A119288 -id:A119288 - cp's OEIS frontend

A360768 Numbers k that are neither prime powers nor squarefree, such that k/rad(k) >= q, where rad(k) = A007947(k) and prime q = A119288(k).

18, 24, 36, 48, 50, 54, 72, 75, 80, 90, 96, 98, 100, 108, 112, 120, 126, 135, 144, 147, 150, 160, 162, 168, 180, 189, 192, 196, 198, 200, 216, 224, 225, 234, 240, 242, 245, 250, 252, 264, 270, 288, 294, 300, 306, 312, 320, 324, 336, 338, 342, 350, 352, 360, 363, 375, 378, 384, 392, 396, 400, 405, 408

Offset: 1

Michael De Vlieger, Feb 22 2023

nonn

Proper subsequence of A126706.

Numbers k such that there exists j such that 1 < j < k and rad(j) = rad(k), but j does not divide k.

			a(1) = 18, since 18/6 >= 3. We note that rad(12) = rad(18) = 6, yet 12 does not divide 18.
a(2) = 24, since 24/6 >= 3. Note: rad(18) = rad(24) = 6 and 24 mod 18 = 6.
a(3) = 36, since 36/6 >= 3. Note: rad(24) = rad(36) = 6 and 36 mod 24 = 12.
a(6) = 54, since 54/6 >= 3. Note: m in {12, 24, 36, 48} are such that rad(m) = rad(54) = 6, but none divides 54, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, 1016 pixel square bitmap of indices n = 1..1032256, read left to right, top to bottom, such that A126706(n) in this sequence appears in black and A126706(n) in A360767 in white. Shows a curious "sand ripple" pattern perhaps associated with congruence. (Magnification 3X)
Michael De Vlieger, 1016 pixel square bitmap as described above, at scale 1X.

Cf. A007947, A013929, A024619, A119288, A126706, A355432.

Select[Select[Range[120], Nor[SquareFreeQ[#], PrimePowerQ[#]] &], #1/#2 >= #3 & @@ {#1, Times @@ #2, #2[[2]]} & @@ {#, FactorInteger[#][[All, 1]]} &]

This sequence is { k in A126706 : k/A007947(k) >= A119288(k) }.

A364998 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

18, 24, 90, 120, 126, 150, 168, 180, 198, 234, 264, 306, 312, 342, 408, 414, 456, 522, 552, 558, 630, 666, 696, 738, 744, 774, 840, 846, 888, 954, 984, 990, 1032, 1050, 1062, 1098, 1128, 1170, 1206, 1260, 1272, 1278, 1314, 1320, 1386, 1416, 1422, 1464, 1470, 1494

Offset: 1

Michael De Vlieger, Aug 16 2023

nonn

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 nondivisors m < k such that rad(m) = rad(k); however, m < k, gcd(m,k) > 1 such that both omega(k) > omega(m) and rad(m) | k do not exist.

			Let b(n) = A126706(n), S = A360768, and T = A363082.
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 not in S.
b(2) = a(1) = 18 since p*r = 3*6 = 18 and q*r = 5*6 = 30. Indeed, 18 does not exceed 18 and 30 is larger than 18, hence 18 is in both S and 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 in S but not T.
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, thus 40 is neither in S nor T, etc.

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 A364999 in blue, 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, using 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 a strong quasiperiodic pattern approximately mod 169.

Cf. A007947, A053669, A119288, A126706, A355432, A360432, A360768, A361098, A363082, A364997, A364999.

Select[Select[Range[1500], 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[#]} &]

Intersection of A363082 and A360768.

A360767 Numbers k that are neither prime power nor squarefree, such that k/rad(k) < q, where rad(k) = A007947(k) and prime q = A119288(k).

Original entry on oeis.org

12, 20, 28, 40, 44, 45, 52, 56, 60, 63, 68, 76, 84, 88, 92, 99, 104, 116, 117, 124, 132, 136, 140, 148, 152, 153, 156, 164, 171, 172, 175, 176, 184, 188, 204, 207, 208, 212, 220, 228, 232, 236, 244, 248, 260, 261, 268, 272, 275, 276, 279, 280, 284, 292, 296, 297, 304, 308, 315, 316, 325, 328, 332, 333

Offset: 1

Michael De Vlieger, Feb 28 2023

Proper subsequence of A126706.

Numbers k such that there does not exist j such that 1 < j < k and rad(j) = rad(k), but j does not divide k.

			a(1) = 12, since 12/6 < 3.
a(2) = 20, since 20/10 < 5.
a(3) = 28, since 28/14 < 7.
a(4) = 40, since 40/10 < 5, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007947, A013929, A024619, A119288, A126706, A355432, A360768.

Select[Select[Range[120], Nor[SquareFreeQ[#], PrimePowerQ[#]] &], #1/#2 < #3 & @@ {#1, Times @@ #2, #2[[2]]} & @@ {#, FactorInteger[#][[All, 1]]} &]

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
f(n) = if (isprimepower(n) || (n==1), 1, my(f=factor(n)[, 1]); f[2]); \\ A119288
isok(k) = !isprimepower(k) && !issquarefree(k) && (k/rad(k) < f(k)); \\ Michel Marcus, Mar 01 2023

This sequence is { k in A126706 : k/A007947(k) < A119288(k) } = A126706 \ A360768.

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

Michael De Vlieger, Aug 16 2023

nonn

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).

			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.

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.

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

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[#]} &]

Intersection of A360765 and A360767.

A342479 a(n) is the numerator of the asymptotic density of numbers whose second smallest prime divisor (A119288) is prime(n).

Original entry on oeis.org

0, 1, 1, 1, 46, 44, 288, 33216, 613248, 151296, 391584768, 2383570944, 86830424064, 206470840320, 21270238986240, 987259950858240, 1262040231444480, 3022250536693923840, 3884253754215628800, 1102040800033347993600, 1892288242221318144000, 5616902226049109065728000

Offset: 1

Amiram Eldar, Mar 13 2021

The second smallest prime divisor of a number k is the second member in the ordered list of the distinct prime divisors of k. All the numbers that are not prime powers (A000961) have a second smallest prime divisor.

			The fractions begin with 0, 1/6, 1/10, 1/15, 46/1155, 44/1365, 288/12155, 33216/1616615, 613248/37182145, 151296/11849255, 391584768/33426748355, ...
a(1) = 0 since there are no numbers whose second smallest prime divisor is prime(1) = 2.
a(2)/A342480(2) = 1/6 since the numbers whose second smallest prime divisor is prime(2) = 3 are the positive multiples of 6.
a(3)/A342480(3) = 1/10 since the numbers whose second smallest prime divisor is prime(3) = 5 are the numbers congruent to {10, 15, 20} (mod 30) whose density is 3/30 = 1/10.

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, pp. 337-341.

Amiram Eldar, Table of n, a(n) for n = 1..376
Paul Erdős and Gérald Tenenbaum, Sur les densités de certaines suites d'entiers, Proc. London Math. Soc. (3), Vol. 59, No. 3 (1989), pp. 417-438; alternative link.

Cf. A000961, A038110, A038111, A119288, A342480 (denominators).

f[n_] := Module[{p = Prime[n], q}, q = Select[Range[p - 1], PrimeQ]; Plus @@ (1/(q - 1))*Times @@ ((q - 1)/q)/p]; Numerator @ Array[f, 30]

a(n)/A342480(n) = (1/prime(n)) * Product_{q prime < prime(n)} (1 - 1/q) * Sum_{q prime < prime(n)} 1/(q-1).

Sum_{n>=1} a(n)/A342480(n) = 1 (since the asymptotic density of numbers without a second smallest prime divisor, i.e., the prime powers, is 0).

A342480 a(n) is the denominator of the asymptotic density of numbers whose second smallest prime divisor (A119288) is prime(n).

Original entry on oeis.org

1, 6, 10, 15, 1155, 1365, 12155, 1616615, 37182145, 11849255, 33426748355, 247357937827, 10141675450907, 25652473199353, 2928046583754721, 155186468939000213, 223317113839049087, 558516101711461766587, 796182527971658263007, 241532826894674874877669, 430046252763689411367557

Offset: 1

Amiram Eldar, Mar 13 2021

See A342479 for details.

Amiram Eldar, Table of n, a(n) for n = 1..376

Cf. A038110, A038111, A119288, A342479 (numerators).

f[n_] := Module[{p = Prime[n], q}, q = Select[Range[p - 1], PrimeQ]; Plus @@ (1/(q - 1))*Times @@ ((q - 1)/q)/p]; Denominator @ Array[f, 30]

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

Michael De Vlieger, Aug 16 2023

nonn

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)

			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.

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.

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

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[#]} &]

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

Michael De Vlieger, Jan 28 2024

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.

			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.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

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

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

{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

Michael De Vlieger, Jan 28 2024

nonn

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

			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;
   ...

Michael De Vlieger, Table of n, a(n) for n = 1..10000

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

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

{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

Michael De Vlieger, Mar 10 2024

nonn

			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;
    ...

Michael De Vlieger, Table of n, a(n) for n = 1..10000

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

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}]

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

Intersection of A364998 and A056808.

cp's OEIS Frontend

A360768 Numbers k that are neither prime powers nor squarefree, such that k/rad(k) >= q, where rad(k) = A007947(k) and prime q = A119288(k).

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A364998 Numbers k neither squarefree nor prime power such that rad(k)*A119288(k) <= k but rad(k)*A053669(k) > k.

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A360767 Numbers k that are neither prime power nor squarefree, such that k/rad(k) < q, where rad(k) = A007947(k) and prime q = A119288(k).

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

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A342479 a(n) is the numerator of the asymptotic density of numbers whose second smallest prime divisor (A119288) is prime(n).

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A342480 a(n) is the denominator of the asymptotic density of numbers whose second smallest prime divisor (A119288) is prime(n).

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

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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.

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A364998 Numbers k neither squarefree nor prime power such that rad(k)A119288(k) <= k but rad(k)A053669(k) > k.

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

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