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.

A364997 Numbers k neither squarefree nor prime power such that rad(k)*A119288(k) > k but rad(k)*A053669(k) < k.

Original entry on oeis.org

40, 45, 56, 63, 88, 99, 104, 117, 136, 152, 153, 171, 175, 176, 184, 207, 208, 232, 248, 261, 272, 275, 279, 280, 296, 297, 304, 315, 325, 328, 333, 344, 351, 368, 369, 376, 387, 423, 424, 425, 440, 459, 464, 472, 475, 477, 488, 495, 496, 513, 520, 531, 536, 539
Offset: 1

Views

Author

Michael De Vlieger, Aug 16 2023

Keywords

Comments

Subset of A126706, numbers that are neither squarefree nor prime powers.
For k in this sequence, let p = A119288(k), q = A053669(k), and r = A007947(k).
A355432(k) = 0, A360543(k) > 0. There exist m < k, gcd(m,k) > 1 such that both omega(k) > omega(m) and rad(m) | k, but nondivisors m < k do not exist such that rad(m) = rad(k).

Examples

			Let b(n) = A126706(n), S = A360767, and T = A360765.
b(1) = 12 is not in the sequence since p*r = 3*6 = 18 and q*r = 5*6 = 30; both exceed 12, thus 12 is in S but not in T.
b(2) = 18 is not in the sequence since p*r = 3*6 = 18 and q*r = 5*6 = 30. Indeed, neither is less than 18, hence 18 is not in S but is in T.
b(6) = 36 is not in the sequence since p*r = 3*6 = 18 and q*r = 5*6, and both do not exceed 36, therefore 36 is not in S but is in T.
b(7) = a(1) = 40 since p*r = 5*10 = 50 and q*r = 3*10 = 30. We have both 50 > 40 and 30 < 40, thus 40 is in both S and T, etc.

Links

  • Michael De Vlieger, Table of n, a(n) for n = 1..10000
  • Michael De Vlieger, Annotated plot of b(n) = A126706(n), n = 20*(y-1) + x at (x, -y), for x = 1..20 and y = 1..20, thus for n = 1..400. Terms in this sequence are colored black, those in A364999 in blue, in A364998 in gold, and in A361098 in red.
  • Michael De Vlieger, Plot of b(n), n = 120*(y-1) + x at (x, -y), for x = 1..120 and y = 1..120, thus for n = 1..14400 using the same color scheme as immediately above.
  • Michael De Vlieger, Plot of b(n), with n = 1016*(y-1) + x at (x, -y), for x = 1..1016 and y = 1..1016, thus showing 1032256 terms. Terms b(n) in this sequence are colored black, else white.

Crossrefs

Cf. A007947, A053669, A119288, A126706, A355432, A360432, A360765, A360767, A361098, A364998, A364999.

Programs

  • Mathematica
    Select[Select[Range[500], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], Function[{k, f}, Function[{p, q, r}, And[p r > k, q r < k]] @@ {f[[2, 1]], SelectFirst[Prime@ Range[PrimePi[f[[-1, 1]]] + 1], ! Divisible[k, #] &], Times @@ f[[All, 1]]}] @@ {#, FactorInteger[#]} &]

Formula

Intersection of A360765 and A360767.

A364996 Union of A360767 and A363082.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 52, 56, 60, 63, 68, 76, 84, 88, 90, 92, 99, 104, 116, 117, 120, 124, 126, 132, 136, 140, 148, 150, 152, 153, 156, 164, 168, 171, 172, 175, 176, 180, 184, 188, 198, 204, 207, 208, 212, 220, 228, 232, 234, 236, 244, 248, 260, 261
Offset: 1

Views

Author

Michael De Vlieger, Aug 26 2023

Keywords

Examples

			This sequence is A126706 \ A361098.
Union of A364997, A364998, A364999.

Links

Crossrefs

Cf. A013929, A024619, A126706, A360767, A361098, A363082, A364997, A364998, A364999.

Programs

  • Mathematica
    Select[Select[Range[261], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], Function[{k, f}, Function[{p, q, r}, Or[p r > k, q r > k]] @@ {f[[2, 1]], SelectFirst[Prime@ Range[PrimePi[f[[-1, 1]]] + 1], ! Divisible[k, #] &], Times @@ f[[All, 1]]}] @@ {#, FactorInteger[#]} &]

A364999 Numbers k neither squarefree nor prime power such that both rad(k)*A119288(k) > k and rad(k)*A053669(k) > k.

Original entry on oeis.org

12, 20, 28, 44, 52, 60, 68, 76, 84, 92, 116, 124, 132, 140, 148, 156, 164, 172, 188, 204, 212, 220, 228, 236, 244, 260, 268, 276, 284, 292, 308, 316, 332, 340, 348, 356, 364, 372, 380, 388, 404, 412, 420, 428, 436, 444, 452, 460, 476, 492, 508, 516, 524, 532, 548
Offset: 1

Views

Author

Michael De Vlieger, Aug 16 2023

Keywords

Comments

Subset of A126706, numbers that are neither squarefree nor prime powers.
For k in this sequence, let p = A119288(k), q = A053669(k), and r = A007947(k).
A355432(k) = A360543(k) = 0. There exist neither nondivisor m < k such that rad(m) = rad(k), nor m < k, gcd(m,k) > 1 such that both omega(k) > omega(m) and rad(m) | k.
Apparently this is A081770 without the leading 4. - R. J. Mathar, Sep 05 2023
From Peter Munn, Mar 05 2024: (Start)
The preceding observation is true for the whole sequence, for reasons outlined below.
To qualify for this sequence, a number k must be smaller than 2 different multiples of rad(k): one based on a divisor, A119288(k): the other on a nondivisor, A053669(k).
For k that is not a prime power, straightforward calculations show (1) if k = 2 * rad(k) then k satisfies both of these comparisons, whereas (2) for k >= 3 * rad(k), k fails the divisor-based comparison if k is a multiple of 6 and fails the nondivisor-based comparison otherwise.
(End)

Examples

			Let b(n) = A126706(n), S = A360767, and T = A363082.
b(1) = a(1) = 12 since p*r = 3*6 = 18 and q*r = 5*6 = 30, and both exceed 12. Indeed, 12 is in both S and T.
b(2) = 18 is not in the sequence since p*r = 3*6 = 18; 18 is not in S.
b(6) = 36 is not in the sequence since p*r = 3*6 = 18 and q*r = 5*6, and both do not exceed 36.
b(7) = 40 is not in the sequence since p*r = 5*10 = 50 and q*r = 3*10 = 30. Though 50 > 40, 30 < 40, and is not in T, etc.

Links

  • Michael De Vlieger, Table of n, a(n) for n = 1..10000
  • Michael De Vlieger, Annotated plot of b(n) = A126706(n), with n = 20*(y-1) + x at (x, -y), for x = 1..20 and y = 1..20, thus showing 400 terms. Terms in this sequence are colored black, those in A364998 in gold, in A364997 in green, and in A361098 in red.
  • Michael De Vlieger, Plot of b(n), with n = 120*(y-1) + x at (x, -y), for x = 1..120 and y = 1..120, thus showing 14400 terms. This uses the same color scheme as described immediately above.
  • Michael De Vlieger, Plot of b(n), with n = 1016*(y-1) + x at (x, -y), for x = 1..1016 and y = 1..1016, thus showing 1032256 terms. Terms in this sequence are colored black, else white. Demonstrates fairly constant density of a(n) in A126706 as well as a slight quasiperiodic pattern approximately mod 169.

Crossrefs

Cf. A007947, A039956, A053669, A081770, A088860, A092742, A119288, A126706, A355432, A360432, A360767, A361098, A363082, A364998, A364999.

Programs

  • Mathematica
    Select[Select[Range[500], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], Function[{k, f}, Function[{p, q, r}, And[p r > k, q r > k]] @@ {f[[2, 1]], SelectFirst[Prime@ Range[PrimePi[f[[-1, 1]]] + 1], ! Divisible[k, #] &], Times @@ f[[All, 1]]}] @@ {#, FactorInteger[#]} &]

Formula

Intersection of A363082 and A360767.
From Peter Munn, Feb 21 2024: (Start)
a(n) = 2*A039956(n+1).
Asymptotic density is 1/Pi^2 = 0.101321183642337... (A092742). (End)
From Michael De Vlieger, Mar 08 2024: (Start)
{a(n)} = A366825 \ A366460, i.e., even terms in A366825.
A088860 = {a(n)} intersect A025487 = {a(n)} intersect A055932, where A088860(k) = 2*A002110(k). (End)

A369540 Numbers k neither squarefree nor prime powers such that A119288(k) <= k/A007947(k) < A053669(k) and A007947(k) is a primorial P(i) = A002110(i) for some i.

Original entry on oeis.org

18, 24, 90, 120, 150, 180, 630, 840, 1050, 1260, 1470, 1680, 1890, 2100, 6930, 9240, 11550, 13860, 16170, 18480, 20790, 23100, 25410, 27720, 90090, 120120, 150150, 180180, 210210, 240240, 270270, 300300, 330330, 360360, 390390, 420420, 450450, 480480, 1531530
Offset: 1

Views

Author

Michael De Vlieger, Jan 28 2024

Keywords

Comments

Nonsquarefree numbers k such that omega(k) > 1, whose squarefree kernel rad(k) is a primorial, with second least prime factor not greater than k/rad(k), and k/rad(k) is smaller than the smallest nondivisor prime.
Definition implies the following:
1.) A119288(k) = 3 since all terms are even, hence 6 | k.
2.) k is a product m * P(n), n > 1, such that rad(m) | P(n) and 3 <= m < prime(n+1).
Superset of A369541.

Examples

			Seen as a table T(n,k) of rows n where P(n) | T(n,k)
2:    18,   24;
3:    90,  120,   150,   180;
4:   630,  840,  1050,  1260,  1470,  1680,  1890,  2100;
5:  6930, 9240, 11550, 13860, 16170, 18480, 20790, 23100, 25410, 27720;
     ...
12 is not in the sequence since 3 <= 12/6 < 5 is false.
18 is in the sequence since 3 <= 18/6 < 5 is true.
36 is not in the sequence since 3 <= 36/6 < 5 is false.
Generally, 2*P(i) is not in the sequence since 3 <= 2*P(i)/P(i) < prime(i+1) is false.

Links

Crossrefs

Cf. A002110, A007947, A053669, A055932, A060735, A119288, A126706, A364998, A369541.

Programs

  • Mathematica
    P = 2; Table[P *= Prime[n]; Array[# P &, Prime[n + 1] - 3, 3], {n, 2, 6}] // Flatten

Formula

{a(n)} = { m × P(n) : 3 <= m < q, n >= 2 }.
Intersection of A364998 and A055932.
A060735 without primorials P(i) and 2*P(i).

A369541 Numbers k neither squarefree nor prime powers that are products of primorials such that A119288(k) <= k/A007947(k) < A053669(k).

Original entry on oeis.org

24, 120, 180, 840, 1260, 1680, 9240, 13860, 18480, 27720, 120120, 180180, 240240, 360360, 480480, 2042040, 3063060, 4084080, 6126120, 8168160, 38798760, 58198140, 77597520, 116396280, 155195040, 892371480, 1338557220, 1784742960, 2677114440, 3569485920, 5354228880
Offset: 1

Views

Author

Michael De Vlieger, Jan 28 2024

Keywords

Comments

Proper subset of A369540, itself contained in A060735, which in turn is a subset of A055932.

Examples

			Seen as an irregular triangle T(n,k) of rows n where P(n) | T(n,k)
2:      24;
3:     120,     180;
4:     840,    1260,    1680;
5:    9240,   13860,   18480,   27720;
6:  120120,  180180,  240240,  360360,  480480;
7: 2042040, 3063060, 4084080, 6126120, 8168160;
   ...

Links

Crossrefs

Cf. A002110, A007947, A025487, A053669, A055932, A060735, A119288, A364998, A369540.

Programs

  • Mathematica
    P = 2; nn = 10;
 s = Select[Range[4, Prime[nn], 2],
   Or[IntegerQ@ Log2[#],
     And[Union@ Differences@ PrimePi[#1] == {1},
        AllTrue[Differences[#2], # <= 0 &]] & @@
        Transpose@ FactorInteger[#]] &];
 Table[P *= Prime[n];
   P*TakeWhile[s, # <= Prime[n + 1] &], {n, 2, nn}] // Flatten

Formula

{a(n)} = { m × P(n) : 3 <= m < q, n >= 2, m not in A025487 }.
Intersection of A364998 and A025487.

A369419 Numbers k that are neither squarefree nor prime powers such that A119288(k) <= k/A007947(k) < A053669(k) and A007947(k) is not a primorial.

Original entry on oeis.org

18, 90, 150, 630, 1050, 1470, 1890, 2100, 6930, 11550, 16170, 20790, 23100, 25410, 90090, 150150, 210210, 270270, 300300, 330330, 390390, 420420, 450450, 1531530, 2552550, 3573570, 4594590, 5105100, 5615610, 6636630, 7147140, 7657650, 8678670, 9189180, 29099070
Offset: 1

Views

Author

Michael De Vlieger, Mar 10 2024

Keywords

Examples

			Seen as an irregular triangle T(n,k) of rows n where T(n,k) = P(n)*k, and k < prime(n+1) is in A369361.
n\k    3       5       7       9      10      11
------------------------------------------------
2:    18;
3:    90,    150;
4:   630,   1050,   1470,   1890,   2100;
5:  6930,  11550,  16170,  20790,  23100,  25410;
    ...

Links

Crossrefs

Cf. A002110, A003557, A007947, A025487, A053669, A055932, A056808, A060735, A119288, A364998, A369361, A369540, A369541.

Programs

  • Mathematica
    P = 2; nn = 8;
s = Select[Range[3, Prime[nn+1]],
  Nor[IntegerQ@ Log2[#],
      And[EvenQ[#1], Union@ Differences@ PrimePi[#2[[All, 1]]] == {1},
          AllTrue[Differences@ #2[[All, -1]], # <= 0 &]]] & @@
    {#, FactorInteger[#]} &];
Table[P *= Prime[n]; P*TakeWhile[s, # < Prime[n + 1] &], {n, 2, nn}]

Formula

This sequence is { k = m*P(i) : 3 <= m < prime(i), i > 1, m in A369361 }.
Intersection of A364998 and A056808.

A380473 Numbers k neither squarefree nor prime power (i.e., in A126706) such that A119288(k) <= A003557(k) < A053669(k) < A006530(k).

Original entry on oeis.org

126, 168, 198, 234, 264, 306, 312, 342, 408, 414, 456, 522, 552, 558, 666, 696, 738, 744, 774, 846, 888, 954, 984, 990, 1032, 1062, 1098, 1128, 1170, 1206, 1272, 1278, 1314, 1320, 1386, 1416, 1422, 1464, 1494, 1530, 1560, 1602, 1608, 1638, 1650, 1704, 1710, 1746
Offset: 1

Views

Author

Michael De Vlieger, Jul 22 2025

Keywords

Comments

Let rad = A007947, p = A119288, q = A053669, g = A006530, and r = A003557.
Numbers k in A126706 such that p <= r < q < g.
Terms are products k of a number s in A033845 and a number t in A007310 with at least one prime power factor p^m | k such that m > 1.

Examples

			Table of n, a(n) for select n:
   n    a(n)                       r   q
  --------------------------------------
   1    126 = 2 * 3^2 * 7          3   5
   2    168 = 2^3 * 3 * 7          4   5
   3    198 = 2 * 3^2 * 11         3   5
   4    234 = 2 * 3^2 * 13         3   5
   5    264 = 2^3 * 3 * 11         4   5
   6    306 = 2 * 3^2 * 17         3   5
   7    312 = 2^3 * 3 * 13         4   5
  24    990 = 2 * 3^2 * 5 * 11     3   7
  29   1170 = 2 * 3^2 * 5 * 13     3   7
  45   1650 = 2 * 3 * 5^2 * 11     5   7
  57   1980 = 2^2 * 3^2 * 5 * 11   6   7
  68   2340 = 2^2 * 3^2 * 5 * 13   6   7

Links

Crossrefs

Cf. A003557, A007310, A007947, A033845, A053669, A055932, A080259, A119288, A126706, A364998, A369540.

Programs

  • Mathematica
    a053669[x_] := Block[{q = 2}, While[Divisible[x, q], q = NextPrime[q] ]; q];
s = Select[Range[2^12], Nor[SquareFreeQ[#], PrimePowerQ[#]] &];
Select[s, And[#3 < #4 < #2[[-1, 1]], #2[[2, 1]] <= #3] & @@
  {#1, #2, #1/Apply[Times, #2[[All, 1]]], a053669[#1]} & @@
  {#, FactorInteger[#]} &]

Formula

Intersection of A364998 and A080259 = A364998 \ A055932 = A364998 \ A369540.
Showing 1-7 of 7 results.

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