A360769 -id:A360769 - cp's OEIS frontend

A363217 Odd powerful numbers that are not powers of primes.

225, 441, 675, 1089, 1125, 1225, 1323, 1521, 2025, 2601, 3025, 3087, 3249, 3267, 3375, 3969, 4225, 4563, 4761, 5625, 5929, 6075, 6125, 7225, 7569, 7803, 8281, 8575, 8649, 9025, 9261, 9747, 9801, 10125, 11025, 11907, 11979, 12321, 13225, 13689, 14161, 14283, 15125, 15129, 16641, 16875, 17689, 18225, 19773

Offset: 1

Michael De Vlieger, May 21 2023

nonn

This sequence is { A286708 INTERSECT A005408 } = { A001694 INTERSECT A360769 }.

Subset of A001694, A062739, A126706, and A360769.

			a(1) = 225 = 3^2 * 5^2, the smallest odd number with multiple distinct prime factors, each of which have multiplicity exceeding 1.
a(2) = 441 = 3^2 * 7^2,
a(3) = 675 = 3^3 * 5^2, etc.

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

Cf. A000961, A001694, A062739, A126706, A286708, A363216.

Cf. A002117, A013661, A013664, A082695, A136141.

With[{nn = 20000}, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], And[OddQ[#], ! PrimePowerQ[#]] &] ]

isok(k) = (k>1) && (k%2) && ispowerful(k) && !isprimepower(k); \\ Michel Marcus, May 28 2023

This sequence is { k = a^2*b^3 : a >= 1, b >= 1, omega(k) > 1, k mod 2 = 1 }.

Sum_{n>=1} 1/a(n) = 2*zeta(2)*zeta(3)/(3*zeta(6)) - 1/2 - Sum_{p prime} 1/(p*(p-1)) = (2/3) * A082695 - 1/2 - A136141 = 0.0225742... . - Amiram Eldar, May 28 2023

A377590 Numbers k neither squarefree nor prime powers such that there exist no numbers m such that rad(m) | k and Omega(m) > Omega(k), where rad = A007947 and Omega = A001222.

Original entry on oeis.org

12, 24, 45, 48, 63, 75, 96, 135, 175, 189, 192, 225, 245, 275, 325, 384, 405, 425, 475, 539, 567, 575, 605, 637, 675, 768, 833, 847, 875, 931, 1127, 1183, 1215, 1225, 1375, 1421, 1519, 1536, 1573, 1625, 1701, 1715, 1813, 1859, 1925, 2009, 2023, 2025, 2057, 2107

Offset: 1

Michael De Vlieger, Nov 02 2024

This sequence contains numbers k in A126706 for which A376846(k) = 0; A376846(k) = 0 for prime powers k or squarefree numbers k (i.e., k in A303554).
It is sufficient to determine floor(log k / log p) <= Omega(k) for p = lpf(k) = A020639(k).
Sequence contains numbers k of the form 2^j*3, j > 1, i.e., A007283 \ {3, 6} is a proper subset of this sequence, since 2^(j+1) < 2^j*3 and j+1 = Omega(2^j*3).
The numbers k that remain in the sequence ({a(n)} \ A007283) are odd, that is, in A360769. For k = 2^j*p, prime p > 3, we have j+floor(log_2 p) > j+1, since log_2 p > 2, therefore we see m = 2^(j+floor(log_2 p)) < 2^j*p, with Omega(m) > Omega(k).

			12 is in the sequence since 2^3 < 12, and Omega(2^3) = Omega(12) = 3.
20 is not in the sequence since 2^4 < 20 and Omega(2^4) = 4, but Omega(20) = 3.
45 is in the sequence since 3^3 < 45, and Omega(3^3) = Omega(45) = 3.
375 = 3*5^3 is not in the sequence since 3^5 < 375 and Omega(3^5) = 5, but Omega(345) = 4.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Numbers k for which floor(log k / log lpf(k)) <= bigomega(k), 2024.

Cf. A001222, A007283, A007947, A020639, A126706, A303554, A360769.

Select[Select[Range[4000], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], Function[{n, k}, NoneTrue[FactorInteger[n][[All, 1]], Floor@ Log[#, n] > k &]] @@ {#, PrimeOmega[#]} &]

A363101 Even numbers that are neither prime powers nor squarefree.

Original entry on oeis.org

12, 18, 20, 24, 28, 36, 40, 44, 48, 50, 52, 54, 56, 60, 68, 72, 76, 80, 84, 88, 90, 92, 96, 98, 100, 104, 108, 112, 116, 120, 124, 126, 132, 136, 140, 144, 148, 150, 152, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188, 192, 196, 198, 200, 204, 208, 212, 216, 220, 224, 228, 232, 234, 236, 240, 242

Offset: 1

Michael De Vlieger, May 19 2023

nonn

Even numbers k such that A001222(k) > A001221(k) > 1.

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

Cf. A001221, A001222, A005418, A024619, A126706, A360769.

Intersection of A013929 and A177712.

Select[Range[2, 242, 2], Nor[PrimePowerQ[#], SquareFreeQ[#]] &]

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

A363217 Odd powerful numbers that are not powers of primes.

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A377590 Numbers k neither squarefree nor prime powers such that there exist no numbers m such that rad(m) | k and Omega(m) > Omega(k), where rad = A007947 and Omega = A001222.

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A363101 Even numbers that are neither prime powers nor squarefree.

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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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