A087797 -id:A087797 - cp's OEIS frontend

A168363 Squares and cubes of primes.

4, 8, 9, 25, 27, 49, 121, 125, 169, 289, 343, 361, 529, 841, 961, 1331, 1369, 1681, 1849, 2197, 2209, 2809, 3481, 3721, 4489, 4913, 5041, 5329, 6241, 6859, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12167, 12769, 16129, 17161, 18769, 19321, 22201

Offset: 1

Franklin T. Adams-Watters, Nov 23 2009

nonn

Primitive elements for powerful numbers; every powerful is product of these numbers. The representation is not necessarily unique.

Cf. A001694, A053810, A001248, A030078, A087797, A178254.

m=30000;Union[Prime[Range[PrimePi[m^(1/2)]]]^2,Prime[Range[PrimePi[m^(1/3)]]]^3] (* Vladimir Joseph Stephan Orlovsky, Apr 11 2011 *)
With[{nn=50},Take[Union[Flatten[Table[{n^2,n^3},{n,Prime[Range[ nn]]}]]],nn]] (* Harvey P. Dale, Feb 26 2015 *)

for(n=1,40000,fm=factor(n);if(matsize(fm)[1]==1&(fm[1,2]==2||fm[1,2]==3),print1(n",")))

is(n)=my(k=isprimepower(n)); k && k<4 \\ Charles R Greathouse IV, May 24 2013

from math import isqrt
from sympy import primepi, integer_nthroot
def A168363(n):
    def f(x): return n+x-primepi(isqrt(x))-primepi(integer_nthroot(x,3)[0])
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return int(m) # Chai Wah Wu, Aug 09 2024

A178254(a(n)) = 2. - Reinhard Zumkeller, May 24 2010

Sum_{n>=1} 1/a(n) = P(2) + P(3) = 0.6270100593..., where P is the prime zeta function. - Amiram Eldar, Dec 21 2020

A322885 Number of 3-generated Abelian groups of order n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 7, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 5, 1, 2, 2, 4, 1, 1

Offset: 1

Álvar Ibeas, Dec 29 2018

Groups generated by fewer than 3 elements are not excluded. The number of Abelian groups with 3 invariant factors is a(n) - A046951(n).
Sum of the first three columns from A249770 (for n > 1).
Dirichlet convolution of A061704 and A010052. Dirichlet convolution of A046951 and A010057.
The number of unordered factorizations of n into biquadratefree power of primes (1 and primes, squares of primes and cubes of primes, A087797). - Amiram Eldar, Jun 12 2025

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001399, A010052, A010057, A046951, A061704, A087797, A249770.

f:= proc(n) local t;
  mul(round((t[2]+3)^2/12),t=ifactors(n)[2])
end proc:
map(f, [$1..200]); # Robert Israel, May 20 2019

a[n_] := Times @@ (Round[(# + 3)^2/12]& /@ FactorInteger[n][[All, 2]]);
Array[a, 102] (* Jean-François Alcover, Jan 02 2019 *)

a(n) = vecprod(apply(x -> round((x+3)^2/12), factor(n)[, 2])); \\ Amiram Eldar, Jun 12 2025

Multiplicative with a(p^e) = A001399(e).
Dirichlet g.f.: zeta(s) * zeta(2s) * zeta(3s).
Sum_{k=1..n} a(k) ~ Pi^2*zeta(3)*n/6 + zeta(1/2)*zeta(3/2)*sqrt(n) + zeta(1/3)*zeta(2/3)*n^(1/3). - Vaclav Kotesovec, Feb 02 2019

A277187 Numbers n such that A001158(n) == 1 (mod n).

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 36, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 289, 293

Offset: 1

Ilya Gutkovskiy, Oct 04 2016

Essentially the same as A087797. - Ilya Gutkovskiy, Dec 26 2016

			a(1) = 2 because sigma_3(2) = 1^3 + 2^3 = 9 and 9 == 1 (mod 2);
a(2) = 3 because sigma_3(3) = 1^3 + 3^3 = 28 and 28 == 1 (mod 3);
a(3) = 4 because sigma_3(4) = 1^3 + 2^3 + 4^3 = 73 and 73 == 1 (mod 4), etc.

Cf. A000040, A000430, A001158, A001248, A030078, A087797.

Select[Range[300], Mod[DivisorSigma[3, #1], #1] == 1 & ]

Edited by Ilya Gutkovskiy, Dec 26 2016

A280715 Expansion of Product_{k>=1} 1/((1 - x^prime(k))(1 - x^(prime(k)^2))(1 - x^(prime(k)^3))).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 4, 6, 7, 9, 12, 15, 19, 23, 29, 36, 44, 53, 65, 78, 94, 112, 134, 159, 189, 222, 263, 307, 361, 420, 491, 569, 661, 764, 883, 1017, 1170, 1343, 1539, 1761, 2011, 2293, 2611, 2968, 3369, 3819, 4323, 4887, 5518, 6222, 7007, 7883, 8857, 9942, 11144, 12483, 13964, 15609, 17426, 19440, 21664

Offset: 0

Ilya Gutkovskiy, Jan 07 2017

nonn

Number of partitions of n into parts that are primes (A000040), squares of primes (A001248) or cubes of primes (A030078).

			a(8) = 6 because we have [8], [5, 3], [4, 4], [4, 2, 2], [3, 3, 2], [2, 2, 2, 2].

Index entries for related partition-counting sequences

Cf. A000040, A000607, A001248, A023893, A030078, A087797, A090677, A111901, A279760, A280125.

nmax = 61; CoefficientList[Series[Product[1/((1 - x^Prime[k]) (1 - x^Prime[k]^2) (1 - x^Prime[k]^3)), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/((1 - x^prime(k))*(1 - x^(prime(k)^2))*(1 - x^(prime(k)^3))).

cp's OEIS Frontend

A168363 Squares and cubes of primes.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Formula

A322885 Number of 3-generated Abelian groups of order n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A277187 Numbers n such that A001158(n) == 1 (mod n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A280715 Expansion of Product_{k>=1} 1/((1 - x^prime(k))*(1 - x^(prime(k)^2))*(1 - x^(prime(k)^3))).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A280715 Expansion of Product_{k>=1} 1/((1 - x^prime(k))(1 - x^(prime(k)^2))(1 - x^(prime(k)^3))).