(a(n)) = 1 if and only if n is in A356862.
Omega(a(n)) = A051903(n) * A362611(n).
a(n!) = A060818(n) for n != 3.
Sum_{k=1..n} a(k) ~ c * n^2, where c = 3/Pi^2 = 0.303963... (A104141).

A376140 The number of divisors of n whose prime factorization has maximum exponent that is smaller than the maximum exponent in the prime factorization of n.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Sep 11 2024

Keywords

Comments

The maximum exponent in the prime factorization of 1 is considered to be A051903(1) = 0.

Links

Crossrefs

Cf. A000005, A005117, A051903, A005361, A362611, A375931, A375932.

Programs

  • Mathematica
    a[n_] := Module[{e = FactorInteger[n][[;;,2]], m}, m = Max[e]; Times@@ ((Min[#, m-1] & /@ e) + 1)]; a[1] = 0; Array[a, 100]
  • PARI
    a(n) = if(n == 1, 0, my(e = factor(n)[,2], m = vecmax(e)); vecprod(apply(x -> 1 + min(x, m-1), e)));

Formula

a(n) = Sum_{d|n} [m(d) = m(n)], where m(n) = A051903(n) and [] is the Iverson bracket.
If n = Product_{i} p_i^e_i (where p_i are distinct primes), then a(n) = Product_{i} (e_i + [e_i < Max_{i}(e_i)]).
a(n) <= 1 if and only if n is squarefree (A005117), and a(n) = 0 only for n = 1.
a(n) < A000005(n).
a(n) = A000005(A375932(n)) * A005361(A375931(n)) for n >= 2.
a(n) = A000005(n) * (A051903(n)/(A051903(n)+1))^A362611(n) for n >= 2.
Showing 1-3 of 3 results.

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