

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
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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 3perfect (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, 123, 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.  JuriStepan 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



