A059956 -id:A059956 - cp's OEIS frontend

A018805 Number of elements in the set {(x,y): 1 <= x,y <= n, gcd(x,y)=1}.

1, 3, 7, 11, 19, 23, 35, 43, 55, 63, 83, 91, 115, 127, 143, 159, 191, 203, 239, 255, 279, 299, 343, 359, 399, 423, 459, 483, 539, 555, 615, 647, 687, 719, 767, 791, 863, 899, 947, 979, 1059, 1083, 1167, 1207, 1255, 1299, 1391, 1423, 1507, 1547, 1611, 1659, 1763

Offset: 1

David W. Wilson

Number of positive rational numbers of height at most n, where the height of p/q is max(p, q) when p and q are relatively prime positive integers. - Charles R Greathouse IV, Jul 05 2012
The number of ordered pairs (i,j) with 1<=i<=n, 1<=j<=n, gcd(i,j)=d is a(floor(n/d)). - N. J. A. Sloane, Jul 29 2012
Equals partial sums of A140434 (1, 2, 4, 4, 8, 4, 12, 8, ...) and row sums of triangle A143469. - Gary W. Adamson, Aug 17 2008
Number of distinct solutions to k*x+h=0, where 1 <= k,h <= n. - Giovanni Resta, Jan 08 2013
a(n) is the number of rational numbers which can be constructed from the set of integers between 1 and n, without combination of multiplication and division. a(3) = 7 because {1, 2, 3} can only create {1/3, 1/2, 2/3, 1, 3/2, 2, 3}. - Bernard Schott, Jul 07 2019

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 110-112.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954. See Theorem 332.

Olivier Gérard, Table of n, a(n) for n = 1..100000 [Replaces an earlier b-file from Charles R Greathouse IV]
Jin-Yi Cai and Eric Bach, On testing for zero polynomials by a set of points with bounded precision, Theoret. Comput. Sci. 296 (2003), no. 1, 15-25. MR1965515 (2004m:68279).
Pieter Moree, Counting carefree couples, arXiv:math/0510003 [math.NT], 2005-2014.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
Eric Weisstein's World of Mathematics, Carefree Couple

Cf. A015614, A002088, A100613 (gcd > 1), A071778 (triples), A143469, A140434, A013661, A059956, A137243, A171503.
Cf. A177853 (partial sums).
The main diagonal of A331781, also of A333295.

a018805 n = length [()| x <- [1..n], y <- [1..n], gcd x y == 1]
-- Reinhard Zumkeller, Jan 21 2013

/* based on the first formula */ A018805:=func< n | 2*&+[ EulerPhi(k): k in [1..n] ]-1 >; [ A018805(n): n in [1..60] ]; // Klaus Brockhaus, Jan 27 2011

/* based on the second formula */ A018805:=func< n | n eq 1 select 1 else n^2-&+[ $$(n div j): j in [2..n] ] >; [ A018805(n): n in [1..60] ]; // Klaus Brockhaus, Feb 07 2011

N:= 1000; # to get the first N entries
P:= Array(1..N,numtheory:-phi);
A:= map(t -> 2*round(t)-1, Statistics:-CumulativeSum(P));
convert(A,list); # Robert Israel, Jul 16 2014

FoldList[ Plus, 1, 2 Array[ EulerPhi, 60, 2 ] ] (* Olivier Gérard, Aug 15 1997 *)
Accumulate[2*EulerPhi[Range[60]]]-1 (* Harvey P. Dale, Oct 21 2013 *)

PARI
```
a(n)=sum(k=1,n,moebius(k)*(n\k)^2)
```

A018805(n)=2 *sum(j=1, n, eulerphi(j)) - 1;
for(n=1, 99, print1(A018805(n), ", ")); /* show terms */

a(n)=my(s); forsquarefree(k=1,n, s+=moebius(k)*(n\k[1])^2); s \\ Charles R Greathouse IV, Jan 08 2018

from sympy import sieve
def A018805(n): return 2*sum(t for t in sieve.totientrange(1,n+1)) - 1 # Chai Wah Wu, Mar 23 2021

from functools import lru_cache
@lru_cache(maxsize=None)
def A018805(n): # based on second formula
    if n == 0:
        return 0
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A018805(k1)
        j, k1 = j2, n//j2
    return n*(n-1)-c+j # Chai Wah Wu, Mar 24 2021

a(n) = 2*(Sum_{j=1..n} phi(j)) - 1.
a(n) = n^2 - Sum_{j=2..n} a(floor(n/j)).
a(n) = 2*A015614(n) + 1. - Reinhard Zumkeller, Apr 08 2006
a(n) = 2*A002088(n) - 1. - Hugo van der Sanden, Nov 22 2008
a(n) ~ (1/zeta(2)) * n^2 = (6/Pi^2) * n^2 as n goes to infinity (zeta is the Riemann zeta function, A013661, and the constant 6/Pi^2 is 0.607927..., A059956). - Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 18 2001
a(n) ~ 6*n^2/Pi^2 + O(n*log n). - N. J. A. Sloane, May 31 2020
a(n) = Sum_{k=1..n} mu(k)*floor(n/k)^2. - Benoit Cloitre, May 11 2003
a(n) = A000290(n) - A100613(n) = A015614(n) + A002088(n). - Reinhard Zumkeller, Jan 21 2013
a(n) = A242114(floor(n/k),1), 1<=k<=n; particularly a(n) = A242114(n,1). - Reinhard Zumkeller, May 04 2014
a(n) = 2 * A005728(n) - 3. - David H Post, Dec 20 2016
a(n) ~ 6*n^2/Pi^2, cf. A059956. [Hardy and Wright] - M. F. Hasler, Jan 20 2017
G.f.: (1/(1 - x)) * (-x + 2 * Sum_{k>=1} mu(k) * x^k / (1 - x^k)^2). - Ilya Gutkovskiy, Feb 14 2020

More terms from Reinhard Zumkeller, Apr 08 2006

Link to Moree's paper corrected by Peter Luschny, Aug 08 2009

A068781 Lesser of two consecutive numbers each divisible by a square.

Original entry on oeis.org

8, 24, 27, 44, 48, 49, 63, 75, 80, 98, 99, 116, 120, 124, 125, 135, 147, 152, 168, 171, 175, 188, 207, 224, 242, 243, 244, 260, 275, 279, 288, 296, 315, 324, 332, 342, 343, 350, 351, 360, 363, 368, 375, 387, 404, 423, 424, 440, 459, 475, 476, 495, 507, 512

Offset: 1

Robert G. Wilson v, Mar 04 2002

nonn

Also numbers m such that mu(m)=mu(m+1)=0, where mu is the Moebius-function (A008683); A081221(a(n))>1. - Reinhard Zumkeller, Mar 10 2003
The sequence contains an infinite family of arithmetic progressions like {36a+8}={8,44,80,116,152,188,...} ={4(9a+2)}. {36a+9} provides 2nd nonsquarefree terms. Such AP's can be constructed to any term by solution of a system of linear Diophantine equation. - Labos Elemer, Nov 25 2002
1. 4k^2 + 4k is a member for all k; i.e., 8 times a triangular number is a member. 2. (4k+1) times an odd square - 1 is a member. 3. (4k+3) times odd square is a member. - Amarnath Murthy, Apr 24 2003
The asymptotic density of this sequence is 1 - 2/zeta(2) + Product_{p prime} (1 - 2/p^2) = 1 - 2 * A059956 + A065474 = 0.1067798952... (Matomäki et al., 2016). - Amiram Eldar, Feb 14 2021
Maximum of the n-th maximal anti-run of nonsquarefree numbers (A013929) differing by more than one. For runs instead of anti-runs we have A376164. For squarefree instead of nonsquarefree we have A007674. - Gus Wiseman, Sep 14 2024

			44 is in the sequence because 44 = 2^2 * 11 and 45 = 3^2 * 5.
From _Gus Wiseman_, Sep 14 2024: (Start)
Splitting nonsquarefree numbers into maximal anti-runs gives:
  (4,8)
  (9,12,16,18,20,24)
  (25,27)
  (28,32,36,40,44)
  (45,48)
  (49)
  (50,52,54,56,60,63)
  (64,68,72,75)
  (76,80)
  (81,84,88,90,92,96,98)
  (99)
The maxima are a(n). The corresponding pairs are (8,9), (24,25), (27,28), (44,45), etc.
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Kaisa Matomäki, Maksym Radziwiłł and Terence Tao, Sign patterns of the Liouville and Möbius functions, Forum of Mathematics, Sigma, Vol. 4. (2016), e14.

Cf. A068780, A068140, A068781, A068782, A068783, A068784, A068785.
Cf. A049535, A077647, A078143, A045882.
Subsequence of A261869.
Cf. A007674, A373409, A373410, A373412, A375707, A376164.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.
Cf. A049094, A061399, A072284, A120992, A294242, A373573, A375709.

a068781 n = a068781_list !! (n-1)
a068781_list = filter ((== 0) . a261869) [1..]
-- Reinhard Zumkeller, Sep 04 2015

Select[ Range[2, 600], Max[ Transpose[ FactorInteger[ # ]] [[2]]] > 1 && Max[ Transpose[ FactorInteger[ # + 1]] [[2]]] > 1 &]
f@n_:= Flatten@Position[Partition[SquareFreeQ/@Range@2000,n,1], Table[False,{n}]]; f@2 (* Hans Rudolf Widmer, Aug 30 2022 *)
Max/@Split[Select[Range[100], !SquareFreeQ[#]&],#1+1!=#2&]//Most (* Gus Wiseman, Sep 14 2024 *)

isok(m) = !moebius(m) && !moebius(m+1); \\ Michel Marcus, Feb 14 2021

A261869(a(n)) = 0. - Reinhard Zumkeller, Sep 04 2015

A082695 Decimal expansion of zeta(2)*zeta(3)/zeta(6).

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Apr 12 2003

Equals the Dirichlet zeta-function Sum_{n>=1} A001615(n)/n^s at s=3. - R. J. Mathar, Apr 02 2011
Dressler shows that this is the average value of A014197, that is, the number of values m such that phi(m) <= n is asymptotically n times this constant. Erdős had shown earlier that this limit exists. - Charles R Greathouse IV, Nov 26 2013
From Stanislav Sykora, Nov 14 2014: (Start)
Equals lim_{n->infinity} (Sum_{k=1..n} k/phi(k))/n, i.e., the limit mean value of k/phi(k), where phi(k) is Euler's totient function.
Also equals lim_{n->infinity} (Sum_{k=1..n} 1/phi(k))/log(n).
Proofs are trivial using the formulas for Sum_{k=1..n} k/phi(k) and Sum_{k=1..n} 1/phi(k) listed in the Wikipedia link.
For the limit mean value of phi(k)/k, see A059956. (End)
The asymptotic mean of A005361. - Amiram Eldar, Apr 13 2020

			1.94359643682075920505707036257476343718785850176780571602663568890 ...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.7, p. 116.
Joe Roberts, Lure of the Integers, Mathematical Association of America, 1992. See p. 74.

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Paul T. Bateman, The distribution of values of the Euler function, Acta Arithmetica 21:1 (1972), pp. 329-345.
Olivier Bordellès and Benoit Cloitre, An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions, J. Int. Seq. 16 (2013) #13.6.3.
Robert E. Dressler, A density which counts multiplicity, Pacific J. Math. 34 (1970), pp. 371-378.
Paul Erdős, Some remarks on Euler's ϕ function and some related problems, Bull. Amer. Math. Soc. 51 (1945), pp. 540-544.
J. von zur Gathen et al., Average order in cyclic groups, J. Theor. Nombres Bordeaux, 16 (2004), 107-123. Lists several other papers where this constant arises.
S. W. Golomb, Powerful numbers, Amer. Math. Monthly, Vol. 77 (1970), 848-852.
D. Handelman, Invariants for critical dimension groups and permutation-Hermite equivalence, arXiv preprint arXiv:1309.7417 [math.AC], 2013.
Eric Weisstein's World of Mathematics, Totient Summatory Function.
Eric Weisstein's World of Mathematics, Powerful Number.
Wikipedia, Euler's totient function.

Cf. A000010, A001615, A001694, A002117, A005117, A005361, A013661, A013664, A014197, A059956, A068468, A070243, A082696 (continued fraction), A157292.

First@RealDigits[ Zeta[2]*Zeta[3]/Zeta[6], 10, 100]
RealDigits[ 315 Zeta[3]/(2 Pi^4), 10, 111][[1]] (* Robert G. Wilson v, Aug 11 2014 *)

PARI
```
zeta(3)*315/2/Pi^4
```

Decimal expansion of Product_{p prime} (1+1/p/(p-1)) = zeta(2)*zeta(3)/zeta(6) = 1.94359643682075920505707...
The sum of the reciprocals of the powerful numbers, A001694. - T. D. Noe, May 03 2006
Equals A013661 * A002117 / A013664 = 1 / A068468 = zeta(3) * 315/(2*Pi^4) = zeta(3) * A157292.
Equals Sum_{k>=1} mu(k)^2/(k*phi(k)) (the sum of reciprocals of the squarefree numbers multiplied by their Euler totient function values, A000010). - Amiram Eldar, Aug 18 2020

New definition from Eric W. Weisstein, May 04 2006

A306016 Decimal expansion of -Zeta'(2)/Zeta(2).

Original entry on oeis.org

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

Offset: 0

Peter Luschny, Jun 17 2018

			Equals 0.569960993094532806399864360019730002403482280806930979558...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 179.
J. B. Rosser, L. Schoenfeld, Approximate formulas for some functions of prime numbers, Ill. J. Math. 6 (1) (1962) 64-94, Table IV.

Cf. A059956, A073002.

Maple
```
-(6/Pi^2)*Zeta(1,2): evalf(%,100);
```

RealDigits[- Zeta'[2] / Zeta[2], 10, 87][[1]]

-zeta'(2)/zeta(2) \\ Michel Marcus, Jan 11 2019

Equals -(6/Pi^2)*Zeta'(2).
Equals 1 - 12*Zeta'(-1) - gamma - log(2*Pi).
From Amiram Eldar, Aug 14 2020: (Start)
Equals Sum_{k>=1} Lambda(k)/k^2, where Lambda is the Mangoldt function.
Equals Sum_{p prime} log(p)/(p^2 - 1). (End)

A067259 Cubefree numbers which are not squarefree.

Original entry on oeis.org

4, 9, 12, 18, 20, 25, 28, 36, 44, 45, 49, 50, 52, 60, 63, 68, 75, 76, 84, 90, 92, 98, 99, 100, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 164, 169, 171, 172, 175, 180, 188, 196, 198, 204, 207, 212, 220, 225, 228

Offset: 1

Reinhard Zumkeller, Feb 20 2002

nonn

a(n)=m iff A051903(m)=2.

The asymptotic density of this sequence is 1/zeta(3) - 1/zeta(2) = A088453 - A059956 = 0.22398... - Amiram Eldar, Jul 09 2020

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

Cf. A004709, A005117, A059956, A088453.

a067259 n = a067259_list !! (n-1)
a067259_list = filter ((== 2) . a051903) [1..]
-- Reinhard Zumkeller, May 27 2012

f[n_]:=Union[Last/@FactorInteger[n]][[ -1]]; lst={}; Do[If[f[n]==2,AppendTo[lst,n]],{n,2,6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 12 2010 *)
Select[Range[500], Not[SquareFreeQ[#]] && FreeQ[FactorInteger[#], {,k /;k>2}]&] (* Vaclav Kotesovec, Jul 09 2020 *)

is(n)=n>3 && vecmax(factor(n)[,2])==2 \\ Charles R Greathouse IV, Oct 15 2015

from math import isqrt
from sympy import mobius, integer_nthroot
def A067259(n):
    def f(x): return n+x+sum(mobius(k)*(x//k**2-x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))+sum(mobius(k)*(x//k**2) for k in range(integer_nthroot(x,3)[0]+1,isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 05 2024

A212793(a(n)) * (1 - A008966(a(n))) = 1. - Reinhard Zumkeller, May 27 2012

Unrelated comment removed by Jason Yuen, Apr 04 2025

A073247 Squarefree numbers k such that k-1 and k+1 are not squarefree.

Original entry on oeis.org

17, 19, 26, 51, 53, 55, 89, 91, 97, 127, 149, 151, 161, 163, 170, 197, 199, 233, 235, 241, 249, 251, 269, 271, 293, 295, 305, 307, 337, 339, 341, 349, 362, 377, 379, 413, 415, 449, 451, 485, 487, 489, 491, 521, 523, 530, 551, 557, 559, 577, 579, 593, 595

Offset: 1

Reinhard Zumkeller, Jul 22 2002

nonn

Probably 11*n < a(n) < 12*n for n > 189. - Charles R Greathouse IV, Nov 05 2017

The asymptotic density of this sequence is 1/zeta(2) - 2 * Product_{p prime} (1 - 2/p^2) + Product_{p prime} (1 - 3/p^2) = A059956 - 2*A065474 + A206256 = 0.088145884881346585838... . - Amiram Eldar, Aug 30 2024

Christopher E. Thompson, Table of n, a(n) for n = 1..10000

Cf. A005117, A073249, A073248, A007675.
Cf. A268331, A268332, A268333, A268334 (squarefree numbers isolated by more than 2, 3, etc.).
Cf. A059956, A065474, A206256.

sf:= select(numtheory:-issqrfree,[$1..1000]):
map(t -> `if`(sf[t-1]=sf[t]-1 or sf[t+1]=sf[t]+1,NULL,sf[t]), [$2..nops(sf)-1]); # Robert Israel, Feb 01 2016

Reap[For[n = 0, n <= 1000, n++, If[SquareFreeQ[n] && !SquareFreeQ[n-1] && !SquareFreeQ[n+1], Sow[n]]]][[2, 1]] (* Jean-François Alcover, Feb 26 2019 *)

is(n)=!issquarefree(n-1) && issquarefree(n) && !issquarefree(n+1) \\ Charles R Greathouse IV, Nov 05 2017

list(lim)=my(v=List(),l1,l2); forfactored(k=9,lim\1+1, if(!issquarefree(k) && !issquarefree(l2) && issquarefree(l1), listput(v,l1[1])); l2=l1; l1=k); Vec(v) \\ Charles R Greathouse IV, Nov 27 2024

A229099 Decimal expansion of 1 - 6/Pi^2.

Original entry on oeis.org

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

Offset: 0

Charles R Greathouse IV, Sep 13 2013

Probability that a random number is not squarefree; probability that two random numbers have a common divisor greater than 1.

			0.39207289814597337133672322074163416657384735196652070692634580863496...

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers

Cf. A000796, A002388, A013661, A059956, A063035.

RealDigits[1-1/Zeta[2],10,90][[1]] (* Stefano Spezia, Feb 21 2025 *)

PARI
```
1-6/Pi^2
```

Equals 1 - 1/zeta(2). - Stefano Spezia, Feb 21 2025

A104141 Decimal expansion of 3/Pi^2.

Original entry on oeis.org

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

Offset: 0

Lekraj Beedassy, Mar 07 2005

3/Pi^2 is the limit of (Sum_{k=1..n} phi(k))/n^2, where phi(k) is Euler's totient A000010(k), i.e., of A002088(n)/A000290(n) as n tends to infinity.
The previous comment in the context of Farey series means that the length of the n-th Farey series can be approximated by multiplying this constant by n^2, "and that the approximation gets proportionally better as n gets larger", according to Conway and Guy. - Alonso del Arte, May 28 2011
The asymptotic density of the sequences of squarefree numbers with even number of prime factors (A030229), odd number of prime factors (A030059), and coprime to 6 (A276378). - Amiram Eldar, May 22 2020

			3/Pi^2 = 0.303963550927013314331638389629...

J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1995, p. 156.
L. E. Dickson, History of the Theory of Numbers, Vol. I pp. 126 Chelsea NY 1966.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 184.

Eric Weisstein's World of Mathematics, Farey Sequence
Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.
Index entries for transcendental numbers.

Cf. A000010, A002088, A000290.
Cf. A046642, A030229, A030059, A039956, A276378.
Cf. A013661, A195055, A306633, A082020, A088246.

l = RealDigits[N[3/Pi^2, 100]]; Prepend[First[l], Last[l]] (* Ryan Propper, Aug 04 2005 *)

3/Pi^2 \\ Charles R Greathouse IV, Mar 08 2013

Equals Sum_{n>=1} 1/A039956(n)^2. - Amiram Eldar, May 22 2020
From Terry D. Grant, Oct 31 2020: (Start)
Equals (-1)*zeta(0)/zeta(2).
Equals 1/(zeta(2)/2).
Equals 1/A195055.
Equals (1/2)*Sum_{k>=1} mu(k)/k^2. (End)
From Hugo Pfoertner, Apr 23 2024: (Start)
Equals A059956/2.
Equals A082020/5. (End)

More terms from Ryan Propper, Aug 04 2005

A109395 Denominator of phi(n)/n = Product_{p|n} (1 - 1/p); phi(n)=A000010(n), the Euler totient function.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 15, 2, 17, 3, 19, 5, 7, 11, 23, 3, 5, 13, 3, 7, 29, 15, 31, 2, 33, 17, 35, 3, 37, 19, 13, 5, 41, 7, 43, 11, 15, 23, 47, 3, 7, 5, 51, 13, 53, 3, 11, 7, 19, 29, 59, 15, 61, 31, 7, 2, 65, 33, 67, 17, 69, 35, 71, 3, 73, 37, 15, 19, 77, 13, 79, 5, 3

Offset: 1

Franz Vrabec, Aug 26 2005

a(n)=2 iff n=2^k (k>0); otherwise a(n) is odd. If p is prime, a(p)=p; the converse is false, e.g.: a(15)=15. It is remarkable that this sequence often coincides with A006530, the largest prime P dividing n. Theorem: a(n)=P if and only if for every prime p < P in n there is some prime q in n with p|(q-1). - Franz Vrabec, Aug 30 2005

			a(10) = 10/gcd(10,phi(10)) = 10/gcd(10,4) = 10/2 = 5.

Antti Karttunen, Table of n, a(n) for n = 1..16384 (terms 1..1000 from T. D. Noe)
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A076512 for the numerator.
Cf. A000010, A009195, A054741, A059956, A082695, A318304, A318305, A323170.
Phi(m)/m = k: A000079 \ {1} (k=1/2), A033845 (k=1/3), A000244 \ {1} (k=2/3), A033846 (k=2/5), A000351 \ {1} (k=4/5), A033847 (k=3/7), A033850 (k=4/7), A000420 \ {1} (k=6/7), A033848 (k=5/11), A001020 \ {1} (k=10/11), A288162 (k=6/13), A001022 \ {1} (12/13), A143207 (k=4/15), A033849 (k=8/15), A033851 (k=24/35).

Table[Denominator[EulerPhi[n]/n], {n, 81}] (* Alonso del Arte, Sep 03 2011 *)

a(n)=n/gcd(n,eulerphi(n)) \\ Charles R Greathouse IV, Feb 20 2013

A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ From A173557
A109395(n) = denominator(A173557(n)/A007947(n)); \\ Antti Karttunen, Feb 09 2019

a(n) = n/gcd(n, phi(n)) = n/A009195(n).
From Antti Karttunen, Feb 09 2019: (Start)
a(n) = denominator of A173557(n)/A007947(n).
a(2^n) = 2 for all n >= 1.
(End)
From Amiram Eldar, Jul 31 2020: (Start)
Asymptotic mean of phi(n)/n: lim_{m->oo} (1/m) * Sum_{n=1..m} A076512(n)/a(n) = 6/Pi^2 (A059956).
Asymptotic mean of n/phi(n): lim_{m->oo} (1/m) * Sum_{n=1..m} a(n)/A076512(n) = zeta(2)*zeta(3)/zeta(6) (A082695). (End)

A215267 Decimal expansion of 90/Pi^4.

Original entry on oeis.org

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

Offset: 0

Jimmy Zotos, Aug 07 2012

Decimal expansion of 1/zeta(4), the inverse of A013662. This is the probability that 4 randomly chosen natural numbers are relatively prime.
Also the asymptotic probability that a random integer is 4-free. See equivalent comments in A088453, A059956. - Balarka Sen, Aug 08 2012
The probability that the greatest common divisor of two numbers selected at random is squarefree (Christopher, 1956). - Amiram Eldar, May 23 2020

			0.92393840292159016702375049404068247276450216682744364463512319...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 231.

Karl-Heinz Hofmann, Table of n, a(n) for n = 0..10000
John Christopher, The Asymptotic Density of Some k-Dimensional Sets, The American Mathematical Monthly, Vol. 63, No. 6 (1956), pp. 399-401.
Math Forums, Probability that a number is composite, Aug 2012.
Index entries for transcendental numbers

Cf. A013662, A046100 (4-free numbers), A059956 (1/zeta(2)).

RealDigits[90/Pi^4, 10, 100][[1]] (* Bruno Berselli, Aug 07 2012 *)

90/%pi^4; /* Balarka Sen, Aug 08 2012 */

90/Pi^4 \\ Charles R Greathouse IV, Aug 07 2012

Reciprocal of A013662.

1/zeta(4) = 90/Pi^4 = Product_{k>=1} (1 - 1/prime(k)^4) = Sum_{n>=1} mu(n)/n^4, a Dirichlet series for the Möbius function mu. See the examples in Apostol, here for s = 4. - Wolfdieter Lang, Aug 07 2019

cp's OEIS Frontend

A018805 Number of elements in the set {(x,y): 1 <= x,y <= n, gcd(x,y)=1}.

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A068781 Lesser of two consecutive numbers each divisible by a square.

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A082695 Decimal expansion of zeta(2)*zeta(3)/zeta(6).

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A306016 Decimal expansion of -Zeta'(2)/Zeta(2).

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A067259 Cubefree numbers which are not squarefree.

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A073247 Squarefree numbers k such that k-1 and k+1 are not squarefree.

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