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%I A030078 #118 Aug 06 2024 09:56:06
%S A030078 8,27,125,343,1331,2197,4913,6859,12167,24389,29791,50653,68921,79507,
%T A030078 103823,148877,205379,226981,300763,357911,389017,493039,571787,
%U A030078 704969,912673,1030301,1092727,1225043,1295029,1442897,2048383,2248091,2571353,2685619,3307949
%N A030078 Cubes of primes.
%C A030078 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
%C A030078 Intersection of A014612 and A000578. Intersection of A014612 and A030513. - _Wesley Ivan Hurt_, Sep 10 2013
%C A030078 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
%C A030078 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
%D A030078 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.
%H A030078 T. D. Noe, <a href="/A030078/b030078.txt">Table of n, a(n) for n = 1..1000</a>
%H A030078 Xavier Gourdon and Pascal Sebah, <a href="http://numbers.computation.free.fr/Constants/Miscellaneous/constantsNumTheory.html">Some Constants from Number theory</a>.
%H A030078 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimePower.html">Prime Power</a>.
%H A030078 Wikipedia, <a href="https://en.wikipedia.org/wiki/P-group">p-group</a>, Classification.
%H A030078 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%F A030078 n such that A062799(n) = 3. - _Benoit Cloitre_, Apr 06 2002
%F A030078 a(n) = A000040(n)^3. - _Omar E. Pol_, Jul 27 2009
%F A030078 A064380(a(n)) = A000010(a(n)). - _Vladimir Shevelev_, Apr 19 2010
%F A030078 A003415(a(n)) = A079705(n). - _Reinhard Zumkeller_, Jun 26 2011
%F A030078 A056595(a(n)) = 2. - _Reinhard Zumkeller_, Aug 15 2011
%F A030078 A000005(a(n)) = 4. - _Wesley Ivan Hurt_, Sep 10 2013
%F A030078 a(n) = A119959(n) * A008864(n) -1.- _R. J. Mathar_, Aug 13 2019
%F A030078 Sum_{n>=1} 1/a(n) = P(3) = 0.1747626392... (A085541). - _Amiram Eldar_, Jul 27 2020
%F A030078 From _Amiram Eldar_, Jan 23 2021: (Start)
%F A030078 Product_{n>=1} (1 + 1/a(n)) = zeta(3)/zeta(6) (A157289).
%F A030078 Product_{n>=1} (1 - 1/a(n)) = 1/zeta(3) (A088453). (End)
%e A030078 a(3) = 125; since the 3rd prime is 5, a(3) = 5^3 = 125.
%t A030078 Array[Prime[ # ]^3&, 5! ] (* _Vladimir Joseph Stephan Orlovsky_, Sep 01 2008 *)
%o A030078 (Sage)
%o A030078 [p**3 for p in prime_range(100)] # _Zerinvary Lajos_, May 15 2007
%o A030078 (Haskell)
%o A030078 a030078 = a000578 . a000040
%o A030078 a030078_list = map a000578 a000040_list -- _Reinhard Zumkeller_, May 26 2012
%o A030078 (PARI) a(n)=prime(n)^3 \\ _Charles R Greathouse IV_, Mar 20 2013
%o A030078 (Magma) [p^3: p in PrimesUpTo(300)]; // _Vincenzo Librandi_, Mar 27 2014
%o A030078 (Python)
%o A030078 from sympy import prime, primerange
%o A030078 def aupton(terms): return [p**3 for p in primerange(1, prime(terms)+1)]
%o A030078 print(aupton(35)) # _Michael S. Branicky_, Aug 27 2021
%Y A030078 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).
%Y A030078 Cf. A060800, A131991, A000578, subsequence of A046099.
%Y A030078 Subsequence of A007422 and of A054397.
%Y A030078 Cf. A036966, A085541, A088453, A157289, A258600.
%K A030078 nonn,easy
%O A030078 1,1
%A A030078 _Patrick De Geest_

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