A195086 - OEIS

A195086

Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 2.

8, 24, 27, 36, 40, 54, 56, 88, 100, 104, 120, 125, 135, 136, 152, 168, 180, 184, 189, 196, 225, 232, 248, 250, 252, 264, 270, 280, 296, 297, 300, 312, 328, 343, 344, 351, 375, 376, 378, 396, 408, 424, 440, 441, 450, 456, 459, 468, 472, 484, 488

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OFFSET

1,1

COMMENTS

From Amiram Eldar, Nov 07 2020: (Start)

Numbers whose powerful part (A057521) is either a cube of a prime (A030078) or a square of a squarefree semiprime (A085986).

The asymptotic density of this sequence is (6/Pi^2) * (Sum_{p prime} 1/(p^2*(p+1)) + Sum_{p<q primes} 1/(p*(p+1)*q*(q+1))) = (1/zeta(2)) * (2*P(3) + Sum_{k>=4} (-1)^(k+1)*(k-1)*P(k) + (Sum_{k>=2} (-1)^k*P(k))^2))/2 = 0.0963023158..., where P is the prime zeta function. (End)

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Alfred Rényi, On the density of certain sequences of integers, Publications de l'Institut Mathématique, Vol. 8 (1955), pp. 157-162.

Index entries for sequences computed from exponents in factorization of n

FORMULA

A001222(a(n)) - A001221(a(n)) = 2.

A046660(a(n)) = 2. - Reinhard Zumkeller, Nov 29 2015

MATHEMATICA

Select[Range[500], PrimeOmega[#]-PrimeNu[#]==2&]

PROG

(PARI) is(n)=bigomega(n)-omega(n)==2 \\ Charles R Greathouse IV, Sep 14 2015

(PARI) is(n)=my(f=factor(n)[, 2]); vecsum(f)==#f+2 \\ Charles R Greathouse IV, Aug 01 2016