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A030515 Numbers with exactly 6 divisors. 14
12, 18, 20, 28, 32, 44, 45, 50, 52, 63, 68, 75, 76, 92, 98, 99, 116, 117, 124, 147, 148, 153, 164, 171, 172, 175, 188, 207, 212, 236, 242, 243, 244, 245, 261, 268, 275, 279, 284, 292, 316, 325, 332, 333, 338, 356, 363, 369, 387, 388, 404, 412 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Numbers which are either the 5th power of a prime or the product of a prime and the square of a different prime, i.e., numbers which are in A050997 (5th powers of primes) or A014612 (product of three primes) but not in A030078 (cubes of primes) or A007304 (product of three distinct primes). - Henry Bottomley, Apr 25 2000

Also numbers which are the square root of the product of their proper divisors. - Amarnath Murthy, Apr 21 2001

Numbers of the form p^5 or p^2*q, where p and q are distinct primes. Such numbers are multiplicatively 3-perfect (i.e., the product of divisors of a(n) equals a(n)^3). - Lekraj Beedassy, Jul 13 2005

Since A119479(6)=5, there are never more than 5 consecutive terms. Quintuples of consecutive terms start at 10093613546512321, 14414905793929921, 266667848769941521, ... (A141621). No such quintuple contains a term of the form p^5. - Ivan Neretin, Feb 08 2016

REFERENCES

Amarnath Murthy, A note on the Smarandache Divisor sequences, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring 2000.

LINKS

R. J. Mathar, Table of n, a(n) for n = 1..1000

Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4, 1.12.

Eric Weisstein's World of Mathematics, Divisor Product

FORMULA

Union of A050997 and A054753. - Lekraj Beedassy, Jul 13 2005

A000005(a(n))=6. - Juri-Stepan Gerasimov, Oct 10 2009

MAPLE

N:= 1000: # to get all terms <= N

Primes:= select(isprime, {2, seq(i, i=3..floor(N/4))}):

S:= select(`<=`, {seq(p^5, p = Primes), seq(seq(p*q^2, p=Primes minus {q}), q=Primes)}, N):

sort(convert(S, list)); # Robert Israel, Feb 10 2016

MATHEMATICA

f[n_]:=Length[Divisors[n]]==6; lst={}; Do[If[f[n], AppendTo[lst, n]], {n, 6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 14 2009 *)

Select[Range[500], DivisorSigma[0, #]==6&] (* Harvey P. Dale, Oct 02 2014 *)

PROG

(PARI) is(n)=numdiv(n)==6 \\ Charles R Greathouse IV, Jan 23 2014

CROSSREFS

Cf. A061117.

Sequence in context: A263838 A217856 A253388 * A162947 A070011 A084679

Adjacent sequences:  A030512 A030513 A030514 * A030516 A030517 A030518

KEYWORD

nonn,easy

AUTHOR

Jeff Burch

EXTENSIONS

Definition clarified by Jonathan Sondow, Jan 23 2014

STATUS

approved

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Last modified October 15 11:01 EDT 2019. Contains 328026 sequences. (Running on oeis4.)