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.

A334215 T(n, k) is the greatest positive integer m such that m^k divides n; square array T(n, k), n, k > 0 read by antidiagonals downwards.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 5, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 2, 9, 1, 1, 1, 1, 1, 1, 1, 2, 3, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Rémy Sigrist, Apr 19 2020

Keywords

Examples

			Square array starts:
  n\k|   1  2  3  4  5  6  7  8  9 10
  ---+-------------------------------
    1|   1  1  1  1  1  1  1  1  1  1
    2|   2  1  1  1  1  1  1  1  1  1
    3|   3  1  1  1  1  1  1  1  1  1
    4|   4  2  1  1  1  1  1  1  1  1
    5|   5  1  1  1  1  1  1  1  1  1
    6|   6  1  1  1  1  1  1  1  1  1
    7|   7  1  1  1  1  1  1  1  1  1
    8|   8  2  2  1  1  1  1  1  1  1
    9|   9  3  1  1  1  1  1  1  1  1
   10|  10  1  1  1  1  1  1  1  1  1
   11|  11  1  1  1  1  1  1  1  1  1
   12|  12  2  1  1  1  1  1  1  1  1
   13|  13  1  1  1  1  1  1  1  1  1
   14|  14  1  1  1  1  1  1  1  1  1
   15|  15  1  1  1  1  1  1  1  1  1
   16|  16  4  2  2  1  1  1  1  1  1

Crossrefs

Cf. A000188, A051903, A053150, A053164, A060175, A261969, A334215, A334216.

Programs

  • PARI
    T(n,k) = { my (f=factor(n)); prod (i=1, #f~, f[i,1]^(f[i,2]\k)) }

Formula

T(n, 1) = n.
T(n, 2) = A000188(n).
T(n, 3) = A053150(n).
T(n, 4) = A053164(n).
T(n, A051903(n)) = A261969(n).
T(n, k) = 1 for any k > A051903(n).
T(n^k, k) = n.

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).

A334217 Irregular table T(n, k) read by rows, n > 0 and k = 1..A334216(n); n-th row corresponds to distinct terms of n-th row of A334215, in ascending order.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 6, 1, 7, 1, 2, 8, 1, 3, 9, 1, 10, 1, 11, 1, 2, 12, 1, 13, 1, 14, 1, 15, 1, 2, 4, 16, 1, 17, 1, 3, 18, 1, 19, 1, 2, 20, 1, 21, 1, 22, 1, 23, 1, 2, 24, 1, 5, 25, 1, 26, 1, 3, 27, 1, 2, 28, 1, 29, 1, 30, 1, 31, 1, 2, 4, 32, 1, 33
Offset: 1

Views

Author

Rémy Sigrist, Apr 19 2020

Keywords

Examples

			The first rows are:
  n     n-th row
  --    -------------
   1    [1]
   2    [1, 2]
   3    [1, 3]
   4    [1, 2, 4]
   5    [1, 5]
   6    [1, 6]
   7    [1, 7]
   8    [1, 2, 8]
   9    [1, 3, 9]
  10    [1, 10]
  11    [1, 11]
  12    [1, 2, 12]
  13    [1, 13]
  14    [1, 14]
  15    [1, 15]
  16    [1, 2, 4, 16]

Crossrefs

Cf. A000188, A261969, A334215, A334216.

Programs

  • PARI
    row(n) = { my (f=factor(n)); Set(apply (k -> prod (i=1, #f~, f[i,1]^(f[i,2]\k)), [1..1+if (n==1, 0, vecmax(f[,2]~))])) }

Formula

T(n, 1) = 1.
T(n, 2) = A261969(n) for any n > 1.
T(n, A334216(n)-1) = A000188(n) for any n > 1.
T(n, A334216(n)) = n.
Showing 1-3 of 3 results.

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