A079705 -id:A079705 - 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)

A192015 Arithmetic derivative of prime powers: a(n) = A003415(A000961(n)).

Original entry on oeis.org

0, 1, 1, 4, 1, 1, 12, 6, 1, 1, 32, 1, 1, 1, 10, 27, 1, 1, 80, 1, 1, 1, 1, 14, 1, 1, 1, 192, 1, 1, 1, 1, 108, 1, 1, 1, 1, 1, 1, 1, 1, 22, 75, 1, 448, 1, 1, 1, 1, 1, 1, 1, 1, 26, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 405, 1, 1024, 1, 1, 1, 1, 1, 1, 1, 34

Offset: 1

Reinhard Zumkeller, Jun 26 2011

nonn

a(A000040(n)) = 1; a(A002808(n)) > 1;
A001787, A027471, A100484, A079705 and A051674 are subsequences;
A001787 and A024622 give record values and where they occur;
A192016(n) = A003415(a(n)).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Power

a192015 = a003415 . a000961  -- Reinhard Zumkeller, Apr 16 2014

Join[{0}, Reap[For[n = 1, n <= 300, n++, f = FactorInteger[n]; If[Length[f] == 1, Sow[n*Total[Apply[#2/#1&, f, {1}]]]]]][[2, 1]]] (* Jean-François Alcover, Feb 21 2014 *)

from sympy import primepi, integer_nthroot, factorint
def A192015(n):
    if n == 1: return 0
    def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return sum((m*e//p for p,e in factorint(m).items())) # Chai Wah Wu, Aug 15 2024

a(n) = A025474(n) * A025473(n)^(A025474(n) - 1).

A192083 Arithmetic derivative of squares of prime powers: a(n) = A003415(A056798(n)).

Original entry on oeis.org

0, 4, 6, 32, 10, 14, 192, 108, 22, 26, 1024, 34, 38, 46, 500, 1458, 58, 62, 5120, 74, 82, 86, 94, 1372, 106, 118, 122, 24576, 134, 142, 146, 158, 17496, 166, 178, 194, 202, 206, 214, 218, 226, 5324, 18750, 254, 114688, 262, 274, 278, 298, 302, 314, 326, 334

Offset: 1

Reinhard Zumkeller, Jun 26 2011

nonn

A001787 and A024622 give record values and where they occur.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A003415, A024622, A025473, A025474, A056798, A192084.

Subsequences: A001787, A027471, A051674, A079705, A100484.

s[n_] := If[PrimePowerQ[n], f = FactorInteger[n][[1]]; 2*f[[2]]*n^(2 - 1/f[[2]]), Nothing]; s[1] = 0; Array[s, 200] (* Amiram Eldar, Apr 06 2025 *)

a(n) = 2 * A025474(n) * A025473(n)^(2*A025474(n) - 1).

A192084(n) = A003415(a(n)).

A338519 Integers that can be expressed as a product d*tau(d), where tau is the number of divisors function, in a single way.

Original entry on oeis.org

1, 4, 6, 10, 12, 14, 22, 24, 26, 27, 32, 34, 38, 40, 46, 56, 58, 60, 62, 72, 74, 75, 80, 82, 84, 86, 88, 94, 104, 106, 118, 120, 122, 132, 134, 136, 140, 142, 146, 147, 152, 156, 158, 166, 168, 178, 184, 194, 202, 204, 206, 214, 218, 220, 226, 228, 232, 240, 248, 254

Offset: 1

Michel Marcus, Nov 01 2020

nonn

Integers m such that A327166(m) = 1.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000005, A038040, A327166, A338382.
Subsequences: A100484 (2*p), A079705 (3*p^2) that gives odd terms.
Cf. A338520 (similar for sum of divisors).

f(n) = sumdiv(n, d, d*numdiv(d) == n); \\ A327166
isok(n) = f(n)==1;

A349986 Numbers that can be represented as p^2 + p*q + q^2 where p and q are primes.

Original entry on oeis.org

12, 19, 27, 39, 49, 67, 75, 79, 109, 147, 163, 199, 201, 217, 247, 259, 309, 327, 349, 363, 399, 403, 427, 433, 457, 481, 507, 543, 579, 597, 607, 669, 679, 691, 739, 777, 867, 903, 937, 973, 997, 1011, 1027, 1063, 1083, 1093, 1141, 1209, 1227, 1281, 1327, 1387, 1423, 1447, 1489, 1533, 1579, 1587

Offset: 1

J. M. Bergot and Robert Israel, Jan 09 2022

nonn

The only square in this sequence is 49.

			a(3) = 27 is a term because 27 = 3^2+3*3+3^2.
a(4) = 39 is a term because 39 = 2^2+2*5+5^2.

Robert Israel, Table of n, a(n) for n = 1..10000
Mathematics StackExchange, Largest square written as p^2+pq+q^2 where p, q are primes?

Contains A079705, A244146, A349987.

Subsequence of A024614.

N:= 10^4: # for terms <= N
P:= select(isprime, [2,seq(i,i=3..floor(sqrt(N)),2)]):
nP:= nops(P):
S:= {}:
for i from 1 to nP do
  for j from 1 to i do
    x:= P[i]^2 + P[i]*P[j]+P[j]^2;
    if x > N then break fi;
    S:= S union {x};
od od:
sort(convert(S,list));

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A030078 Cubes of primes.

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A192015 Arithmetic derivative of prime powers: a(n) = A003415(A000961(n)).

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A192083 Arithmetic derivative of squares of prime powers: a(n) = A003415(A056798(n)).

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A338519 Integers that can be expressed as a product d*tau(d), where tau is the number of divisors function, in a single way.

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PARI

A349986 Numbers that can be represented as p^2 + p*q + q^2 where p and q are primes.

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