A065036 -id:A065036 - cp's OEIS frontend

A307342 Products of four primes, except fourth powers of primes.

24, 36, 40, 54, 56, 60, 84, 88, 90, 100, 104, 126, 132, 135, 136, 140, 150, 152, 156, 184, 189, 196, 198, 204, 210, 220, 225, 228, 232, 234, 248, 250, 260, 276, 294, 296, 297, 306, 308, 315, 328, 330, 340, 342, 344, 348, 350, 351, 364, 372, 375, 376, 380, 390

Offset: 1

Kalle Siukola, Apr 02 2019

Numbers with exactly four prime factors (counted with multiplicity) and more than one distinct prime factor.

Numbers n such that bigomega(n) = 4 and omega(n) > 1.

Setwise difference of A014613 and A030514.
Union of A046386, A065036, A085986 and A085987.
Cf. A307682.

Select[Range@ 400, And[! PrimePowerQ@ #, PrimeOmega@ # == 4] &] (* Michael De Vlieger, Apr 21 2019 *)
Select[Range[400],PrimeOmega[#]==4&&PrimeNu[#]>1&] (* Harvey P. Dale, Aug 27 2021 *)

isok(n) = (bigomega(n)==4) && (omega(n) > 1); \\ Michel Marcus, Apr 03 2019

import sympy
def bigomega(n): return sympy.primeomega(n)
def omega(n): return len(sympy.primefactors(n))
print([n for n in range(1, 1000) if bigomega(n) == 4 and omega(n) > 1])

A307682 Products of four primes, two of which are distinct.

Original entry on oeis.org

24, 36, 40, 54, 56, 88, 100, 104, 135, 136, 152, 184, 189, 196, 225, 232, 248, 250, 296, 297, 328, 344, 351, 375, 376, 424, 441, 459, 472, 484, 488, 513, 536, 568, 584, 621, 632, 664, 676, 686, 712, 776, 783, 808, 824, 837, 856, 872, 875, 904, 999, 1016, 1029

Offset: 1

Kalle Siukola, Apr 21 2019

nonn

Numbers with exactly four prime factors (counted with multiplicity) and exactly two distinct prime factors.
Numbers n such that bigomega(n) = 4 and omega(n) = 2.
Products of a prime and the cube of a different prime (pq^3) together with squares of squarefree semiprimes (p^2*q^2).

Union of A065036 and A085986.
Intersection of A007774 and A067801.
Intersection of A007774 and A195086.
Intersection of A014613 and A067801.
Intersection of A014613 and A195086.
Cf. A307342.

Select[Range@ 1050, And[PrimeNu@ # == 2, PrimeOmega@ # == 4] &] (* Michael De Vlieger, Apr 21 2019 *)

isok(n) = (bigomega(n) == 4) && (omega(n) == 2); \\ Michel Marcus, Apr 22 2019

import sympy
def bigomega(n): return sympy.primeomega(n)
def omega(n): return len(sympy.primefactors(n))
print([n for n in range(1, 1000) if bigomega(n) == 4 and omega(n) == 2])

A381316 Numbers whose powerful part (A057521) is a power of a prime with an exponent >= 3 (A246549).

Original entry on oeis.org

8, 16, 24, 27, 32, 40, 48, 54, 56, 64, 80, 81, 88, 96, 104, 112, 120, 125, 128, 135, 136, 152, 160, 162, 168, 176, 184, 189, 192, 208, 224, 232, 240, 243, 248, 250, 256, 264, 270, 272, 280, 296, 297, 304, 312, 320, 328, 336, 343, 344, 351, 352, 368, 375, 376, 378

Offset: 1

Amiram Eldar, Feb 19 2025

First differs from A344653 and A345193 at n = 17: a(17) = 120 is not a term of these sequences.
Numbers whose prime signature (A118914) is of the form {1, 1, ..., m} with m >= 3, i.e., any number (including zero) of 1's and then a single number >= 3.
The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} 1/(p*(p^2-1)) = A369632 / A013661 = 0.13463358553764438661... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Complement of A060687 within A190641.
Cf. A013661, A057521, A246549, A369632.
Cf. A344653, A345193.
Subsequences: A030078, A030514, A030516, A030629, A030631, A050997, A056824, A065036, A079395, A092759, A138031, A178739, A178740, A179644, A179645, A179664, A179665, A179667, A179668, A179670, A179672, A179692, A179693, A179696, A179704, A179747, A189975, A189984, A189987, A190110, A190292, A190378, A190383, A190388, A190473, A381312.

q[n_] := Module[{e = ReverseSort[FactorInteger[n][[;; , 2]]]}, e[[1]] > 2 && (Length[e] == 1 || e[[2]] == 1)]; Select[Range[1000], q]

isok(k) = if(k == 1, 0, my(e = vecsort(factor(k)[, 2], , 4)); e[1] > 2 && (#e == 1 || e[2] == 1));

A382208 Numbers k for which pi(bigomega(k)) = omega(k).

Original entry on oeis.org

1, 4, 9, 12, 18, 20, 24, 25, 28, 36, 40, 44, 45, 49, 50, 52, 54, 56, 63, 68, 75, 76, 88, 92, 98, 99, 100, 104, 116, 117, 120, 121, 124, 135, 136, 147, 148, 152, 153, 164, 168, 169, 171, 172, 175, 180, 184, 188, 189, 196, 207, 212, 225, 232, 236, 240, 242, 244, 245

Offset: 1

Felix Huber, Mar 30 2025

nonn

Numbers k for which A000720(A001222(k)) = A001221(k).
Numbers k = p_1^e_1 * ... * p_j^e_j for which pi(Sum_{i=1..j} e_i) = j where pi = A000720.
Supersequence of A001248, A054753, A065036, A085986, A162143, A179643, A179644, A179693, A179700, A179704.

			240 = 2^4*3*5 is in the sequence because pi(Omega(240)) = pi(6) = 3 = omega(240).

Felix Huber, Table of n, a(n) for n = 1..10000

Cf. A000720, A001221, A001222, A001248, A046386, A054753, A065036, A085986, A162143, A179644, A179693, A179700, A179704.

with(NumberTheory):
A382208:=proc(n)
    option remember;
    local k;
    if n=1 then
        1
    else
        for k from procname(n-1)+1 do
            if pi(Omega(k))=Omega(k,distinct) then
                return k
            fi
        od
    fi;
end proc;
seq(A382208(n),n=1..59);
# second Maple program:
q:= n-> (l-> is(numtheory[pi](add(i[2], i=l))=nops(l)))(ifactors(n)[2]):
select(q, [$1..245])[];  # Alois P. Heinz, Apr 05 2025

Select[Range[250], PrimePi[PrimeOmega[#]] == PrimeNu[#] &] (* Amiram Eldar, Apr 05 2025 *)

isok(k) = primepi(bigomega(k)) == omega(k); \\ Michel Marcus, Apr 05 2025

a(1) inserted by Michel Marcus, Apr 05 2025

A331669 List of distinct numbers that occur in A318366 (the Dirichlet convolution square of bigomega).

Original entry on oeis.org

0, 1, 2, 4, 8, 10, 12, 20, 24, 34, 35, 40, 48, 52, 56, 70, 72, 84, 95, 104, 112, 116, 120, 130, 156, 160, 164, 165, 168, 180, 189, 212, 220, 224, 238, 240, 258, 280, 284, 286, 300, 304, 322, 330, 344, 348, 352, 364, 380, 420, 438, 440, 455, 460, 464, 472, 477, 480

Offset: 1

Torlach Rush, Jan 23 2020

nonn

There is a strong correlation between values of this function and values of other arithmetic functions. In other words, a(n) correlates to a single distinct value from one or more of the arithmetic functions.
Terms of this sequence select from the positive integers as follows:
A318366(k) = a(1), 1 followed by the primes (A008578).
A318366(k) = A008836(k) = A001221(k) = a(2), primes squared (A001248).
A318366(k) = A001221(k) = a(3), squarefree semiprimes (A006881).
A318366(k) = A000005(k) = a(4), primes cubed (A030078).
A318366(k) = a(5), a prime squared times a prime (A054753).
A318366(k) = a(6), primes to the fourth power (A030514).
A318366(k) = a(7), sphenic numbers (A007304).
A318366(k) = a(8), union of A050997 and A065036.
A318366(k) = a(9), squarefree semiprimes squared (A085986).
A318366(k) = a(10), product of four primes, three distinct (A085987).
A318366(k) = a(11), primes to the sixth power (A030516).
A318366(k) = a(12), product of prime to fourth power and a different prime (A178739).
A318366(k) = a(13), product of four distinct primes (A046386).
...

			0 is a term because the only divisors of a prime (p) are 1 and a prime itself and bigomega(1) * bigomega(p) + bigomega(p) * bigomega(1) = 0 * 1 + 1 * 0 = 0.
1 is a term because a prime squared gives bigomega(1) * bigomega(p^2) + bigomega(p) * bigomega(p) + bigomega(p^2) * bigomega(1) = 0 * 2 + 1 * 1 + 2 * 0 = 1.

Cf. A001222, A318366.

Cf. also A101296.

More terms, using A318366 extended b-file, from Michel Marcus, Jan 24 2020

A351380 Table read by rows: T(n,k) is the number of integers in the interval [2^(n-1), 2^n - 1] that have the k-th least prime signature.

Original entry on oeis.org

1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 3, 1, 1, 0, 5, 1, 3, 1, 3, 1, 1, 1, 0, 7, 1, 11, 0, 5, 0, 3, 1, 1, 1, 1, 1, 0, 13, 1, 19, 1, 9, 1, 2, 7, 0, 1, 2, 3, 1, 2, 1, 1, 0, 23, 1, 39, 0, 14, 0, 8, 16, 1, 2, 3, 9, 0, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 0, 43, 2, 73, 1, 27, 0, 11, 37, 0, 2, 6, 20, 0, 2, 3, 8, 0, 2, 4, 2, 4, 0, 1, 1, 1, 2, 1, 1, 1, 1

Offset: 1

Jon E. Schoenfield, Feb 09 2022

In rows n = 4 and n = 6..19, T(n,4) is the largest term in the row, i.e., squarefree semiprimes (A006881) outnumber the integers of each of the other prime signatures, but T(20,4) = 106408 < 109245 = T(20,9): among 20-bit numbers, sphenic numbers (A007304) (i.e., products of three distinct primes) are more numerous than squarefree semiprimes.

			The first 7 rows are shown in the body of the table below. Across the top of the table are the terms of A025487, whose k-th term is the smallest integer having the k-th prime signature.
.
  A025487(k)| 1  2  4  6  8 12 16 24 30 32 36 48 60 64 72 96 120 ...
  ----------+-------------------------------------------------------
         n\k| 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17  ...
  ----------+-------------------------------------------------------
         1  | 1
         2  | 0  2
         3  | 0  2  1  1
         4  | 0  2  1  3  1  1
         5  | 0  5  1  3  1  3  1  1  1
         6  | 0  7  1 11  0  5  0  3  1  1  1  1  1
         7  | 0 13  1 19  1  9  1  2  7  0  1  2  3  1  2  1  1
.
E.g., the 9 terms in row n=5 are 0, 5, 1, 3, 1, 3, 1, 1, 1 because, of the 16 integers in the interval [2^(5-1), 2^5 - 1] = [16, 31]:
  - 0 have prime signature 1 (since all are > 1)
  - 5 are primes
  - 1 is the square of a prime
  - 3 are squarefree semiprimes
etc., as shown below (where p, q, and r represent distinct primes):
.
.                  prime     OEIS
  k  A025487(k)  signature  Annnnnn  integers in [16, 31]  T(5,k)
  -  ----------  ---------  -------  --------------------  ------
  1       1          1         -            (none)            0
  2       2          p      A000040   17, 19, 23, 29, 31      5
  3       4         p^2     A001248           25              1
  4       6        p * q    A006881       21, 22, 26          3
  5       8         p^3     A030078           27              1
  6      12       p^2 * q   A054753       18, 20, 28          3
  7      16         p^4     A030514           16              1
  8      24       p^3 * q   A065036           24              1
  9      30      p * q * r  A007304           30              1

Cf. A006881, A007304, A025487, A162145.

Sum_{k>=1} T(n,k) = 2^n.

T(n,2) = A162145(n) for n > 1.

A355571 Complement of A007956: numbers not of the form P(k)/k where P(n) is the product of the divisors of n.

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 24, 25, 28, 30, 32, 36, 40, 42, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 121, 124, 126, 128, 130, 132, 135, 136, 138, 140, 147, 148, 150, 152

Offset: 1

Luca Onnis, Jul 07 2022

nonn

There are no primes in the sequence, since A007956(p^2) = p for all primes p.
There are infinitely many terms, in fact p^2 is a term for all primes p.
If 8k+1 is not a perfect square, then p^k is a term for all primes p.
Depends only on the prime signature: n is in this sequence if and only if A046523(n) is in this sequence. - Charles R Greathouse IV, Jul 08 2022
Contains all the weak numbers (A052485) aside from the primes (A000040) and squarefree semiprimes (A006881). - Charles R Greathouse IV, Jul 08 2022

			4 is a term of this sequence because there are no numbers k such that A007956(k) = 4.
2^10 is not a term of this sequence because A007956(32) = 1024 (Note that 8*10+1=81=9^2 is a perfect square).
p^4 belongs to this sequence for all primes p, in fact 8*4+1=33 is not a perfect square, so there are no numbers h such that A007956(h) = p^4.

Wacław Sierpiński, Elementary Theory of Numbers, Ex. 2 p. 174, Warsaw, 1964.

Subsequences by prime signature: A001248 (p^2), A054753 (p^2*q), A030514 (p^4), A065036 (p^3*q), A007304 (p*q*r), A050997 (p^5), A085986 (p^2*q^2).

Cf. A007956, A052485, A106543.

Complement[Complement[Table[n, {n, 2, 1000}], Select[NumericalSort[Table[Times @@ Most[Divisors[n]], {n, 1000000}]], # != 1 && # < 1000 &]], Select[Table[Prime[n], {n, 1, 1000}], # < 1000 &]]

a(n) = n + O(n log log n/log n). - Charles R Greathouse IV, Jul 08 2022

cp's OEIS Frontend

A307342 Products of four primes, except fourth powers of primes.

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A307682 Products of four primes, two of which are distinct.

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A381316 Numbers whose powerful part (A057521) is a power of a prime with an exponent >= 3 (A246549).

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A382208 Numbers k for which pi(bigomega(k)) = omega(k).

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A331669 List of distinct numbers that occur in A318366 (the Dirichlet convolution square of bigomega).

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A351380 Table read by rows: T(n,k) is the number of integers in the interval [2^(n-1), 2^n - 1] that have the k-th least prime signature.

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A355571 Complement of A007956: numbers not of the form P(k)/k where P(n) is the product of the divisors of n.

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