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

4, 8, 9, 16, 24, 25, 27, 32, 36, 40, 49, 54, 56, 64, 81, 88, 96, 100, 104, 121, 125, 128, 135, 136, 144, 152, 160, 169, 184, 189, 196, 216, 224, 225, 232, 240, 243, 248, 250, 256, 289, 296, 297, 324, 328, 336, 343, 344, 351, 352, 360, 361, 375, 376, 384, 400, 416, 424, 441, 459

Offset: 1

Author

Stanislav Sykora, Jul 11 2014

Keywords

• nonn

Comments

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.

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).

There are 40134838 terms in the range 10^9 <= k <= 2*10^9. - Hugo Pfoertner, Oct 28 2024

Examples

			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).
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..20000
• Wikipedia, Arithmetic functions

Crossrefs

Cf. A000040, A001597, A067340, A070011.

Programs

Mathematica

Select[Range[500], Divisible[PrimeOmega[#], PrimeNu[#]] && PrimeNu[#] != PrimeOmega[#] &] (* Kritsada Moomuang, Oct 27 2024 *)

PARI

OmegaTest(n)=(bigomega(n)>omega(n))&&(bigomega(n)%omega(n)==0);
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);}
Ntest(20000,OmegaTest)

PARI

is_a245080(n) = my(F=factor(n), o=omega(F), O=bigomega(F)); O>o && O%o==0; \\ Hugo Pfoertner, Oct 28 2024