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
This sequence is A126706 \ A361098.
Union of A364997, A364998, A364999.
- Michael De Vlieger, Table of n, a(n) for n = 1..10000
- Michael De Vlieger, List of A126706(1..400), 20 in a row. Terms in this sequence are circled, while those in small colored circles appear in A361098. Blue represents numbers in A364702, purple A359280, and magenta A303606.
- Michael De Vlieger, Plot b(n) at (x,y) = (n mod 1024, -floor(n/1024)), where terms in this sequence are shown in black, and those in A361098 appear in white.
-
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[#]} &]
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).
Original entry on oeis.org
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
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.
-
Select[Select[Range[120], Nor[SquareFreeQ[#], PrimePowerQ[#]] &], #1/#2 >= #3 & @@ {#1, Times @@ #2, #2[[2]]} & @@ {#, FactorInteger[#][[All, 1]]} &]
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
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[#]} &]
Original entry on oeis.org
0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 3, 1, 1, 3, 1, 4, 3, 3, 1, 4, 1, 3, 1, 4, 1, 7, 1, 1, 3, 3, 3, 4, 1, 3, 3, 5, 1, 7, 1, 4, 4, 3, 1, 4, 1, 2, 3, 4, 1, 1, 3, 5, 3, 3, 1, 10, 1, 3, 4, 1, 3, 7, 1, 4, 3, 7, 1, 4, 1, 3, 3, 4, 3, 7, 1, 5, 1, 3, 1, 10, 3, 3, 3
Offset: 1
Table of a(n), b(n) = A000005(n), and c(n) = A008479(n) for n <= 12:
n b(n) c(n) a(n)
------------------
1 1 1 0
2 2 1 1
3 2 1 1
4 3 2 1
5 2 1 1
6 4 1 3
7 2 1 1
8 4 3 1
9 3 2 1
10 4 1 3
11 2 1 1
12 6 2 4
a(12) = 4 since 12 has 6 divisors {1, 2, 3, 4, 6, 12}, and row 12 of A369609 has 2 terms {6, 12}.
a(18) = 3 since 18 has 6 divisors {1, 2, 3, 6, 9, 18}, and row 18 of A369609 has 3 terms {6, 12, 18}.
a(50) = 2 since 50 has 6 divisors {1, 2, 5, 10, 25, 50}, and row 50 of A369609 has 4 terms {10, 20, 40, 50}
a(162) = -2 since 162 has 10 divisors {1,2,3,6,9,18,27,54,81,162} but row 162 of A369609 has 12 terms {6,12,18,24,36,48,54,72,96,108,144,162}.
a(500) = 0 since 500 has as many divisors {1,2,4,5,10,20,25,50,100,125,250,500} as terms in row 500 of A369609 {10,20,40,50,80,100,160,200,250,320,400,500}.
Cf.
A000005,
A003557,
A008479,
A095960,
A119288,
A162306,
A183093,
A303554,
A355432,
A360767,
A369609.
-
rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; DivisorSigma[0, n] - Count[Range[n/r], _?(Divisible[r, rad[#]] &)], {n, 120}]
-
a(n) = my(f=factor(n)[, 1], s); forvec(v=vector(#f, i, [1, logint(n, f[i])]), if(prod(i=1, #f, f[i]^v[i])<=n, s++)); numdiv(n) - s; \\ after A008479 \\ Michel Marcus, Jun 03 2024
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
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[#]} &]
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.
Original entry on oeis.org
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
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.
-
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[#]} &]
Showing 1-6 of 6 results.
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