A157290 -id:A157290 - cp's OEIS frontend

A008966 a(n) = 1 if n is squarefree, otherwise 0.

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

Offset: 1

N. J. A. Sloane

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3, 1).

The infinite lower triangular matrix with A008966 on the main diagonal and the rest zeros is the square of triangle A143255. - Gary W. Adamson, Aug 02 2008

Daniel Forgues, Table of n, a(n) for n = 1..100000
Eric Weisstein's World of Mathematics, Moebius Function
Index entries for characteristic functions
Index entries for sequences computed from exponents in factorization of n

Cf. A005117, A008836 (Dirichlet inverse), A013928 (partial sums).
Cf. A179211, A179212, A179213, A179214, A179215.
Cf. A020639, A073576, A087188.
Cf. A124010, A212793, A085357, A156552.
Parity of A002033.
Cf. A047999, A036987
Cf. A082020 (Dgf at s=2), A157289 (Dgf at s=3), A157290 (Dgf at s=4).

a008966 = abs . a008683
-- Reinhard Zumkeller, Dec 13 2015, Dec 15 2014, May 27 2012, Jan 25 2012

Magma
```
[ Abs(MoebiusMu(n)) : n in [1..100]];
```

A008966 := proc(n) if numtheory[issqrfree](n) then 1 ; else 0 ; end if; end proc: # R. J. Mathar, Mar 14 2011

A008966[n_] := Abs[MoebiusMu[n]]; Table[A008966[n], {n, 100}] (* Enrique Pérez Herrero, Apr 15 2010 *)
Table[If[SquareFreeQ[n],1,0],{n,100}] (* or *) Boole[SquareFreeQ/@ Range[ 100]] (* Harvey P. Dale, Feb 28 2015 *)

MuPAD
```
func(abs(numlib::moebius(n)), n):
```
PARI
```
a(n)=if(n<1,0,direuler(p=2,n,1+X))[n]
```

a(n)=issquarefree(n) \\ Michel Marcus, Feb 22 2015

from sympy import factorint
def A008966(n): return int(max(factorint(n).values(),default=1)==1) # Chai Wah Wu, Apr 05 2023

Dirichlet g.f.: zeta(s)/zeta(2s).
a(n) = abs(mu(n)), where mu is the Moebius function (A008683).
a(n) = 0^(bigomega(n) - omega(n)), where bigomega(n) and omega(n) are the numbers of prime factors of n with and without repetition (A001222, A001221, A046660). - Reinhard Zumkeller, Apr 05 2003
Multiplicative with p^e -> 0^(e - 1), p prime and e > 0. - Reinhard Zumkeller, Jul 15 2003
a(n) = 0^(A046951(n) - 1). - Reinhard Zumkeller, May 20 2007
a(n) = 1 - A107078(n). - Reinhard Zumkeller, Oct 03 2008
a(n) = floor(rad(n)/n), where rad() is A007947. - Enrique Pérez Herrero, Nov 13 2009
A175046(n) = a(n)*A073311(n). - Reinhard Zumkeller, Apr 05 2010
a(n) = floor(A000005(n^2)/A007425(n)). - Enrique Pérez Herrero, Apr 15 2010
a(A005117(n)) = 1; a(A013929(n)) = 0; a(n) = A013928(n + 1) - A013928(n). - Reinhard Zumkeller, Jul 05 2010
a(n) * A112526(n) = A063524(n). - Reinhard Zumkeller, Sep 16 2011
a(n) = mu(n) * lambda(n) = A008836(n) * A008683(n). - Enrique Pérez Herrero, Nov 29 2013
a(n) = Sum_{d|n} 2^omega(d)*mu(n/d). - Geoffrey Critzer, Feb 22 2015
a(n) = A085357(A156552(n)). - Antti Karttunen, Mar 06 2017
Limit_{n->oo} (1/n)*Sum_{j=1..n} a(j) = 6/Pi^2. - Andres Cicuttin, Aug 13 2017
a(1) = 1; a(n) = -Sum_{d|n, d < n} (-1)^bigomega(n/d) * a(d). - Ilya Gutkovskiy, Mar 10 2021

Deleted an unclear comment. - N. J. A. Sloane, May 30 2021

A030514 a(n) = prime(n)^4.

Original entry on oeis.org

16, 81, 625, 2401, 14641, 28561, 83521, 130321, 279841, 707281, 923521, 1874161, 2825761, 3418801, 4879681, 7890481, 12117361, 13845841, 20151121, 25411681, 28398241, 38950081, 47458321, 62742241, 88529281, 104060401, 112550881, 131079601, 141158161

Offset: 1

Jeff Burch

Numbers with 5 divisors (1, p, p^2, p^3, p^4, where p is the n-th prime). - Alexandre Wajnberg, Jan 15 2006
Subsequence of A036967. - Reinhard Zumkeller, Feb 05 2008
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, May 06 2008
The general product formula for even s is: product_{p = A000040} (p^s-1)/(p^s+1) = 2*Bernoulli(2s)/( binomial(2s, s)*Bernoulli^2(s)), where the infinite product is over all primes. Here, with s = 4, product_{n = 1, 2, ...} (a(n)-1)/(a(n)+1) = 6/7. In A030516, where s = 6, the product of the ratios is 691/715. For s = 8, the 8th row in A120458, the corresponding product of ratios is 7234/7293. - R. J. Mathar, Feb 01 2009
Except for the first three terms, all others are congruent to 1 mod 240. - Robert Israel, Aug 29 2014

R. J. Mathar, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Prime Power.
OEIS Wiki, Index entries for number of divisors
Index to sequences related to prime signature

Cf. A030078, A085964, A131991, A131992, A000005, A000040, A001248, A157290, A215267.

Cf. A258601.

a030514 = (^ 4) . a000040
a030514_list = map (^ 4) a000040_list
-- Reinhard Zumkeller, Jun 03 2015

[NthPrime(n)^4: n in [1..100] ]; // Vincenzo Librandi, Apr 22 2011

map(p -> p^4, select(isprime,[2,seq(2*i+1,i=1..100)])); # Robert Israel, Aug 29 2014

Array[Prime[#]^4 &, 5!] (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)

a(n)=prime(n)^4 \\ Charles R Greathouse IV, Mar 21 2013

[p**4 for p in prime_range(100)]
# Zerinvary Lajos, May 15 2007

a(n) = A000040(n)^(5-1) = A000040(n)^4, where 5 is the number of divisors of a(n). - Omar E. Pol, May 06 2008
A000005(a(n)) = 5. - Alexandre Wajnberg, Jan 15 2006
A056595(a(n)) = 2. - Reinhard Zumkeller, Aug 15 2011
Sum_{n>=1} 1/a(n) = P(4) = 0.0769931397... (A085964). - Amiram Eldar, Jul 27 2020
From Amiram Eldar, Jan 23 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(4)/zeta(8) = 105/Pi^4 (A157290).
Product_{n>=1} (1 - 1/a(n)) = 1/zeta(4) = 90/Pi^4 (A215267). (End)

Description corrected by Eric W. Weisstein

A082020 Decimal expansion of 15/Pi^2.

Original entry on oeis.org

1, 5, 1, 9, 8, 1, 7, 7, 5, 4, 6, 3, 5, 0, 6, 6, 5, 7, 1, 6, 5, 8, 1, 9, 1, 9, 4, 8, 1, 4, 5, 9, 1, 4, 5, 8, 3, 5, 6, 5, 3, 8, 1, 6, 2, 0, 0, 8, 3, 6, 9, 8, 2, 3, 2, 6, 8, 4, 1, 3, 5, 4, 7, 8, 4, 1, 2, 5, 9, 6, 8, 1, 4, 4, 3, 3, 5, 3, 1, 6, 1, 7, 8, 6, 8, 1, 3, 9, 1, 0, 8, 8, 8, 4, 3, 2, 7, 5, 6

Offset: 1

N. J. A. Sloane, May 09 2003

3/(2*Pi^2) (the same decimal expansion with an offset 0) is the probability that the greatest common divisor of two numbers selected at random is 2 (Christopher, 1956). - Amiram Eldar, May 23 2020
Sum of 1/n^2 over all squarefree n, see Penn link. - Charles R Greathouse IV, Jan 01 2022
Equals the asymptotic mean of the abundancy index of the cubefree numbers (A004709) (Jakimczuk and Lalín, 2022). - Amiram Eldar, May 12 2023

			1.51981775463506657...

G. C. Greubel, Table of n, a(n) for n = 1..1000
John Christopher, The Asymptotic Density of Some k-Dimensional Sets, The American Mathematical Monthly, Vol. 63, No. 6 (1956), pp. 399-401.
Werner Hürlimann, Dedekind's arithmetic function and primitive four squares counting functions, Journal of Algebra, Number Theory: Advances and Applications, Vol. 14, No. 2 (2015), pp. 73-88.
Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (1).
Michael Penn, Irregular numbers and the coolest sum I have ever seen!, 2021 video.
S. Ramanujan, Irregular numbers, J. Indian Math. Soc., Vol. 5 (1913), pp. 105-106.
V. Sitaramaiah and M. V. Subbarao, Some asymptotic formulae involving powers of arithmetic functions, Number Theory, Madras 1987, Springer, 1989, pp. 201-234, alternative link (p. 230).
Eric Weisstein's World of Mathematics, Prime Sums.
Eric Weisstein's World of Mathematics, Moebius Function.
Eric Weisstein's World of Mathematics, Prime Products.
Index entries for transcendental numbers.

Cf. A001615, A004709, A005117, A007434, A013661, A013662, A157290.

SetDefaultRealField(RealField(100)); R:= RealField(); 15/Pi(R)^2; // G. C. Greubel, Oct 18 2019

evalf(15/Pi^2, 120); # G. C. Greubel, Oct 18 2019

A082020[digits_] := First[RealDigits[Zeta[2]/Zeta[4],10,digits]]; A082020[100] (* Enrique Pérez Herrero, Jan 15 2012 *)
RealDigits[15/Pi^2,10,120][[1]] (* Harvey P. Dale, Jun 23 2019 *)

PARI
```
15/Pi^2 \\ Michel Marcus, Oct 18 2019
```

numerical_approx(15/pi^2, digits=100) # G. C. Greubel, Oct 18 2019

Product_{n >= 1} (1+1/prime(n)^2) = 15/Pi^2 (Ramanujan).
Equals zeta(2)/zeta(4) = A013661/A013662 = Sum_{n>=1} mu(n)^2/n^2 = Sum_{n>=1} |mu(n)|/n^2 . - Enrique Pérez Herrero, Jan 15 2012
Equals Sum_{n>=1} 1/A005117(n)^2 . - Enrique Pérez Herrero, Mar 30 2012
Equals lim_{n->oo} (1/n) * Sum_{k=1..n} psi(k)/k, where psi(k) is the Dedekind psi function (A001615). - Amiram Eldar, May 12 2019.
Equals Sum_{k>=1} A007434(k)/k^4. - Amiram Eldar, Jan 25 2024

A157294 Decimal expansion of 1575/Pi^6.

Original entry on oeis.org

1, 6, 3, 8, 2, 5, 4, 3, 2, 0, 4, 4, 0, 9, 6, 7, 3, 6, 6, 3, 4, 1, 4, 9, 4, 2, 7, 4, 9, 8, 9, 8, 7, 3, 5, 5, 4, 9, 1, 8, 7, 0, 2, 5, 2, 6, 6, 4, 4, 3, 4, 4, 7, 1, 8, 0, 7, 2, 9, 0, 0, 6, 7, 4, 8, 9, 2, 5, 0, 4, 2, 3, 5, 5, 7, 4, 4, 7, 9, 0, 4, 1, 3, 4, 8, 3, 1, 5, 9, 2, 4, 6, 3, 0, 4, 9, 2, 3, 6, 9, 2, 5, 6, 9, 1

Offset: 1

R. J. Mathar, Feb 26 2009

Equals the asymptotic mean of the abundancy index of the 7-free numbers (numbers that are not divisible by a 7th power other than 1) (Jakimczuk and Lalín, 2022). - Amiram Eldar, May 12 2023

			1.63825432044096736634149427498... = (1+1/2^2+1/2^4+1/2^6)*(1+1/3^2+1/3^4+1/3^6)*(1+1/5^2+1/5^4+1/5^6)*(1+1/7^2+1/7^4+1/7^6)*...

Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (1).
Index entries for transcendental numbers.

Cf. A001248, A030514, A030516, A082020, A013661, A013666, A092746, A157290.

Maple
```
evalf(1575/Pi^6) ;
```

RealDigits[1575/Pi^6,10,120][[1]] (* Harvey P. Dale, May 26 2019 *)

1575/Pi^6 \\ Charles R Greathouse IV, Oct 01 2022

Equals Product_{p = primes = A000040} (1+1/p^2+1/p^4+1/p^6).

Equals A013661/A013666 = A082020*A157290 = Product_{i>=1} (1+1/A001248(i)+1/A030514(i)+1/A030516(i)) = 1575*A092746.

A269404 Decimal expansion of Product_{k >= 1} (1 + 1/prime(k)^6).

Original entry on oeis.org

1, 0, 1, 7, 0, 9, 2, 7, 6, 9, 1, 3, 0, 4, 9, 9, 2, 7, 6, 6, 4, 3, 2, 7, 2, 1, 3, 3, 0, 9, 7, 9, 0, 9, 9, 2, 0, 4, 9, 2, 2, 1, 9, 0, 7, 9, 4, 9, 4, 1, 0, 1, 1, 3, 4, 6, 6, 4, 6, 5, 1, 7, 9, 3, 8, 1, 8, 9, 3, 5, 3, 3, 5, 8, 3, 4, 2, 2, 7, 9, 4, 3, 1, 8, 1, 5, 1, 5, 9, 6, 4, 7, 8, 5, 0, 6, 6, 8, 9, 7, 8, 4, 5, 4, 6, 5, 1, 0, 6, 4, 0, 2, 6, 1, 3, 3, 6, 9, 3, 0

Offset: 1

Ilya Gutkovskiy, Feb 25 2016

More generally, Product_{k >= 1} (1 + 1/prime(k)^m) = zeta(m)/zeta(2*m), where zeta(m) is the Riemann zeta function.

			1.0170927691304992766432721330979099204922190794941...

Eric Weisstein's World of Mathematics, Prime Products.

Cf. A005117, A082020, A113851, A157289, A157290, A157291.

RealDigits[Zeta[6]/Zeta[12], 10, 120][[1]]
RealDigits[675675/(691 Pi^6), 10, 120][[1]]

zeta(6)/zeta(12) \\ Amiram Eldar, Jun 11 2023

Equals zeta(6)/zeta(12).
Equals 675675/(691*Pi^6).
Equals Sum_{k>=1} 1/A005117(k)^6 = 1 + Sum_{k>=1} 1/A113851(k). - Amiram Eldar, Jun 27 2020

cp's OEIS Frontend

A008966 a(n) = 1 if n is squarefree, otherwise 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

MuPAD

PARI

PARI

Python

Formula

Extensions

A030514 a(n) = prime(n)^4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A082020 Decimal expansion of 15/Pi^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A157294 Decimal expansion of 1575/Pi^6.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A269404 Decimal expansion of Product_{k >= 1} (1 + 1/prime(k)^6).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula