A360768 -id:A360768 - cp's OEIS frontend

A363082 Numbers k neither squarefree nor prime power such that q*r > k, where q = A053669(k) is the smallest prime that does not divide k and r = A007947(k) is the squarefree kernel.

12, 18, 20, 24, 28, 44, 52, 60, 68, 76, 84, 90, 92, 116, 120, 124, 126, 132, 140, 148, 150, 156, 164, 168, 172, 180, 188, 198, 204, 212, 220, 228, 234, 236, 244, 260, 264, 268, 276, 284, 292, 306, 308, 312, 316, 332, 340, 342, 348, 356, 364, 372, 380, 388, 404, 408, 412, 414, 420

Offset: 1

Michael De Vlieger, Jul 29 2023

nonn

			a(1) = 12 since 12 is the smallest number that is neither squarefree nor a prime power. Additionally, 12 < 5*6.
a(2) = 18 since it is in A126706, and like 12, 18 < 5*6.
a(3) = 20 since it is neither squarefree nor prime power, and 20 < 3*10.
36 is not in this sequence since 36 > 5*6.
40 is not in this sequence since 40 > 3*10, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Plot b(n) = A126706(n) at (x, y) for n = ym + x = 1..1032256, m = 1016 and x = 1..m, y = 0..m-1, showing b(n) in A360765 in white, and b(n) in this sequence in other colors, where red indicates b(n) also in A360767, and blue indicates b(n) also in A360768.

Cf. A007947, A053669, A126706, A360765, A360767, A360768.

Select[Select[Range[452], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], Function[{k, f}, Function[{q, r}, q r > k] @@ {SelectFirst[Prime@ Range[PrimePi[f[[-1, 1]]] + 1], ! Divisible[k, #] &], Times @@ f[[All, 1]]} ] @@ {#, FactorInteger[#]} &]

This sequence is A126706 \ A360765.

A361235 a(n) = number of k < n, such that k does not divide n, omega(k) < omega(n) and rad(k) | rad(n), where omega(n) = A001221(n) and rad(n) = A007947(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 2, 1, 0, 0, 3, 0, 2, 1, 3, 0, 2, 0, 3, 0, 2, 0, 10, 0, 0, 2, 4, 1, 4, 0, 4, 2, 3, 0, 11, 0, 3, 2, 4, 0, 3, 0, 4, 2, 3, 0, 4, 1, 3, 2, 4, 0, 14, 0, 4, 2, 0, 1, 14, 0, 4, 2, 12, 0, 4, 0, 5, 2, 4, 1, 15, 0, 3, 0, 5, 0, 16, 1, 5, 3, 3, 0, 19, 1, 4, 3, 5, 1, 4, 0, 5

Offset: 1

Michael De Vlieger, Mar 06 2023

nonn

a(n) = 0 for prime powers, since the definition implies omega(n) >= 2.

			a(6) = 1 since k = 4 is such that rad(4)|rad(6) = 2|6 and omega(4) < omega(6).
a(10) = 2 since k = 4 is such that rad(4)|rad(10) = 2|10 and omega(4) < omega(10), and k = 8 is such that rad(8)|rad(10) = 2|10 and omega(8) < omega(10).
a(12) = 2 since the following satisfies definition: {8, 9}.
a(14) = 2, i.e., {4, 8}.
a(15) = 1, i.e., {9}.
a(18) = 3, i.e., {8, 9, 16}.
a(30) = 10, i.e., {4, 8, 9, 12, 16, 18, 20, 24, 25, 27}, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Diagram showing k <= n, n = 1..36, where a(n) is the number of numbers k in row n shown in gold. Numbers k in magenta in row n are counted by A355432(n). Together, gold and magenta numbers are counted by A243822(n) and appear in row n of A272618. Dots at (k, n) in red are divisors, and in green and blue in row n are counted by A243823(n).
Michael De Vlieger, Plot k < n at (x, y) = (k, -n) for n = 1..2^10, where black represents k such that k mod n != 0, such that omega(k) < omega(n) and rad(k) | rad(n).

Cf. A000961, A001221, A007947, A010846, A013929, A045763, A051953, A243822, A243823, A272618, A355432.

nn = 2^10;
rad[n_] := rad[n] = Times @@ FactorInteger[n][[All, 1]];
{0}~Join~Table[
   If[PrimePowerQ[n], 0,
    q = PrimeNu[n]; r = rad[n];
    Count[ DeleteCases[ Range[n],
     _?(Or[Divisible[n, #], CoprimeQ[#, n], ! Divisible[r, rad[#]]] &)],
     _?(PrimeNu[#] < q &)]],
   {n, 2, nn}]

a(n) = A243822(n) - A355432(n).
a(n) = A045763(n) - A243823(n) - A355432(n).
a(n) = A051953(n) - A000005(n) - A243823(n) - A355432(n) + 1.
a(n) = A010846(n) - A000005(n) - A355432(n).
a(n) = 0 for n in A000961.
a(n) > 0 for n in A013929.
a(n) = A243822(n) for n not in A360768.

A372972 Numbers k such that A372720(k) is negative.

Original entry on oeis.org

162, 250, 324, 384, 486, 648, 686, 768, 972, 1152, 1250, 1296, 1372, 1458, 1536, 1728, 1875, 1944, 2058, 2250, 2304, 2430, 2500, 2560, 2592, 2662, 2738, 2916, 3000, 3072, 3362, 3402, 3456, 3698, 3750, 3840, 3888, 3993, 4050, 4116, 4374, 4394, 4418, 4500, 4608

Offset: 1

Michael De Vlieger, Jun 02 2024

nonn

Let tau = A000005, let omega = A001221, let f = A008479, and let g = A372720.
For squarefree k, A372720(k) >= 0, since A008479(k) = 1 while tau(k) = 2^omega(k).
For prime power p^m, A372720(p^m) = 1, since A008479(p^m) = m while tau(k) = m+1.
Therefore, apart from a(1) = 1, this sequence is a proper subset of A126706.
In the sequence R = {k = m*s : rad(m) | s, s > 1 in A120944}, there is a smallest term k such that g(k) <= 0 and a largest term k such that g(k) is positive. For instance, in A033845 where s = 6, only {6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 192, 216, 288, 432, 576, 864} are such that g(k) > 0.
For s > 1, an infinite number of k in R are such that g(k) is negative. For example, with s = 6, all terms k > 864 in A033845 are in this sequence.
Conjecture: proper subset of A361098, hence of A360765 and A360768. This is to say that k = a(n) is such that A003557(k) >= A119288(k), i.e., k/rad(k) >= second smallest prime factor of k, and A003557(k) > A053669(k), where A053669(k) is the smallest prime q that does not divide k.

			a(1) = 162 = 2*3^4, since tau(162) - f(162)
     = (1+1)*(4+1) - card(A369609(162))
     = 10 - 12 = -2.
a(2) = 250 = 2*5^3, since tau(250) - f(250)
     = (1+1)*(3+1) - card(A369609(250))
     = 8 - 9 = -1.
a(3) = 324 = 2^2*3^4, since tau(324) - f(324)
     = (2+1)*(4+1) - card(A369609(324))
     = 15 - 16 = -1, etc.

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

Cf. A000005, A001221, A007947, A008479, A126706, A372720, A372864.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Position[Table[r = rad[n]; DivisorSigma[0, n] - Count[Range[n/r], ?(Divisible[r, rad[#]] &)], {n, 5000}], ?(# < 0 &)][[All, 1]]

A372864 Numbers k such that A372720(k) = 0.

Original entry on oeis.org

1, 500, 578, 722, 750, 1058, 1500, 1682, 1922, 2646, 2744, 3430, 3645, 4800, 5202, 5346, 5476, 5488, 5625, 6318, 6400, 6724, 7168, 7396, 8000, 8836, 10092, 10976, 11236, 11532, 11979, 12005, 13068, 13924, 14450, 14884, 15309, 16810, 16875, 16896, 18050, 18225

Offset: 1

Michael De Vlieger, Jun 02 2024

nonn

Let tau = A000005, let omega = A001221, let f = A008479, and let g = A372720.
For squarefree k, A372720(k) >= 0, since f(k) = 1 while tau(k) = 2^omega(k).
For prime power p^m, A372720(p^m) = 1, since f(p^m) = m while tau(k) = m+1.
Therefore, apart from a(1) = 1, this sequence is a proper subset of A126706.
In the sequence R = {k = m*s : rad(m) | s, s > 1 in A120944}, there is a smallest term k such that g(k) <= 0 and a largest term k such that g(k) is positive. For instance, in A033845 where s = 6, only {6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 192, 216, 288, 432, 576, 864} are such that g(k) > 0.
Apart from terms in this sequence, all the rest of the terms k in R are such that g(k) is negative.
There are no 3-smooth numbers k > 1 in this sequence, however there are 3 terms {500, 6400, 8000} in A033846 (with s = rad(k) = 10). For s = 2*3*23, there are 6 terms {19044, 25392, 38088, 70656, 536544, 953856}.
Conjecture: proper subset of A361098, hence of A360765 and A360768. This is to say that k = a(n) is such that A003557(k) >= A119288(k), i.e., k/rad(k) >= second smallest prime factor of k, and A003557(k) > A053669(k), where A053669(k) is the smallest prime q that does not divide k.

			a(1) = 1 since tau(1) - f(1) = 1 - 1 = 0.
a(2) = 500 = 2^2 * 5*3, since tau(500) - f(500)
     = (2+1)*(3+1) - card({10,20,40,50,80,100,160,200,250,320,400,500})
     = 12 - 12 = 0.
a(3) = 578 = 2*17^2, since tau(578) - f(578)
     = (1+1)*(2+1) - card({34,68,136,272,544,578})
     = 6 - 6 = 0, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..2000

Cf. A000005, A001221, A007947, A008479, A010846, A372720, A372972.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Position[Table[r = rad[n]; DivisorSigma[0, n] - Count[Range[n/r], ?(Divisible[r, rad[#]] &)], {n, 20000}], ?(# == 0 &)][[All, 1]]

A361487 Odd 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).

Original entry on oeis.org

75, 135, 147, 189, 225, 245, 363, 375, 405, 441, 507, 525, 567, 605, 675, 735, 825, 845, 847, 867, 875, 891, 945, 975, 1029, 1053, 1083, 1089, 1125, 1183, 1215, 1225, 1275, 1323, 1375, 1377, 1425, 1445, 1485, 1521, 1539, 1575, 1587, 1617, 1625, 1701, 1715, 1725, 1755, 1805, 1815, 1859, 1863, 1875, 1911

Offset: 1

Michael De Vlieger, Mar 29 2023

nonn

Odd terms in A360768, which itself is a proper subsequence of A126706.

Odd 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) = 75, since 75/15 >= 5. We note that rad(45) = rad(75) = 15, yet 45 does not divide 75.
a(2) = 135, since 135/15 >= 5. Note: rad(75) = rad(135) = 15, yet 45 does not divide 135.
a(3) = 147, since 147/21 >= 7. Note: rad(63) = rad(147) = 21, yet 147 mod 63 = 21.
Chart below shows k < a(n) such that rad(k) = rad(n), yet k does not divide n:
      75 | 45   .
     135 |  .   .  75   .   .
     147 |  .  63   .   .   .   .
     189 |  .   .   .   .   .   . 147   .   .   .
a(n) 225 |  .   .   .   .   . 135   .   .   .   .   .   .
     245 |  .   .   .   .   .   .   .   .   . 175   .   .   .
     363 |  .   .   .  99   .   .   .   .   .   .   .   .   .   .   .   .   . 297
     375 | 45   .   .   .   . 135   .   .   .   .   .   . 225   .   .   .   .   .
     ----------------------------------------------------------------------------
         | 45  63  75  99 117 135 147 153 171 175 189 207 225 245 261 275 279 297
                                        k in A360769

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, 1020 pixel square bitmap of indices n = 1..1040400, read left to right, top to bottom, such that A360768(n) in this sequence appears in black, else white. There is a faint pattern apparently related to that mentioned in A360768.
Michael De Vlieger, Chart showing k < a(n), n = 1..36, rows n contain k such that rad(k) = rad(n), yet k does not divide n. These k are in A360769, the number of k in row a(n) given by A355432(a(n)).

Cf. A005408, A007947, A013929, A024619, A119288, A126706, A355432, A360768, A360769.

Select[Select[Range[1, 2000, 2], Nor[SquareFreeQ[#], PrimePowerQ[#]] &], #1/#2 >= #3 & @@ {#1, Times @@ #2, #2[[2]]} & @@ {#, FactorInteger[#][[All, 1]]} &]

is(k) = { if (k%2, my (f = factor(k)); #f~ > 1 && k/vecprod(f[,1]~) >= f[2, 1], 0); } \\ Rémy Sigrist, Mar 29 2023

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

cp's OEIS Frontend

A363082 Numbers k neither squarefree nor prime power such that q*r > k, where q = A053669(k) is the smallest prime that does not divide k and r = A007947(k) is the squarefree kernel.

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A361235 a(n) = number of k < n, such that k does not divide n, omega(k) < omega(n) and rad(k) | rad(n), where omega(n) = A001221(n) and rad(n) = A007947(n).

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A372972 Numbers k such that A372720(k) is negative.

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A372864 Numbers k such that A372720(k) = 0.

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A361487 Odd 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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