cp's OEIS Frontend

A245080 Numbers such that omega(a(n)) is a proper divisor of bigomega(a(n)).

%I A245080 #39 Nov 18 2024 07:43:28
%S A245080 4,8,9,16,24,25,27,32,36,40,49,54,56,64,81,88,96,100,104,121,125,128,
%T A245080 135,136,144,152,160,169,184,189,196,216,224,225,232,240,243,248,250,
%U A245080 256,289,296,297,324,328,336,343,344,351,352,360,361,375,376,384,400,416,424,441,459
%N A245080 Numbers such that omega(a(n)) is a proper divisor of bigomega(a(n)).
%C A245080 All proper powers of any number greater than 1 (A001597(n), n>1) are a subset of this sequence. On the other hand, this is a subset of A067340 which admits also numbers k for which bigomega(k) = omega(k). In particular, prime numbers are excluded.
%C A245080 The density of these numbers, i.e., the ratio n/a(n), apparently decreases with n, reaching 0.04420... for n = 10000000. Conjecture: n/a(n) might have a nonzero limit below 0.0427 (the density found in the interval 9500000 < n <= 10000000).
%C A245080 There are 40134838 terms in the range 10^9 <= k <= 2*10^9. - _Hugo Pfoertner_, Oct 28 2024
%H A245080 Stanislav Sykora, <a href="/A245080/b245080.txt">Table of n, a(n) for n = 1..20000</a>
%H A245080 Wikipedia, <a href="http://en.wikipedia.org/wiki/Arithmetic_functions">Arithmetic functions</a>
%e A245080 240 is in the sequence because 240=5^1*3^1*2^4. Hence omega(240)=3 (three distinct prime divisors) is a proper divisor of bigomega(240)=6 (six prime divisors with multiplicity).
%t A245080 Select[Range[500], Divisible[PrimeOmega[#], PrimeNu[#]] && PrimeNu[#] != PrimeOmega[#] &] (* _Kritsada Moomuang_, Oct 27 2024 *)
%o A245080 (PARI) OmegaTest(n)=(bigomega(n)>omega(n))&&(bigomega(n)%omega(n)==0);
%o A245080 Ntest(nmax,test)={my(k=1,n=0,v);v=vector(nmax);while(1,n++;if(test(n),v[k]=n;k++;if(k>nmax,break)););return(v);}
%o A245080 Ntest(20000,OmegaTest)
%o A245080 (PARI) is_a245080(n) = my(F=factor(n), o=omega(F), O=bigomega(F)); O>o && O%o==0; \\ _Hugo Pfoertner_, Oct 28 2024
%Y A245080 Cf. A000040, A001597, A067340, A070011.
%K A245080 nonn
%O A245080 1,1
%A A245080 _Stanislav Sykora_, Jul 11 2014