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 1-7 of 7 results.

A377854 Achilles numbers whose squarefree kernel is a primorial.

Original entry on oeis.org

72, 108, 288, 432, 648, 864, 972, 1152, 1800, 1944, 2592, 2700, 3456, 3888, 4500, 4608, 5400, 6912, 7200, 8748, 9000, 10368, 10800, 13500, 15552, 16200, 17496, 18000, 18432, 21600, 23328, 24300, 27648, 28800, 31104, 34992, 36000, 40500, 41472, 43200, 45000, 48600
Offset: 1

Views

Author

Michael De Vlieger, Nov 16 2024

Keywords

Comments

Numbers whose squarefree kernel is a primorial that are powerful but not a perfect power.

Examples

			Prime power decomposition of the first 12 terms:
   a(1) =   72 = 2^3 * 3^2
   a(2) =  108 = 2^2 * 3^3
   a(3) =  288 = 2^5 * 3^2
   a(4) =  432 = 2^4 * 3^3
   a(5) =  648 = 2^3 * 3^4
   a(6) =  864 = 2^5 * 3^3
   a(7) =  972 = 2^2 * 3^5
   a(8) = 1152 = 2^7 * 3^2
   a(9) = 1800 = 2^3 * 3^2 * 5^2
  a(10) = 1944 = 2^3 * 3^5
  a(11) = 2592 = 2^5 * 3^4
  a(12) = 2700 = 2^2 * 3^3 * 5^2

Links

Crossrefs

Cf. A001597, A001694, A002110, A007947, A052486, A055932, A286708, A369374, A378002.

Programs

  • Mathematica
    (* First load function f in the link, then: *)
Select[Rest@ Union@ Flatten@ f[10],
 And[Divisible[#, Apply[Times, #2[[All, 1]] ]^2],
   GCD @@ #2[[All, -1]] == 1] & @@ {#, FactorInteger[#]} &]

Formula

Intersection of A286708 \ A001597 and A055932.
Intersection of A052486 and A055932.
Proper subset of A369374.
Superset of A378002.

A369417 Powerful numbers k with multiple distinct prime factors such that rad(k) is not a primorial, where rad(k) = A007947(k).

Original entry on oeis.org

100, 196, 200, 225, 392, 400, 441, 484, 500, 675, 676, 784, 800, 968, 1000, 1089, 1125, 1156, 1225, 1323, 1352, 1372, 1444, 1521, 1568, 1600, 1764, 1936, 2000, 2025, 2116, 2312, 2500, 2601, 2704, 2744, 2888, 3025, 3087, 3136, 3200, 3249, 3267, 3364, 3375, 3528
Offset: 1

Views

Author

Michael De Vlieger, Jan 22 2024

Keywords

Comments

Numbers k such that Omega(k) > omega(k) > 1, where all prime power factors p^m have exponents m > 1, such that squarefree kernel rad(k) not in A002110, where Omega = A001222 and omega = A001221.

Examples

			Let S = A366413 = {A120944 \ A002110}.
This sequence is the union of the following infinite sets:
S(1)^2 * A003592 = 10^2 * A003592 = {100, 200, 400, 500, 800, 1000, ...}
                 = { m*S(1)^2 : rad(m) | S(1) }.
S(2)^2 * A003591 = 14^2 * A003591 = {196, 392, 784, 1372, 1568, ...}
                 = { m*S(2)^2 : rad(m) | S(2) }.
S(3)^2 * A003593 = 15^2 * A003593 = {225, 675, 1125, 2025, 3375, ...}
                 = { m*S(3)^2 : rad(m) | S(3) }, etc.

Links

Crossrefs

Cf. A001221, A001222, A002110, A055932, A120944, A126706, A286708, A366413, A369374.

Programs

  • Mathematica
    With[{nn = 2^14},
  Select[
    Select[
      Rest@ Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}],
    Not@*PrimePowerQ],
  Nand[EvenQ[#],
    Union@ Differences@ PrimePi[FactorInteger[#][[All, 1]]] == {1}] &] ]

Formula

{a(n)} = { A286708 \ A055932 }.
{a(n)} = { m*s^2 : Omega(s) = omega(s) > 1, s not in A002110, rad(m) | s }.
A286708 is the union of A369374 and this sequence.

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.

A380446 Perfect powers k^m, m > 1, omega(k) > 1, such that A053669(k) > A006530(k), where omega = A001221.

Original entry on oeis.org

36, 144, 216, 324, 576, 900, 1296, 1728, 2304, 2916, 3600, 5184, 5832, 7776, 8100, 9216, 11664, 13824, 14400, 20736, 22500, 26244, 27000, 32400, 36864, 44100, 46656, 57600, 72900, 82944, 90000, 104976, 110592, 129600, 147456, 157464, 176400, 186624, 202500, 216000
Offset: 1

Views

Author

Michael De Vlieger, Jul 25 2025

Keywords

Comments

Perfect powers k^m, m > 1, for k in A055932.
Union of {k^m : rad(k) | P(i), m >= 2}, rad = A007947, P = A002110. Therefore perfect powers in A033845, A143207, A147571, A147572, etc. are proper subsets.
Terms are even. For a(n) such that omega(a(n)) > 2, a(n) mod 10 = 0, where omega = A001221.

Examples

			Table of n, a(n) for select n, showing exponents m of prime power factors p^m | a(n) for primes p listed in the heading. Terms that also appear in A368682 are marked by "#":
                         Exponents
 n      a(n)             2.3.5.7.11
-----------------------------------
 1       36 =    6^2  #  2.2
 2      144 =   12^2  #  4.2
 3      216 =    6^3  #  3.3
 4      324 =   18^2     2.4
 5      576 =   24^2  #  6.2
 6      900 =   30^2  #  2.2.2
 7     1296 =    6^4  #  4.4
 8     1728 =   12^3  #  6.3
 9     2304 =   48^2  #  8.2
10     2916 =   54^2     2.6
11     3600 =   60^2  #  4.2.2
12     5184 =   72^2  #  6.4
26    44100 =  210^2  #  2.2.2.2
90  5336100 = 2310^2  #  2.2.2.2.2

Links

Crossrefs

Cf. A001597, A002110, A006530, A007947, A033845, A053669, A055932, A126706, A131605, A143207, A147571, A147572, A246547, A304250, A365308 (subset), A368682 (subset), A369374 (superset).

Programs

  • Mathematica
    (* Load linked Mathematica algorithm, then: *)
Select[Union@ Flatten[a055932[7][[3 ;; -1, 2 ;; -1]] ], And[Divisible[#1, Apply[Times, #2[[All, 1]] ]^2], GCD @@ #2[[All, -1]] > 1] & @@ {#, FactorInteger[#]} &]

Formula

Intersection of A131605 and A055932 = A304250 \ A246547.

A369420 Powerful numbers k that are not prime powers, such that k has a primorial kernel but is not a product of primorials.

Original entry on oeis.org

108, 324, 648, 972, 1944, 2700, 2916, 3888, 4500, 5832, 8100, 8748, 9000, 11664, 13500, 16200, 17496, 18000, 22500, 23328, 24300, 26244, 34992, 36000, 40500, 45000, 48600, 52488, 67500, 69984, 72000, 72900, 78732, 81000, 90000, 97200, 104976, 112500, 121500, 132300
Offset: 1

Views

Author

Michael De Vlieger, Jan 22 2024

Keywords

Comments

Numbers k such that Omega(k) > omega(k) > 1, prime powers p^m | k are such that m > 1, rad(k) is a primorial, but k is not a product of primorials, where Omega = A001222 and omega = A001221.
Contains no odd numbers as a consequence of being a proper subset of A055932.
Proper subset of A369419, which is in turn a proper subset of A126706.

Examples

			36 = 2^2 * 3^2 is a product of primorials, therefore not in the sequence.
72 = 2^3 * 3^2 is not a term because it is a product of primorials.
100 = 2^2 * 5^2 is not in the sequence since it does not have a primorial kernel.
108 = 2^2 * 3*3 is in the sequence since it is not a product of primorials, but its squarefree kernel is 6, a primorial.
144 = 2^4 * 3^2 is not in the sequence since it is a product of primorials, etc.

Links

Crossrefs

Cf. A001221, A001222, A002110, A025487, A055932, A056808, A126706, A286708, A364930, A369374.

Programs

  • Mathematica
    With[{nn = 2^20},
 Select[
   Select[
     Rest@ Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}],
     Not@*PrimePowerQ],
   And[EvenQ[#1],
     Union@ Differences@ PrimePi[#2[[All, 1]]] == {1}, !
     AllTrue[Differences@ #2[[All, -1]], # <= 0 &]] & @@
     {#, FactorInteger[#]} &] ]

Formula

{a(n)} = {A369374 \ A364930}.
Intersection of A056808 and A286708.

A380452 Perfect powers k^m, m > 1, omega(k) > 1, such that A053669(k) > A006530(k) that are not also products of primorials, where omega = A001221.

Original entry on oeis.org

324, 2916, 5832, 8100, 11664, 22500, 26244, 72900, 90000, 104976, 157464, 202500, 236196, 291600, 360000, 396900, 419904, 562500, 656100, 729000, 944784, 1102500, 1259712, 1440000, 1822500, 1889568, 2125764, 2160900, 2250000, 2624400, 3375000, 3572100, 3779136
Offset: 1

Views

Author

Michael De Vlieger, Jul 25 2025

Keywords

Comments

Perfect powers k^m, m > 1, for composite k in A056808.
Terms are even. For a(n) such that omega(a(n)) > 2, a(n) mod 10 = 0, where omega = A001221.

Examples

			Table of n, a(n) for select n, showing exponents m of prime power factors p^m | a(n) for primes p listed in the heading:
                      Exponents
 n      a(n)          2.3.5
-------------------------------
 1      324 =  18^2   2.4
 2     2916 =  54^2   2.6
 3     5832 =  18^3   3.6
 4     8100 =  90^2   2.4.2
 5    11664 = 108^2   4.6
 6    22500 = 150^2   2.2.4
 7    26244 = 162^2   2.8
 8    72900 = 270^2   2.6.2
 9    90000 = 300^2   4.2.4
10   104976 =  18^4   4.8
11   157464 =  54^3   3.9
12   202500 = 450^2   2.4.4

Links

Crossrefs

Cf. A001597, A002110, A006530, A007947, A053669, A055932, A131605, A246547, A304250, A368682, A369374, A380446.

Programs

  • Mathematica
    (* Load linked Mathematica algorithm, then: *)
Select[Union@ Flatten[a055932[7][[3 ;; -1, 2 ;; -1]] ], And[Divisible[#1, Apply[Times, #2[[All, 1]] ]^2], GCD @@ #2[[All, -1]] > 1] & @@ {#, FactorInteger[#]} &]

Formula

Intersection of A131605 and A056808 = A380446 \ A368682.
Set difference A380446 \ A025487.
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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