A030631 -id:A030631 - 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)

A030513 Numbers with 4 divisors.

Original entry on oeis.org

6, 8, 10, 14, 15, 21, 22, 26, 27, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187

Offset: 1

Jeff Burch

Essentially the same as A007422.
Numbers which are either the product of two distinct primes (A006881) or the cube of a prime (A030078).
4*a(n) are the solutions to A048272(x) = Sum_{d|x} (-1)^d = 4. - Benoit Cloitre, Apr 14 2002
Since A119479(4)=3, there are never more than 3 consecutive integers in the sequence. Triples of consecutive integers start at 33, 85, 93, 141, 201, ... (A039833). No such triple contains a term of the form p^3. - Ivan Neretin, Feb 08 2016
Numbers that are equal to the product of their proper divisors (A007956) (proof in Sierpiński). - Bernard Schott, Apr 04 2022

Wacław Sierpiński, Elementary Theory of Numbers, Ex. 2 p. 174, Warsaw, 1964.

Cf. A000005, A007422, A007956, A030515, A035533, A048272, A119479.

Equals the disjoint union of A006881 and A030078.

[n: n in [1..200] | DivisorSigma(0, n) eq 4]; // Vincenzo Librandi, Jul 16 2015

Select[Range[200], DivisorSigma[0,#]==4&] (* Harvey P. Dale, Apr 06 2011 *)

is(n)=numdiv(n)==4 \\ Charles R Greathouse IV, May 18 2015

from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A030513(n):
    def f(x): return int(n+x-primepi(integer_nthroot(x,3)[0])+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 16 2024

{n : A000005(n) = 4}. - Juri-Stepan Gerasimov, Oct 10 2009

Incorrect comments removed by Charles R Greathouse IV, Mar 18 2010

A139572 Numbers with 37 divisors.

Original entry on oeis.org

68719476736, 150094635296999121, 14551915228366851806640625, 2651730845859653471779023381601, 30912680532870672635673352936887453361, 12646218552730347184269489080961456410641

Offset: 1

Omar E. Pol, May 07 2008

36th powers of primes. The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, Index entries for number of divisors

Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139573.

A139572 := proc(n) ithprime(n)^36 ; end proc: seq(A139572(n),n=1..10) ; # R. J. Mathar, Feb 05 2010

Prime[Range[10]]^36 (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

a(n)=prime(n)^36 \\ Charles R Greathouse IV, Jun 19 2016

a(n) = A000040(n)^(37-1) = A000040(n)^36.

More terms from R. J. Mathar, Feb 05 2010

A030634 Numbers with 16 divisors.

Original entry on oeis.org

120, 168, 210, 216, 264, 270, 280, 312, 330, 378, 384, 390, 408, 440, 456, 462, 510, 520, 546, 552, 570, 594, 616, 640, 680, 690, 696, 702, 714, 728, 744, 750, 760, 770, 798, 858, 870, 888, 896, 910, 918, 920, 930, 945, 952, 966, 984, 1000

Offset: 1

Jeff Burch

nonn

Numbers of the form p^15 (subset of A010803), p*q^7, p*q*r^3 or p^3*q^3, or p*q*r*s, where p, q, r and s are distinct primes. - R. J. Mathar, Mar 01 2010

R. J. Mathar, Table of n, a(n) for n = 1..1000
Jérôme Germoni, Nombres à huit diviseurs, Images des Mathématiques, CNRS, 2017 (in French).

Cf. A030626, A030631, A030632, A030633.

[n: n in [1..1000] | DivisorSigma(0, n) eq 16]; // Vincenzo Librandi, Oct 05 2017

Select[Range[3000],DivisorSigma[0,#]==16&] (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

is(n)=numdiv(n)==16 \\ Charles R Greathouse IV, Jun 19 2016

A139574 Numbers with 43 divisors.

Original entry on oeis.org

4398046511104, 109418989131512359209, 227373675443232059478759765625, 311973482284542371301330321821976049, 54763699237492901685126120802225273763666521, 61040881526285814362156628321386486455989674569

Offset: 1

Omar E. Pol, May 09 2008

42nd powers of primes. The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, Index entries for number of divisors

Cf. A000005, A000040, A001248, A030514, A030516, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139575.

Prime[Range[9]]^42 (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

a(n)=prime(n)^42 \\ Charles R Greathouse IV, Jun 19 2016

a(n)=A000040(n)^(43-1)=A000040(n)^42.

More terms from R. J. Mathar, May 11 2008

A139573 Numbers with 41 divisors.

Original entry on oeis.org

1099511627776, 12157665459056928801, 9094947017729282379150390625, 6366805760909027985741435139224001, 452592555681759518058893560348969204658401

Offset: 1

Omar E. Pol, May 07 2008

40th powers of primes. The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, Index entries for number of divisors

Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139572.

Prime[Range[9]]^40 (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

a(n)=prime(n)^40 \\ Charles R Greathouse IV, Jun 19 2016

a(n)=A000040(n)^(41-1)=A000040(n)^40.

More terms from Jon E. Schoenfield, May 18 2010

A139575 Numbers with 47 divisors.

Original entry on oeis.org

70368744177664, 8862938119652501095929, 142108547152020037174224853515625, 749048330965186233494494102694564493649, 801795320536133573571931534665380233173841533961

Offset: 1

Omar E. Pol, May 09 2008

46th powers of primes. The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.

T. D. Noe, Table of n, a(n) for n = 1..1000
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
OEIS Wiki, Index entries for number of divisors

Cf. A000005, A000040, A001248, A030514, A030516, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139574.

Prime[Range[9]]^46 (* Vladimir Joseph Stephan Orlovsky, May 06 2011 *)

a(n)=prime(n)^46 \\ Charles R Greathouse IV, Jun 19 2016

a(n)=A000040(n)^(47-1)=A000040(n)^46.

More terms from R. J. Mathar, May 11 2008

A067004 Number of numbers <= n with same number of divisors as n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 2, 2, 3, 5, 1, 6, 4, 5, 1, 7, 2, 8, 3, 6, 7, 9, 1, 3, 8, 9, 4, 10, 2, 11, 5, 10, 11, 12, 1, 12, 13, 14, 3, 13, 4, 14, 6, 7, 15, 15, 1, 4, 8, 16, 9, 16, 5, 17, 6, 18, 19, 17, 1, 18, 20, 10, 1, 21, 7, 19, 11, 22, 8, 20, 2, 21, 23, 12, 13, 24, 9, 22, 2, 2, 25, 23, 3, 26, 27

Offset: 1

Henry Bottomley, Dec 21 2001

nonn

			a(10)=3 since 6,8,10 each have four divisors. a(11)=5 since 2,3,5,7,11 each have two divisors.

Paul Tek, Table of n, a(n) for n = 1..10000

Cf. A000005, A008479, A058933, A067003. Inverse of A000040, A001248, A030513, A030514, A030515, A030516, A030626, A030627, A030628, A030629, A030630, A030631, A030632, A030633, A030634, A030635, A030636, A030637, A030638 etc.
Cf. also A079788, A138009.
A047983(n) = a(n)-1.

N:= 1000: # to get a(1) to a(N)
R:= Vector(N):
for n from 1 to N do
  v:= numtheory:-tau(n);
  R[v]:= R[v]+1;
  A[n]:= R[v];
od:
seq(A[n],n=1..N); # Robert Israel, May 04 2015

b[_] = 0;
a[n_] := a[n] = With[{t = DivisorSigma[0, n]}, b[t] = b[t]+1];
Array[a, 105] (* Jean-François Alcover, Dec 20 2021 *)

a(n)=my(d=numdiv(n)); sum(k=1,n,numdiv(k)==d) \\ Charles R Greathouse IV, Sep 02 2015

Ordinal transform of A000005. - Franklin T. Adams-Watters, Aug 28 2006

a(A000040(n)^(p-1)) = n if p is prime. - Robert Israel, May 04 2015

A173533 Numbers with 53 divisors.

Original entry on oeis.org

4503599627370496, 6461081889226673298932241, 2220446049250313080847263336181640625, 88124787089723195184393736687912818113311201, 1420429319844313329730664601483335671261683881745483121, 8415003868347247618489696679505181495471801448798649088081

Offset: 1

Omar E. Pol, Oct 16 2010

52nd powers of primes.

The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos

Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139574, A139575.

Prime[Range[9]]^52 (* Vladimir Joseph Stephan Orlovsky, May 06 2011 *)

a(n)=prime(n)^52 \\ Charles R Greathouse IV, Jun 19 2016

a(n) = A000040(n)^(53-1) = A000040(n)^52.

A183062 Numbers with 59 divisors.

Original entry on oeis.org

288230376151711744, 4710128697246244834921603689, 34694469519536141888238489627838134765625, 10367793076318844190248738727596255138212949486449

Offset: 1

Omar E. Pol, Jul 31 2011

Also, 58th powers of primes.

The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.

OEIS Wiki, Index entries for number of divisors

Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139574, A139575, A173533.

a(n)=prime(n)^58 \\ Charles R Greathouse IV, Jul 31 2011

a(n) = A000040(n)^(59-1) = A000040(n)^58.

A000005(a(n)) = 59.

cp's OEIS Frontend

A030078 Cubes of primes.

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A030513 Numbers with 4 divisors.

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A139572 Numbers with 37 divisors.

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A030634 Numbers with 16 divisors.

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A139574 Numbers with 43 divisors.

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A139573 Numbers with 41 divisors.

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