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

1,1

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

Links

Stanislav Sykora, Table of n, a(n) for n = 1..20000

Wikipedia, Arithmetic functions

Example

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).

Mathematica

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

Prog

(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

Crossrefs

Cf. A000040, A001597, A067340, A070011.

Sequence in context: A100657 A372280 A361204 * A212164 A293243 A339740

Adjacent sequences: A245077 A245078 A245079 * A245081 A245082 A245083

Keyword

nonn

Author

Stanislav Sykora, Jul 11 2014

Status

approved