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

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

Author

Torlach Rush, Jan 23 2020

Keywords

• nonn

Comments

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).

...

Examples

			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.
		

Crossrefs

Cf. A001222, A318366.

Cf. also A101296.

Extensions

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