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.

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A377792 Irregular triangle where row n lists squarefree composite k with lpf(k) = prime(n) such that m <= Omega(k), where lpf = A020639, m = floor(log k / log lpf(k)), and Omega = A001222.

Original entry on oeis.org

6, 15, 21, 35, 55, 65, 85, 95, 115, 385, 455, 595, 77, 91, 119, 133, 161, 203, 217, 259, 287, 301, 329, 1001, 1309, 1463, 1547, 1729, 1771, 2093, 2233, 2261, 2387, 143, 187, 209, 253, 319, 341, 407, 451, 473, 517, 583, 649, 671, 737, 781, 803, 869, 913, 979, 1067
Offset: 1

Views

Author

Michael De Vlieger, Nov 07 2024

Keywords

Comments

Permutation of A377713.
Proper subset of A120944.

Examples

			Let b(n) = A377793(n).
In A377713, there are terms k with smallest prime factor prime(n) as follows:
    Prime(n)  | b(n) | k such that floor(log_lpf(k) k) <= Omega(k)
-------------------------------------------------------------------------------
prime(1) =  2 |   1  | 6
prime(2) =  3 |   2  | 15, 27
prime(3) =  5 |   9  | 35, 55, 65, 85, 95, 115, 385, 455, 595
prime(4) =  7 |  21  | 77, 91, 119, 133, 161, 203, 217, 259, 287, 301, 329, 1001,
              |      | 1309, 1463, 1547, 1729, 1771, 2093, 2233, 2261, 2387
prime(5) = 11 | 128  | 143, 187, 209, ..., 1733303

Links

Crossrefs

Cf. A001222, A020639, A120944, A377713, A377793.

Programs

  • Mathematica
    Table[p = Prime[i]; m = p^3;
  Set[{w, t}, {{p, NextPrime[p]}, False}];
  Union@ Reap[
    Do[Set[s, Times @@ w];
      If[s < m,
        AppendTo[w, NextPrime@ Last[w]]; m *= p; Sow[s],
        If[Length[w] < 3, Break[],
          w = Append[w[[;; -3]], NextPrime@ w[[-2]] ]; m /= p] ],
      Infinity] ][[-1, 1]], {i, 10}] // Flatten

A377793 a(n) is the number of squarefree composite k with lpf(k) = prime(n) such that m <= Omega(k), where lpf = A020639, m = floor(log k / log lpf(k)), and Omega = A001222.

Original entry on oeis.org

1, 2, 9, 21, 128, 194, 713, 874, 2276, 11898, 12522, 52469, 103824, 99930, 173685, 534743, 1608864, 1438340, 3894769, 5881191, 5008669, 11802600, 16274460, 36220208, 132526590, 178177142
Offset: 1

Views

Author

Michael De Vlieger, Nov 07 2024

Keywords

Comments

a(n) is the number of terms in A377713 with least prime factor prime(n).

Examples

			In A377713, there are terms k with smallest prime factor prime(n) as follows:
    Prime(n)  | a(n) | k such that floor(log_lpf(k) k) <= Omega(k)
-------------------------------------------------------------------------------
prime(1) =  2 |   1  | 6
prime(2) =  3 |   2  | 15, 27
prime(3) =  5 |   9  | 35, 55, 65, 85, 95, 115, 385, 455, 595
prime(4) =  7 |  21  | 77, 91, 119, 133, 161, 203, 217, 259, 287, 301, 329, 1001,
              |      | 1309, 1463, 1547, 1729, 1771, 2093, 2233, 2261, 2387
prime(5) = 11 | 128  | 143, 187, 209, ..., 1733303

Crossrefs

Cf. A001222, A020639, A120944, A377713, A377792.

Programs

  • Mathematica
    Table[c = 0; p = Prime[i]; m = p^3;
  Set[{w, t}, {{p, NextPrime[p]}, False}];
  Do[Set[s, Times @@ w];
    If[s < m,
      AppendTo[w, NextPrime@ Last[w] ]; m *= p; c++,
      If[Length[w] < 3, Break[],
        w = Append[w[[;; -3]], NextPrime@ w[[-2]] ]; m /= p] ],
    Infinity]; c, {i, 12}]

Formula

a(n) = length of row n of A377792.

A377794 a(n) is the greatest number of prime factors with multiplicity of squarefree composite k such that k has lpf(k) = prime(n) such that m <= a(n), where lpf = A020639, m = floor(log k / log lpf(k)).

Original entry on oeis.org

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

Views

Author

Michael De Vlieger, Nov 07 2024

Keywords

Comments

The smallest k such that lpf(k) = prime(n) with Omega(k) = A001222(k) = a(n) is the product of prime(n..n+a(n)-1).

Examples

			Table relating the first 12 terms with prime decomposition of smallest k in A377713 (or A377792) such that lpf(k) = prime(n) and Omega(k) = a(n):
   n                  k   prime factors of k                         a(n)
  -----------------------------------------------------------------------
   1                  6    2 *  3                                      2
   2                 15    3 *  5                                      2
   3                385    5 *  7 * 11                                 3
   4               1001    7 * 11 * 13                                 3
   5            1062347   11 * 13 * 17 * 19 * 23                       5
   6            2800733   13 * 17 * 19 * 23 * 29                       5
   7          247110827   17 * 19 * 23 * 29 * 31 * 37                  6
   8          595973171   19 * 23 * 29 * 31 * 37 * 41                  6
   9        63392725189   23 * 29 * 31 * 37 * 41 * 43 * 47             7
  10      8618654420261   29 * 31 * 37 * 41 * 43 * 47 * 53 * 59        8
  11     18128893780549   31 * 37 * 41 * 43 * 47 * 53 * 59 * 61        8
  12   2781907990776503   37 * 41 * 43 * 47 * 53 * 59 * 61 * 67 * 71   9

Links

Crossrefs

Cf. A001222, A020639, A120944, A377713, A377792.

Programs

  • Mathematica
    Table[j = 1; While[Times @@ Prime[Range[i + 1, i + j]] < Prime[i]^(j + 1), j++]; j, {i, 120}]
Showing 1-3 of 3 results.

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