A085987 -id:A085987 - 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])

A369209 Numbers whose number of divisors has the largest prime factor 3.

Original entry on oeis.org

4, 9, 12, 18, 20, 25, 28, 32, 36, 44, 45, 49, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 96, 98, 99, 100, 108, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 160, 164, 169, 171, 172, 175, 180, 188, 196, 198, 200, 204, 207, 212, 220, 224, 225, 228

Offset: 1

Amiram Eldar, Jan 16 2024

Subsequence of A059269 and first differs from it at n = 36: A059269(136) = 44 has 15 = 3 * 5 divisors and thus is not a term of this sequence.
Numbers k such that A000005(k) is in A065119.
Numbers k such that A071188(k) = 3.
Equals the complement of A354181, without the terms of A036537 (i.e., complement(A354181) \ A036537).
The asymptotic density of this sequence is Product_{p prime} (1-1/p) * (Sum_{k>=1} 1/p^(A003586(k)-1)) - A327839 = 0.26087647470200496716... .

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

Cf. A000005, A003586, A006530, A036537, A065119, A336595, A071188, A211337, A211338, A327839, A354181.
Subsequence of A013929 and A059269.
Subsequences: A001248, A030627, A050997, A054753, A062503, A067259, A079395, A085986, A085987, A086975, A095990, A096156, A138032, A162143, A179643, A179645.

gpf[n_] := FactorInteger[n][[-1, 1]]; Select[Range[300], gpf[DivisorSigma[0, #]] == 3 &]

gpf(n) = if(n == 1, 1, vecmax(factor(n)[, 1]));
is(n) = gpf(numdiv(n)) == 3;

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

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

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A369209 Numbers whose number of divisors has the largest prime factor 3.

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

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