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.

A380543 Nonsquarefree weak numbers k whose squarefree kernel is a primorial.

Original entry on oeis.org

12, 18, 24, 48, 54, 60, 90, 96, 120, 150, 162, 180, 192, 240, 270, 300, 360, 384, 420, 450, 480, 486, 540, 600, 630, 720, 750, 768, 810, 840, 960, 1050, 1080, 1200, 1260, 1350, 1440, 1458, 1470, 1500, 1536, 1620, 1680, 1890, 1920, 2100, 2160, 2250, 2400, 2430
Offset: 1

Views

Author

Michael De Vlieger, Jul 15 2025

Keywords

Comments

Numbers in this sequence have the following properties:
The number a(n) is such that rad(a(n))^2 does not divide a(n), i.e., a(n) is not powerful (i.e., in A001694), where rad = A007947.
For i > 1, prime(i) | a(n) implies prime(i-1) | a(n).

Examples

			Table of n, a(n) and prime decomposition for n = 1..12:
 n   a(n)  prime decomposition
------------------------------
 1    12   2^2 * 3
 2    18   2   * 3^2
 3    24   2^3 * 3
 4    48   2^4 * 3
 5    54   2   * 3^3
 6    60   2^2 * 3   * 5
 7    90   2   * 3^2 * 5
 8    96   2^5 * 3
 9   120   2^3 * 3   * 5
10   150   2   * 3   * 5^2
11   162   2   * 3^4
12   180   2^2 * 3^2 * 5

Links

Crossrefs

Cf. A001694, A002110, A007947, A013929, A052485, A055932, A126706, A286708, A332785, A369374, A386223, A386224.

Programs

  • Mathematica
    (* Load Fast Mathematica algorithm for A055932 linked at A377854, then: *)
rad[x_] := Times @@ FactorInteger[x][[All, 1]]; Select[Union@ Flatten[f[6][[3 ;; -1, 2 ;; -1]] ], ! Divisible[#, rad[#]^2] &]

Formula

Intersection of A055932 and A332785, where A332785 = A052485 \ A005117 = A126706 \ A001694.
The union of this sequence and A369374 is A126706.

A378002 Achilles numbers that are products of primorials.

Original entry on oeis.org

72, 288, 432, 864, 1152, 1800, 2592, 3456, 4608, 5400, 6912, 7200, 10368, 10800, 15552, 18432, 21600, 27648, 28800, 31104, 41472, 43200, 54000, 55296, 62208, 64800, 73728, 86400, 88200, 93312, 108000, 115200, 124416, 162000, 165888, 172800, 194400, 221184, 259200
Offset: 1

Views

Author

Michael De Vlieger, Nov 16 2024

Keywords

Comments

Products of primorials that are powerful but not perfect powers.

Examples

			Prime power decomposition of the first 12 terms:
   a(1) =   72 = 2^3 * 3^2
   a(2) =  288 = 2^5 * 3^2
   a(3) =  432 = 2^4 * 3^3
   a(4) =  864 = 2^5 * 3^3
   a(5) = 1152 = 2^7 * 3^2
   a(6) = 1800 = 2^3 * 3^2 * 5^2
   a(7) = 2592 = 2^5 * 3^4
   a(8) = 3456 = 2^7 * 3^3
   a(9) = 4608 = 2^9 * 3^2
  a(10) = 5400 = 2^3 * 3^3 * 5^2
  a(11) = 6912 = 2^8 * 3^3
  a(12) = 7200 = 2^5 * 3^2 * 5^2

Links

Crossrefs

Cf. A001597, A001694, A002110, A007947, A052486, A025487, A286708, A364930, A377854.

Programs

  • Mathematica
    (* First load function f in A025487, then: *)
Select[Rest@ Union@ Flatten@ f[14],
 And[Divisible[#, Apply[Times, #2[[All, 1]] ]^2],
   GCD @@ #2[[All, -1]] == 1] & @@ {#, FactorInteger[#]} &]

Formula

Intersection of A286708 \ A001597 and A025487.
Intersection of A052486 and A025487.
Proper subset of A364930, in turn a proper subset of A369374.
Proper subset of A377854.

A386433 Achilles numbers with a primorial squarefree kernel that are not products of primorials.

Original entry on oeis.org

108, 648, 972, 1944, 2700, 3888, 4500, 8748, 9000, 13500, 16200, 17496, 18000, 23328, 24300, 34992, 36000, 40500, 45000, 48600, 52488, 67500, 69984, 72000, 78732, 81000, 97200, 112500, 121500, 132300, 135000, 139968, 144000, 145800, 180000, 209952, 218700, 220500
Offset: 1

Views

Author

Michael De Vlieger, Jul 21 2025

Keywords

Examples

			Table of n, a(n), and A053669(a(n)) for n = 1..12.
 n     a(n)                    A053669(a(n))
--------------------------------------------
 1     108 = 2^2 * 3^3         5
 2     648 = 2^3 * 3^4         5
 3     972 = 2^2 * 3^5         5
 4    1944 = 2^3 * 3^5         5
 5    2700 = 2^2 * 3^3 * 5^2   7
 6    3888 = 2^4 * 3^5         5
 7    4500 = 2^2 * 3^2 * 5^3   7
 8    8748 = 2^2 * 3^7         5
 9    9000 = 2^3 * 3^2 * 5^3   7
10   13500 = 2^2 * 3^3 * 5^3   7
11   16200 = 2^3 * 3^4 * 5^2   7
12   17496 = 2^3 * 3^7         5
Let s = A052486.
The number 12 is not a term since it is not powerful (i.e., not in A001694).
The number 36, though powerful, is not a term since it is a perfect square.
s(1) = 72 is not in this sequence since rad(72) = P(2) = 6 and 72 = 2*6*6 = P(1)*P(2)*P(2).
s(2) = 108 = 3*6*6 is in the sequence since it is not a product of primorials.
The number 144, though powerful, is not a term because it is a perfect square.
s(3) = 200 is not a term because rad(200) = 10 = 2*5 is not also divisible by A053669(200) = 3.
s(4) = 288 is not in this sequence since rad(288) = P(2) = 6 and 288 = 2*2*2*6*6 = P(1)*P(1)*P(1)*P(2)*P(2), etc.

Links

Crossrefs

Cf. A001221, A001597, A001694, A002110, A007947, A025487, A055932, A056808, A377854, A378002.

Programs

  • Mathematica
    (* Load Fast Mathematica algorithm for A055932 linked at A377854, then: *)
Select[Union@ Flatten[f[7][[3 ;; -1, 2 ;; -1]] ], And[Divisible[#1, Apply[Times, #2[[All, 1]] ]^2], GCD @@ #2[[;; , -1]] == 1, Max@ Differences[#2[[All, -1]] ] > 0] & @@ {#, FactorInteger[#]} &]

Formula

Let rad = A007947, omega = A001221, and P = A002110.
rad(a(n)) = P(omega(a(n))).
Intersection of A052486 and A056808 = A377854 \ A378002.

A386434 Achilles numbers k such that A053669(k) < A006530(k).

Original entry on oeis.org

200, 392, 500, 675, 800, 968, 1125, 1323, 1352, 1372, 1568, 2000, 2312, 2888, 3087, 3200, 3267, 3528, 3872, 4000, 4232, 4563, 5000, 5292, 5324, 5408, 5488, 6075, 6125, 6272, 6728, 7688, 7803, 8575, 8712, 8788, 9248, 9747, 9800, 10125, 10584, 10952, 10976, 11552
Offset: 1

Views

Author

Michael De Vlieger, Jul 21 2025

Keywords

Examples

			Let s = A052486, q = A053669, and gpf = A006530.
Table of n, a(n), and q(a(n)) for n = 1..12:
 n    a(n)               q(a(n))
--------------------------------
 1    200 = 2^3 * 5^2    3
 2    392 = 2^3 * 7^2    3
 3    500 = 2^2 * 5^3    3
 4    675 = 3^3 * 5^2    2
 5    800 = 2^5 * 5^2    3
 6    968 = 2^3 * 11^2   3
 7   1125 = 3^2 * 5^3    2
 8   1323 = 3^3 * 7^2    2
 9   1352 = 2^3 * 13^2   3
10   1372 = 2^2 * 7^3    3
11   1568 = 2^5 * 7^2    3
12   2000 = 2^4 * 5^3    3
The number 12 is not a term since it is not powerful (i.e., not in A001694).
The number 36, though powerful, is not a term since it is a perfect square.
s(1) = 72 is not in the sequence since q(72) > gpf(72), i.e., 5 > 3.
s(2) = 108 is not in the sequence since q(108) > gpf(108), i.e., 5 > 3.
a(1) = s(3) = 200 because q(200) < gpf(200), i.e., 3 < 5.
a(2) = s(4) = 392 because q(392) < gpf(392), i.e., 3 < 7, etc.

Links

Crossrefs

Cf. A001597, A001694, A006530, A052486, A053669, A052486, A080259, A377854.

Programs

  • Mathematica
    (* Load Fast Mathematica algorithm for A055932 linked at A377854, then: *)
nn = 6; mm = Times @@ Prime@ Range[nn]; Complement[Select[Union@ Flatten@ Table[a^2*b^3, {b, Surd[mm, 3]}, {a, Sqrt[mm/b^3]}], And[Length[#2] > 1, GCD @@ #2 == 1] & @@ {#, FactorInteger[#][[;; , -1]]} &], Union@ Flatten[f[nn][[3 ;; -1, 2 ;; -1]] ] ]

Formula

Intersection of A052486 and A080259 = A052486 \ A377854.
Showing 1-4 of 4 results.

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