A065471 -id:A065471 - cp's OEIS frontend

A112526 Characteristic function for powerful numbers.

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0

Offset: 1

Franklin T. Adams-Watters, Sep 09 2005

A signed multiplicative variant is defined by b(n) = a(n)*mu(n) with mu = A008683, such that b(p^e)=0 if e=1 and b(p^e)= -1 if e>1. This has Dirichlet series Sum_{n>=1} b(n)/n = A005596 and Sum_{n>=1} b(n)/n^2 = A065471. - R. J. Mathar, Apr 04 2011

			a(72) = 1 because 72 = 2^3*3^2 has all exponents > 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Powerful Number.
Index entries for characteristic functions.

Differs from characteristic function of perfect powers A075802 at Achilles numbers A052486.
Cf. A001694 (powerful numbers), A124010, A001221, A027746.
Cf. A008683, A008966, A005596, A065471, A082695, A063524.
Cf. A002117, A078434, A005361.

a112526 1 = 1
a112526 n = fromEnum $ (> 1) $ minimum $ a124010_row n
-- Reinhard Zumkeller, Jun 03 2015, Sep 16 2011

cfpn[n_]:=If[n==1||Min[Transpose[FactorInteger[n]][[2]]]>1,1,0]; Array[ cfpn,120] (* Harvey P. Dale, Jul 17 2012 *)

for(n=1, 100, print1(direuler(p=2, n, (1+X^3)/(1-X^2))[n], ", ")) \\ Vaclav Kotesovec, Jul 15 2022

a(n) = ispowerful(n); \\ Amiram Eldar, Jul 02 2025

from sympy import factorint
def A112526(n): return int(all(e>1 for e in factorint(n).values())) # Chai Wah Wu, Sep 15 2024

Multiplicative with a(p^e) = 1 - 0^(e-1), e > 0 and p prime.
Dirichlet g.f.: zeta(2*s)*zeta(3*s)/zeta(6*s), e.g., A082695 at s=1.
a(n) * A008966(n) = A063524(n). - Reinhard Zumkeller, Sep 16 2011
a(n) = {m: Min{A124010(m,k): k=1..A001221(m)} > 1}. - Reinhard Zumkeller, Jun 03 2015
Sum_{k=1..n} a(k) ~ zeta(3/2)*sqrt(n)/zeta(3) + 6*zeta(2/3)*n^(1/3)/Pi^2. - Vaclav Kotesovec, Feb 08 2019
a(n) = Sum_{d|n} A005361(d)*A008683(n/d). - Ridouane Oudra, Jul 03 2025

A138402 a(n) = (n-th prime)^4-(n-th prime)^2.

Original entry on oeis.org

12, 72, 600, 2352, 14520, 28392, 83232, 129960, 279312, 706440, 922560, 1872792, 2824080, 3416952, 4877472, 7887672, 12113880, 13842120, 20146632, 25406640, 28392912, 38943840, 47451432, 62734320, 88519872, 104050200, 112540272

Offset: 1

Artur Jasinski, Mar 19 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index to sequences related to prime powers.

Cf. A001248, A036689, A036690, A065471, A084920.

[NthPrime((n))^4 - NthPrime((n))^2: n in [1..30] ]; // Vincenzo Librandi, Jun 17 2011

a = {}; Do[p = Prime[n]; AppendTo[a, p^4 - p^2], {n, 1, 50}]; a
#^4-#^2&/@Prime[Range[30]] (* Harvey P. Dale, Sep 19 2018 *)

forprime(p=2,1e3,print1(p^4-p^2", ")) \\ Charles R Greathouse IV, Jun 16 2011

apply(p->p^4-p^2, primes(100)) \\ Charles R Greathouse IV, Apr 17 2015

Product_{n>=1} (1 - 1/a(n)) = A065471.
From Amiram Eldar, Nov 22 2022: (Start)
a(n) = A001248(n) * A084920(n).
a(n) = A036689(n) * A036690(n). (End)

A078086 Continued fraction expansion of Product_{p prime}(1 - 1/(p^2*(p^2-1))).

Original entry on oeis.org

0, 1, 9, 5, 10, 1, 6, 1, 1, 1, 11, 1, 5, 1, 10, 1, 17, 5, 5, 6, 1, 13, 6, 3, 1, 1, 4, 1, 1, 2, 1, 3, 3, 1, 5, 7, 1, 17, 5, 2, 1, 10, 2, 2, 1, 14, 62, 3, 3, 4, 1, 1, 4, 8, 1041, 86, 1, 6, 1, 2, 6, 12, 1, 45, 2, 45, 2, 3, 2, 1, 1, 2, 2, 1, 4, 1, 1, 21, 5, 6, 4, 1, 1, 4, 1, 2, 2, 3, 2, 6, 1

Offset: 0

Benoit Cloitre, Dec 02 2002

Cf. A065471 (decimal expansion).

contfrac(prodeulerrat(1 - 1/(p^2*(p^2-1)))) \\ Amiram Eldar, Mar 16 2021

Offset changed by Andrew Howroyd, Jul 05 2024

cp's OEIS Frontend

A112526 Characteristic function for powerful numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

Python

Formula

A138402 a(n) = (n-th prime)^4-(n-th prime)^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A078086 Continued fraction expansion of Product_{p prime}(1 - 1/(p^2*(p^2-1))).

Views

Author

Keywords

Crossrefs

Programs

PARI

Extensions