A030078 Cubes of primes.

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, 3307949

Offset: 1

Author

Patrick De Geest

Keywords

• nonn
• easy

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

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

There exist 5 groups of order p^3, when p prime, so this is a subsequence of A054397. Three of them are abelian: C_p^3, C_p^2 X C_p and C_p X C_p X C_p = (C_p)^3. For 8 = 2^3, the 2 nonabelian groups are D_8 and Q_8; for odd prime p, the 2 nonabelian groups are (C_p x C_p) : C_p, and C_p^2 : C_p (remark, for p = 2, these two semi-direct products are isomorphic to D_8). Here C, D, Q mean Cyclic, Dihedral, Quaternion groups of the stated order; the symbols X and : mean direct and semidirect products respectively. - Bernard Schott, Dec 11 2021

Examples

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

References

• Edmund 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
• Xavier Gourdon and Pascal Sebah, Some Constants from Number theory.
• Eric Weisstein's World of Mathematics, Prime Power.
• Wikipedia, p-group, Classification.
• Index to sequences related to prime signature

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 and of A054397.

Cf. A036966, A085541, A088453, A157289, A258600.

Programs

Haskell

a030078 = a000578 . a000040
a030078_list = map a000578 a000040_list -- Reinhard Zumkeller, May 26 2012

Magma

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

Mathematica

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

PARI

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

Python

from sympy import prime, primerange
def aupton(terms): return [p**3 for p in primerange(1, prime(terms)+1)]
print(aupton(35)) # Michael S. Branicky, Aug 27 2021

Sage

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

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

From Amiram Eldar, Jan 23 2021: (Start)

Product_{n>=1} (1 + 1/a(n)) = zeta(3)/zeta(6) (A157289).

Product_{n>=1} (1 - 1/a(n)) = 1/zeta(3) (A088453). (End)