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A245080 Numbers such that omega(a(n)) is a proper divisor of bigomega(a(n)). 2
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 (list; graph; refs; listen; history; text; internal format)
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).

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).

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)

CROSSREFS

Cf. A000040, A001597, A067340, A070011.

Sequence in context: A280387 A243180 A100657 * A212164 A293243 A140104

Adjacent sequences:  A245077 A245078 A245079 * A245081 A245082 A245083

KEYWORD

nonn

AUTHOR

Stanislav Sykora, Jul 11 2014

STATUS

approved

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Last modified December 16 09:06 EST 2019. Contains 330020 sequences. (Running on oeis4.)