login
Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 10.
12

%I #31 Sep 25 2024 10:28:38

%S 2048,6144,9216,10240,13824,14336,20736,22528,25600,26624,30720,31104,

%T 34816,38912,43008,46080,46656,47104,50176,59392,63488,64000,64512,

%U 67584,69120,69984,71680,75776,76800,79872,83968,88064,96256,96768,101376,103680,104448

%N Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 10.

%C The asymptotic density of this sequence is (6/Pi^2) * Sum_{k>=1} f(a(k)) = 0.0003698..., where f(k) = A112526(k) * Product_{p|k} p/(p+1). - _Amiram Eldar_, Sep 25 2024

%H Reinhard Zumkeller, <a href="/A195069/b195069.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.

%F A046660(a(n)) = 10. - _Reinhard Zumkeller_, Nov 29 2015

%e 14336 = 2^11 * 7^1, so it has 12 prime factors (counted with multiplicity) and 2 distinct prime factors, and 12-2 = 10.

%p op(select(n->bigomega(n)-nops(factorset(n))=10, [$1..104448])); # _Paolo P. Lava_, Jul 03 2018

%t Select[Range[200000], PrimeOmega[#] - PrimeNu[#] == 10&]

%o (Haskell)

%o a195069 n = a195069_list !! (n-1)

%o a195069_list = filter ((== 10) . a046660) [1..]

%o -- _Reinhard Zumkeller_, Nov 29 2015

%o (PARI) isok(n) = bigomega(n) - omega(n) == 10; \\ _Michel Marcus_, Jul 03 2018

%Y Cf. A025487, A060687, A195087, A195088, A195089, A195090, A195091, A195092, A195093.

%Y Cf. A046660, A059956, A112526, A257851, A261256, A264959.

%K nonn,easy

%O 1,1

%A _Harvey P. Dale_, Sep 08 2011