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A030078 Cubes of primes. 146
8, 27, 125, 343, 1331, 2197, 4913, 6859, 12167, 24389, 29791, 50653, 68921, 79507, 103823, 148877, 205379, 226981, 300763, 357911, 389017, 493039, 571787, 704969, 912673, 1030301, 1092727, 1225043, 1295029, 1442897, 2048383, 2248091, 2571353, 2685619 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Numbers with exactly three factorizations: A001055(a(n)) = 3 (e.g., a(4) = 1*343 = 7*49 = 7*7*7). - Reinhard Zumkeller, Dec 29 2001

Solutions of the equation n'=3*n^(2/3), where n' is the arithmetic derivative of n. - Paolo P. Lava, Oct 31 2012

Intersection of A014612 and A000578. Intersection of A014612 and A030513. - Wesley Ivan Hurt, Sep 10 2013

Let r(n) = (a(n)-1)/(a(n)+1)) if a(n) mod 4 = 1, (a(n)+1)/(a(n)-1)) otherwise; then Product_{n>=1} r(n) = (9/7) * (28/26) * (124/126) * (344/342) * (1332/1330) * ... = 48/35. - Dimitris Valianatos, Mar 06 2020

REFERENCES

E. Landau, Elementary Number Theory, translation by Jacob E. Goodman of Elementare Zahlentheorie (Vol. I_1 (1927) of Vorlesungen über Zahlentheorie), by Edmund Landau, with added exercises by Paul T. Bateman and E. E. Kohlbecker, Chelsea Publishing Co., New York, 1958, pp. 31-32.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

X. Gourdon and P. Sebah, Some Constants from Number theory

Eric Weisstein's World of Mathematics, Prime Power

Index to sequences related to prime signature

FORMULA

n such that A062799(n) = 3. - Benoit Cloitre, Apr 06 2002

a(n) = A000040(n)^3. - Omar E. Pol, Jul 27 2009

A064380(a(n)) = A000010(a(n)). - Vladimir Shevelev, Apr 19 2010

A003415(a(n)) = A079705(n). - Reinhard Zumkeller, Jun 26 2011

A056595(a(n)) = 2. - Reinhard Zumkeller, Aug 15 2011

A000005(a(n)) = 4. - Wesley Ivan Hurt, Sep 10 2013

a(n) = A119959(n) * A008864(n) -1.- R. J. Mathar, Aug 13 2019

Sum_{n>=1} 1/a(n) = P(3) = 0.1747626392... (A085541). - Amiram Eldar, Jul 27 2020

EXAMPLE

a(3) = 125; since the 3rd prime is 5, a(3) = 5^3 = 125.

MATHEMATICA

Array[Prime[ # ]^3&, 5! ] (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)

PROG

(Sage)

[p**3 for p in prime_range(100)] # Zerinvary Lajos, May 15 2007

(Haskell)

a030078 = a000578 . a000040

a030078_list = map a000578 a000040_list -- Reinhard Zumkeller, May 26 2012

(PARI) a(n)=prime(n)^3 \\ Charles R Greathouse IV, Mar 20 2013

(MAGMA) [p^3: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 27 2014

CROSSREFS

Other sequences that are k-th powers of primes are: A000040 (k=1), A001248 (k=2), this sequence (k=3), A030514 (k=4), A050997 (k=5), A030516 (k=6), A092759 (k=7), A179645 (k=8), A179665 (k=9), A030629 (k=10), A079395 (k=11), A030631 (k=12), A138031 (k=13), A030635 (k=16), A138032 (k=17), A030637 (k=18).

Cf. A060800, A131991, A000578, subsequence of A046099.

Subsequence of A007422.

Cf. A036966, A085541, A258600.

Sequence in context: A240859 A277047 A046452 * A051751 A133042 A181361

Adjacent sequences:  A030075 A030076 A030077 * A030079 A030080 A030081

KEYWORD

nonn,easy

AUTHOR

Patrick De Geest

STATUS

approved

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Last modified August 15 08:53 EDT 2020. Contains 336487 sequences. (Running on oeis4.)