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.

A328218 Numbers k for which A006255(k) > A328045(k).

Original entry on oeis.org

2, 3, 6, 20, 24, 28, 30, 32, 35, 40, 42, 45, 56, 84, 90, 91, 99, 108, 110, 120, 126, 143, 150, 156, 165, 171, 180, 182, 189, 195, 198, 210, 220, 224, 231, 243, 245, 272, 280, 285, 294, 304, 312, 315, 323, 330, 342, 350, 378, 405, 416, 420, 432, 455, 459, 460
Offset: 1

Views

Author

Peter Kagey, Oct 08 2019

Keywords

Comments

This is a subsequence of A269045.

Crossrefs

Cf. A006255, A269045, A328045.

A328143 Number of sequences [(b_1, c_1),...,(b_t, c_t)] such that n = b_1 < b_2 < ... < b_t = A328045(n), all c_i are positive integers less than 4, and b_1^c_1*b_2^c_2*...*b_t^c_t is a fourth power.

Original entry on oeis.org

3, 3, 2, 2, 1, 12, 2, 12, 12, 1, 12, 192, 12, 768, 12, 12, 3, 12288, 12, 49152, 2, 6, 48
Offset: 0

Views

Author

Peter Kagey, Oct 04 2019

Keywords

Comments

When does a(n) = 3*4^A260510(n)? It does for n = 0, 1, 5, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, ...
a(n) = 1 if n is square but not a fourth power.
a(k^4) = 3.
a(24) = 2, a(25) = 1, a(26) = 48, a(27) = 3, and a(28) = 2.

Examples

			For n = 21 the a(21) = 6 solutions are
21^2 *               27^2 * 28^2 =  126^4,
21^3 * 24^2 *        27^1 * 28^1 =  252^4,
21^2 *        25^2 * 27^2 * 28^2 =  630^4,
21^3 * 24^2 * 25^2 * 27^1 * 28^1 = 1260^4,
21^1 * 24^2 *        27^3 * 28^3 = 1512^4, and
21^1 * 24^2 * 25^2 * 27^3 * 28^3 = 7560^4.

Crossrefs

Cf. A006255, A260510, A277494, A328045.
A259527 is the analog for squares.

A343825 Table read by antidiagonals upward: T(n,k) is the least m such that there exists a sequence k = b_1 <= b_2 <= ... <= b_t = m such that no term appears n or more times and the product of the sequence is of the form c^n, where c is an integer; n >= 1 and k >= 0.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 6, 3, 0, 1, 4, 8, 4, 0, 1, 4, 6, 4, 5, 0, 1, 4, 6, 9, 10, 6, 0, 1, 4, 6, 4, 10, 12, 7, 0, 1, 4, 6, 8, 10, 12, 14, 8, 0, 1, 4, 6, 4, 10, 9, 14, 15, 9, 0, 1, 4, 6, 8, 10, 9, 14, 8, 9, 10, 0, 1, 4, 6, 4, 10, 12, 14, 15, 16, 18, 11, 0, 1, 4
Offset: 1

Views

Author

Peter Kagey, Apr 30 2021

Keywords

Comments

For each prime p, the p-th row is a permutation of the nonprime integers.
T(n,k) <= A343881(n,k).
Conjecture: T(p,k) = A071537(k) for fixed k and sufficiently large prime p.

Examples

			Table begins:
  n\k | 0  1  2  3  4   5   6   7   8   9  10
------+--------------------------------------
    1 | 0, 1, 2, 3, 4,  5,  6,  7,  8,  9, 10
    2 | 0, 1, 6, 8, 4, 10, 12, 14, 15,  9, 18
    3 | 0, 1, 4, 6, 9, 10, 12, 14,  8, 16, 15
    4 | 0, 1, 4, 6, 4, 10,  9, 14, 15,  9, 18
    5 | 0, 1, 4, 6, 8, 10,  9, 14, 12, 15, 16
    6 | 0, 1, 4, 6, 4, 10, 12, 14,  8,  9, 15
    7 | 0, 1, 4, 6, 8, 10,  9, 14, 12, 15, 16
    8 | 0, 1, 4, 6, 4, 10,  9, 14, 12,  9, 16
Specifically,
T(2,3) =  8 because 3   * 6   * 8        = 12^2,
T(3,3) =  6 because 3   * 4^2 * 6^2      = 12^3,
T(3,5) = 10 because 5   * 6   * 9 * 10^2 = 30^3,
T(4,6) =  9 because 6^2 * 8^2 *      9^3 = 36^4, and
T(4,9) =  9 because 9^2                  =  3^4.

Crossrefs

Row n: A001477 (n=1), A006255 (n=2), A277494 (n=3), A328045 (n=4).
Cf. A071537.
Showing 1-3 of 3 results.

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