A119959 -id:A119959 - cp's OEIS frontend

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

Patrick De Geest

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

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

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.

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

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.

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

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

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

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

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

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

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)

A202535 a(n) = nphi(n)abs( mobius(n) ).

Original entry on oeis.org

1, 2, 6, 0, 20, 12, 42, 0, 0, 40, 110, 0, 156, 84, 120, 0, 272, 0, 342, 0, 252, 220, 506, 0, 0, 312, 0, 0, 812, 240, 930, 0, 660, 544, 840, 0, 1332, 684, 936, 0, 1640, 504, 1806, 0, 0, 1012, 2162, 0, 0, 0, 1632, 0, 2756, 0, 2200, 0, 2052, 1624, 3422

Offset: 1

R. J. Mathar, Dec 20 2011

The inverse Mobius transform is b(n>=1) = 1, 3, 7, 3, 21, 21, 43, 3,7, 63, 11, 21,...., multiplicative with b(p^e) = A002061(p), e>=1 (see A119959). - R. J. Mathar

a(n) > 0 only when n is squarefree. - Alonso del Arte, Dec 20 2011

			a(5) = 20 because 5 * phi(5) * |mu(5)| = 5 * 4 * |(-1)| = 20.

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

Cf. A079579.

Table[n EulerPhi[n] Abs[MoebiusMu[n]], {n, 60}] (* Alonso del Arte, Dec 20 2011 *)
f[p_, e_] := If[e == 1, (p-1)*p, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 26 2020 *)

a(n)=n*eulerphi(n)*abs(moebius(n)) \\ Charles R Greathouse IV, Dec 20 2011

for(n=1, 100, print1(direuler(p=2, n, (1 - p*X + p^2*X))[n], ", ")) \\ Vaclav Kotesovec, Jun 24 2020

a(n) = A002618(n) *A008966(n).
Multiplicative with a(p^e) = (p-1)*p if e=1, a(p^e)=0 if e>1.
Dirichlet g.f.: Sum_(n>=1) a(n)/n^s = Product_{primes p} (1-p^(1-s)+p^(2-s)).
From Vaclav Kotesovec, Jun 24 2020: (Start)
Dirichlet g.f.: zeta(s-2)*Product_{primes p} (1 + p^(3-2*s) - p^(4-2*s) - p^(1-s)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = A065464/3 = 0.142749835... (End)

A083558 p(p^2-p+1) as p runs through the primes.

Original entry on oeis.org

6, 21, 105, 301, 1221, 2041, 4641, 6517, 11661, 23577, 28861, 49321, 67281, 77701, 101661, 146121, 201957, 223321, 296341, 352941, 383761, 486877, 564981, 697137, 903361, 1020201, 1082221, 1213701, 1283257, 1430241, 2032381, 2231061, 2552721, 2666437

Offset: 1

N. J. A. Sloane, Jun 15 2003

nonn

Warning: not all quizzes permit the use of the OEIS!

Discard (from the list of integers) numbers that have exactly 1 factor of prime(n) in their prime factorization. Of those remaining, the proportion that have exactly 2 factors of prime(n) is (prime(n)-1)/a(n). - Peter Munn, Nov 27 2020

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A119959.

[p*(p^2-p+1): p in PrimesUpTo(150)]; // Vincenzo Librandi, Jan 10 2017

Table[p(p^2-p+1),{p,Prime[Range[40]]}] (* Harvey P. Dale, Jan 09 2017 *)

a(n) = A000040(n) * A119959(n). - Peter Munn, Nov 29 2020

A258434 n^2 - phi(n).

Original entry on oeis.org

0, 3, 7, 14, 21, 34, 43, 60, 75, 96, 111, 140, 157, 190, 217, 248, 273, 318, 343, 392, 429, 474, 507, 568, 605, 664, 711, 772, 813, 892, 931, 1008, 1069, 1140, 1201, 1284, 1333, 1426, 1497, 1584, 1641, 1752, 1807, 1916, 2001, 2094, 2163, 2288

Offset: 1

Carlos Eduardo Olivieri, May 30 2015

			a(5) = 21, since 5^2 = 25 and phi(5) = 4.

Cf. A000290, A000010.

[n^2 - EulerPhi(n): n in [1..60]]; // Vincenzo Librandi, Jun 03 2015

Mathematica
```
Table[n^2 - EulerPhi[n], {n, 200}]
```

vector(50, n, n^2-eulerphi(n)) \\ Michel Marcus, May 30 2015

a(n) = A000290(n) - A000010(n).

For p prime, a(p) = A002061(p); A119959 is a subsequence. - Michel Marcus, Jun 06 2015

A067498 Maximum number of reflections for a ray of light which reflects at n points (reflecting more than once at most or all points).

Original entry on oeis.org

1, 3, 7, 11, 21, 27, 43, 51, 69, 83, 111, 123, 157, 171, 199, 227, 273

Offset: 1

Henry Bottomley, Feb 22 2002

The maxima occur when the points of reflection are arranged at the vertices of a regular n-gon and the ray (except entering and leaving) follows the edges and diagonals of the polygon.

M. Oskar van Deventer, Point Mirror Reflection, in The Mathemagician and the Pied Puzzler edited by E. Berlekamp and T. Rodgers, A K Peters Publishers, 1999, pp. 245-251. google books.

Cf. A119959 (subsequence).

For p prime, a(p)=p^2-p+1.
a(n) <= n^2-n+1. [Deventer eq (1)]
a(n) <= n^2-2*n+3 for even n [Deventer eq. (2)]

A121208 Cyclotomic polynomials evaluated at prime number abscissae.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 31, 32, 36, 38, 40, 42, 43, 44, 46, 48, 50, 52, 54, 57, 58, 60, 61, 62, 66, 68, 70, 72, 73, 74, 78, 80, 82, 84, 88, 90, 96, 98, 100, 102, 104, 106, 108, 110, 111, 112, 114, 121, 122, 126, 127

Offset: 1

R. J. Mathar, Aug 20 2006

nonn

Contains as subsets: all A006093 which are Phi_1(p), all A008864 which are Phi_2(p), all A060800 which are Phi_3(p), all A066872 which are Phi_4(p) and all A119959 which are Phi_6(p).

			Contains 50, which is the fourth cyclotomic polynomial evaluated at the prime 7, 50=Phi_4(7).

Eric Weisstein's World of Mathematics, Cyclotomic Polynomials.

Cf. A000040, A006093, A008864, A060800, A066872, A119959.

a(n) = Phi_j(p), p in A000040, j=1,2,3,4...

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A030078 Cubes of primes.

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A202535 a(n) = n*phi(n)*abs( mobius(n) ).

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A083558 p(p^2-p+1) as p runs through the primes.

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Magma

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A258434 n^2 - phi(n).

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Magma

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A067498 Maximum number of reflections for a ray of light which reflects at n points (reflecting more than once at most or all points).

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Formula

A121208 Cyclotomic polynomials evaluated at prime number abscissae.

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Formula

A202535 a(n) = nphi(n)abs( mobius(n) ).