A376846 -id:A376846 - cp's OEIS frontend

A376567 a(n) = binomial(bigomega(n) + omega(n), omega(n)), where bigomega = A001222 and omega = A001221.

1, 2, 2, 3, 2, 6, 2, 4, 3, 6, 2, 10, 2, 6, 6, 5, 2, 10, 2, 10, 6, 6, 2, 15, 3, 6, 4, 10, 2, 20, 2, 6, 6, 6, 6, 15, 2, 6, 6, 15, 2, 20, 2, 10, 10, 6, 2, 21, 3, 10, 6, 10, 2, 15, 6, 15, 6, 6, 2, 35, 2, 6, 10, 7, 6, 20, 2, 10, 6, 20, 2, 21, 2, 6, 10, 10, 6, 20, 2

Offset: 1

Michael De Vlieger, Oct 09 2024

For prime power p^k, a(p^k) = A010846(p^k) = A000005(p^k) = k+1. Therefore, for prime p, a(p) = A010846(p) = A000005(p) = 2.

For n in A024619, a(n) != A010846(n) and A010846(n) > A000005(n).

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

Cf. A000005, A001221, A001222, A007947, A010846, A024619, A068795, A376248, A376846, A376847.

with(NumberTheory):
a := n -> binomial(Omega(n) + Omega(n, distinct), Omega(n, distinct)):
seq(a(n), n = 1..79);  # Peter Luschny, Oct 25 2024

Array[Binomial[#2 + #1, #1] & @@ {PrimeNu[#], PrimeOmega[#]} &, 120]

a(n) = length of row n of A376248.

a(n) = A010846(n) - A376846(n) + A376847(n).

A377713 Squarefree composite k such that floor(log n/log lpf(k)) <= omega(k), where lpf = A020639 and omega = A001221.

Original entry on oeis.org

6, 15, 21, 35, 55, 65, 77, 85, 91, 95, 115, 119, 133, 143, 161, 187, 203, 209, 217, 221, 247, 253, 259, 287, 299, 301, 319, 323, 329, 341, 377, 385, 391, 403, 407, 437, 451, 455, 473, 481, 493, 517, 527, 533, 551, 559, 583, 589, 595, 611, 629, 649, 667, 671, 689

Offset: 1

Michael De Vlieger, Nov 04 2024

Also squarefree composite k such that there exist no numbers m such that rad(m) | k and omega(m) > omega(k).
The only even term is 6.
Let P(i) = A002110(i). Numbers k = prime(i) * P(i+j)/P(i) < prime(i)^(i+j) with j ≥ 1 implies k such that omega(k) = j+1 is in the sequence.
The number k = p*m is a solution where squarefree m with lpf(m) > p is such that m < p^omega(m). For example, k = 5*7 is in the sequence since 7 < 5^2.
The number of a(n) such that lpf(a(n)) = p is finite. For example, the only terms divisible by 3 are {6, 15, 21}.

			6 is in the sequence since floor(log_2 6) = 1+floor(log_2 3) = omega(6) = 2.
10 is not in the sequence since floor(log_2 5) = 2 and omega(10) = 2, thus 1+floor(log_2 5) > omega(10). Seen another way, 2^3 < 10, but omega(8) > omega(10).
15 is in the sequence since floor(log_3 15) = 1+floor(log_3 5) = omega(15) = 2.
21 is in the sequence because 1+floor(log_3 7) = omega(21) = 2.
33 = 3*11 is not in the sequence because 11 > 3^2.
115 = 5*23 is in the sequence because 23 < 5^2.
145 = 5*29 is not in the sequence since 29 > 5^2, etc.

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. A000961, A001221, A002110, A007947, A020639, A120944, A376846, A377590, A377591.

s = Select[Range[1000], And[SquarefreeQ[#], CompositeQ[#]] &];
Select[s, Floor@ Log[FactorInteger[#][[1, 1]], #] <= PrimeOmega[#] &]

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[#]} &]

A376847 Number of m > n such that rad(m) | n and Omega(m) <= Omega(n), where rad = A007947 and Omega = A001222.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 1, 0, 3, 1, 1, 0, 4, 0, 1, 0, 3, 0, 4, 0, 0, 1, 1, 1, 2, 0, 1, 1, 5, 0, 5, 0, 3, 2, 1, 0, 6, 0, 1, 1, 3, 0, 1, 1, 5, 1, 1, 0, 11, 0, 1, 2, 0, 1, 5, 0, 3, 1, 5, 0, 4, 0, 1, 1, 3, 1, 5, 0, 8, 0, 1, 0, 11, 1, 1, 1

Offset: 1

Michael De Vlieger, Oct 13 2024

nonn

			Table of select n such that a(n) > 0:
   n  a(n)  List of m in A376248 such that Omega(m) <= Omega(n)
  -------------------------------------------------------------
   6    1   {9}
  10    1   {25}
  12    2   {18, 27}
  14    1   {49}
  15    1   {25}
  18    1   {27}
  20    3   {25, 50, 125}
  24    4   {27, 36, 54, 81}
  28    3   {49, 98, 343}
  30    4   {45, 50, 75, 125}
  40    5   {50, 100, 125, 250, 625}
  48    6   {54, 72, 81, 108, 162, 243}
  60   11   {75, 81, 90, 100, 125, 135, 150, 225, 250, 375, 625}

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Hasse diagrams of m in select R(n), where R(n) is the union of rows n of A162306 and A376248, indicating in blue those m > n such that Omega(m) <= Omega(n).

Cf. A000961, A001221, A001222, A007947, A162306, A376567, A376846, A068795.

with(NumberTheory):
cond := (m, n) -> irem(n, Radical(m)) = 0 and Omega(m) <= Omega(n):
a := n -> nops(select(m -> cond(m, n), [seq(n+1..A068795(n))])):
seq(a(n), n = 1..87);  # Peter Luschny, Oct 25 2024

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
Table[k = PrimeOmega[n]; w = PrimeNu[n]; Binomial[k + w, w] - Count[Range[n], _?(And[Divisible[n, rad[#]], PrimeOmega[#] > k] &)], {n, 120}]

a(n) = card({m > n : rad(m) | n, Omega(m) <= Omega(n) }).
a(n) = 0 for prime power n (in A000961).
a(n) = card(A376248 \ A162306).
a(n) = A376567(n) - A010846(n) + A376546(n) = binomial(A001222(n) + A001221(n), A001221(n)) - A010846(n) + A376546(n).

A377591 Powerful numbers k that are not 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

225, 675, 1225, 2025, 3025, 5929, 6075, 6125, 8281, 8575, 14161, 15125, 18225, 20449, 30625, 34969, 41503, 42875, 43681, 48841, 54675, 57967, 60025, 61009, 64009, 65219, 75625, 89401, 99127, 101761, 104329, 107653, 116281, 142129, 152881, 153125, 162409, 164025

Offset: 1

Michael De Vlieger, Nov 02 2024

Terms are odd; proper subset of A363217, which is a proper subset of A286708, itself contained in A001694.

Proper subset of A377590.

			36 is not in the sequence since 2^5 < 36, Omega(32) = 5, but Omega(36) = 4.
72 is not in the sequence since 2^6 < 72, but Omega(72) = 5.
225 is in the sequence since 3^4 < 225, Omega(81) = Omega(225) = 4.
441 is not in the sequence since 3^5 < 441, Omega(243) = 5, but Omega(441) = 4, etc.

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, A001694, A007947, A286708, A363217, A376846, A377590.

Select[With[{nn = 200000},
  Rest@ Select[
    Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
    Not@*PrimePowerQ] ],
  Function[{n, k},
    NoneTrue[FactorInteger[n][[All, 1]],
      Floor@ Log[#, n] > k &]] @@ {#, PrimeOmega[#]} &]

cp's OEIS Frontend

A376567 a(n) = binomial(bigomega(n) + omega(n), omega(n)), where bigomega = A001222 and omega = A001221.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A377713 Squarefree composite k such that floor(log n/log lpf(k)) <= omega(k), where lpf = A020639 and omega = A001221.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A376847 Number of m > n such that rad(m) | n and Omega(m) <= Omega(n), where rad = A007947 and Omega = A001222.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A377591 Powerful numbers k that are not 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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica