A057660 -id:A057660 - cp's OEIS frontend

A345295 Decimal expansion of Product_{p primes} (1 - 1/p)(1 + (1 - 1/p^2)Sum_{k>=1} 1/(p^k + p^(-k-1))).

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

Offset: 0

Vaclav Kotesovec, Jun 13 2021

			0.80146969342757733622470493868169850732790583309363219628998277639443297...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 155 (constant C2).
Vaclav Kotesovec, Plot of 1/n * Sum_{k=1..n} k*A000010(k)/A057660(k) for n = 1..100000

Cf. A057660, A174405, A345294.

$MaxExtraPrecision = 1000; m = 500; Do[Clear[f]; f[p_] := (1 - 1/p)*(1 + (1 - 1/p^2)*Sum[1/(p^j + p^(-j - 1)), {j, 1, k}]); cc = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x, m + 1]]; Print[f[2]*Exp[N[Sum[Indexed[cc, n]*(PrimeZetaP[n] - 1/2^n), {n, 2, m}], 110]]], {k, 100, 500, 100}]

Equals lim_{n->infinity} 1/n * Sum_{k=1..n} k*A000010(k)/A057660(k).

A070999 Numbers n such that the denominator of Sum_{k=1..n} 1/gcd(n,k) is not equal to n.

Original entry on oeis.org

6, 15, 18, 21, 30, 33, 35, 42, 44, 45, 48, 51, 54, 60, 66, 69, 70, 78, 84, 87, 90, 99, 102, 105, 114, 119, 120, 123, 126, 132, 133, 135, 138, 140, 141, 144, 147, 150, 153, 159, 162, 165, 168, 174, 177, 180, 186, 195, 198, 204, 207, 210, 213, 217, 220, 221, 222

Offset: 1

Benoit Cloitre, May 18 2002

Does lim_{n->infinity} a(n)/n = 3?
Sum_{k=1..n} 1/gcd(n,k) = (1/n)*Sum_{d|n} phi(d)*d = (1/n)*Sum_{k=1..n} gcd(n,k)*phi(gcd(n,k))/phi(n/gcd(n,k)), where phi = A000010. - Richard L. Ollerton, May 10 2021
Numbers k such that gcd(k, A057660(k)) > 1. - Amiram Eldar, Jun 29 2022

			Sum_{k=1..6} 1/gcd(6,k) = 7/2, hence 6 is in the sequence;
Sum_{k=1..12} 1/gcd(12,k) = 77/12 so 12 is not in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A018804, A057660.

Select[Range[300],Denominator[Sum[1/GCD[#,k],{k,#}]]!=#&] (* Harvey P. Dale, May 07 2022 *)
f[p_, e_] := (p^(2*e + 1) + 1)/(p + 1); s[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[250], !CoprimeQ[#, s[#]] &] (* Amiram Eldar, Jun 29 2022 *)

for(n=1,300,if(denominator(sum(i=1,n,1/gcd(n,i)))

A108223 a(n) = sigma_{2n}(n^2)/sigma_n(n^2), where sigma_n(m) = Sum_{d|m} d^n.

Original entry on oeis.org

1, 13, 703, 61681, 9762501, 2140365529, 678222249307, 280379743338241, 150087010086914941, 99902428887422922553, 81402749386554449442711, 79477293980103609858493681, 91733330193268313783293023757, 123469159731637675342948027295569, 191751045863140709562160603031808243

Offset: 1

Leroy Quet, Jun 28 2005

nonn

			sigma_4(4)/sigma_2(4) =
(1 + 2^4 + 4^4)/(1 + 2^2 + 4^2) = 13.

Cf. A062755.

Cf. A057660, A084218, A084220, A372966.

Table[DivisorSigma[2n, n^2]/DivisorSigma[n, n^2], {n, 10}] (* Ryan Propper, Apr 03 2007 *)

a(n) = sigma(n^2, 2*n)/sigma(n^2, n); \\ Michel Marcus, Sep 06 2019

a(n) = Product_{p=primes} (Sum_{k=0..2*b(n, p)} p^(n*k)*(-1)^k), where p^b(n, p) is the highest power of p dividing n.
From Seiichi Manyama, May 18 2024: (Start)
a(n) = Sum_{1 <= x_1, x_2, ... , x_n <= n} ( n/gcd(x_1, x_2, ... , x_n, n) )^n.
a(n) = Sum_{d|n} mu(n/d) * (n/d)^n * sigma_{2*n}(d). (End)

More terms from Ryan Propper, Apr 03 2007

More terms from Michel Marcus, Sep 06 2019

A127477 Triangle T(n,k) read by rows: matrix product A054522 * A054523.

Original entry on oeis.org

1, 2, 1, 5, 0, 2, 6, 3, 0, 2, 17, 0, 0, 0, 4, 10, 5, 4, 0, 0, 2, 37, 0, 0, 0, 0, 0, 6, 22, 11, 0, 6, 0, 0, 0, 4, 41, 0, 14, 0, 0, 0, 0, 0, 6, 34, 17, 0, 0, 8, 0, 0, 0, 0, 4, 101, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 30, 15, 12, 10, 0, 6, 0, 0, 0, 0, 0, 4, 145, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 74, 37, 0

Offset: 1

Gary W. Adamson, Jan 15 2007

If the two matrices A054522 and A054523 are commuted, the matrix product becomes A127478.

			First few rows of the triangle are:
1;
2, 1;
5, 0, 2;
6, 3, 0, 2;
17, 0, 0, 0, 4;
10, 5, 4, 0, 0, 2;
37, 0, 0, 0, 0, 0, 6;
22, 11, 0, 6, 0, 0, 0, 4;

Cf. A054522, A054523, A057660, A000010, A029939.

A054522 := proc(n,k) if k = 1 then 1; elif n mod k = 0 then numtheory[phi](k) ; else 0 ; fi; end:
A054523 := proc(n,k) if k = n then 1; elif n mod k = 0 then numtheory[phi](n/k) ; else 0 ; fi; end:
A127477 := proc(n,k) add( A054522(n,j)*A054523(j,k), j=k..n) ; end: seq(seq( A127477(n,k),k=1..n),n=1..15) ;

T(n,k) = sum_{j=k..n} A054522(n,j) * A054523(j,k).
sum_{k=1..n} T(n,k) = A057660(n) (row sums).
T(n,n) = A000010(n) (diagonal).
T(n,1) = A029939(n).

Converted comments to formulas, extended - R. J. Mathar, Sep 11 2009

A130211 Triangle read by rows: matrix product A054522 * A000012.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 4, 3, 2, 2, 5, 4, 4, 4, 4, 6, 5, 4, 2, 2, 2, 7, 6, 6, 6, 6, 6, 6, 8, 7, 6, 6, 4, 4, 4, 4, 9, 8, 8, 6, 6, 6, 6, 6, 6, 10, 9, 8, 8, 8, 4, 4, 4, 4, 4, 11, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 12, 11, 10, 8, 6, 6, 4, 4, 4, 4, 4, 4, 13, 12, 12, 12, 12, 12, 12, 12, 12

Offset: 1

Gary W. Adamson, May 17 2007

			First few rows of the triangle are:
1;
2, 1;
3, 2, 2;
4, 3, 2, 2;
5, 4, 4, 4, 4;
6, 5, 4, 2, 2, 2;
7, 6, 6, 6, 6, 6, 6;
8, 7, 6, 6, 4, 4, 4, 4;
...

Cf. A000010, A054522, A130212 (product with swapped order), A057660 (row sums).

A130211 := proc(n,k)
    add( A054522(n,j),j=k..n) ;
end proc:
seq(seq(A130211(n,k),k=1..n),n=1..15) ; # R. J. Mathar, Aug 06 2016

A054522 * A000012 as infinite lower triangular matrices.

T(n,n) = A000010(n).

A226512 a(n) = minimum of sum of orders of all elements of G, where G is a group of order n.

Original entry on oeis.org

1, 3, 7, 7, 21, 13, 43, 15, 25, 31, 111, 31, 157, 57, 147, 31, 273, 43, 343, 71, 85, 133, 507, 67, 121, 183, 79, 157, 813, 177, 931, 63, 777, 307, 903, 111, 1333, 381, 235, 163, 1641, 183, 1807, 377, 525, 553, 2163, 127, 337, 171, 1911, 287, 2757, 133, 331, 351, 457, 871, 3423, 211

Offset: 1

N. J. A. Sloane, Jun 13 2013

nonn

			For n=6 the group S_3 is optimal: it has one element of order 1, 3 of order 2, and 2 of order 3, for a total of a(6) = 13.

G. Nebe, Table of n, a(n) for n = 1..200
Habib Amiri and S. M. Jafarian Amiri, Sum of element orders on finite groups of the same order, J. Algebra Appl. 10 (2011), no. 2, 187--190. MR2795731 (2012d:20050)
Habib Amiri, S. M. Jafarian Amiri, and I. M. Isaacs, Sums of element orders in finite groups Comm. Algebra 37 (2009), no. 9, 2978--2980. MR2554185 (2010i:20022)
Y. Marefat et al., On the sum of element orders of finite simple groups, J. Algebra Applications, 12 (2013), #1350026.

Cf. A057660.

// Program from G. Nebe, Jun 14 2013
SS:=[];
for i in [1..200] do f:=[];
for j in [1..NumberOfSmallGroups(i)] do p:=0; G:=SmallGroup(i,j);
for g in G do p+:= Order(g); end for; Append(~f,p); end for;
Append(~SS,Minimum(f)); end for;
SS;

If p is prime, a(p) = A057660(p).

More terms from G. Nebe, Jun 14 2013

A345294 Decimal expansion of Product_{p primes} (1 - 1/p)(1 + (1 + 1/p)Sum_{k>=1} 1/(p^k + p^(-k-1))).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Jun 13 2021

			1.44386751047063965112504432490384711454555197897178255027794333889778581...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 155 (constant C1).

Cf. A057660, A174405, A345295.

$MaxExtraPrecision = 1000; m = 500; Do[Clear[f]; f[p_] := (1 - 1/p)*(1 + (1 + 1/p)*Sum[1/(p^j + p^(-j - 1)), {j, 1, k}]); cc = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x, m + 1]]; Print[f[2]*Exp[N[Sum[Indexed[cc, n]*(PrimeZetaP[n] - 1/2^n), {n, 2, m}], 110]]], {k, 100, 500, 100}]

Equals lim_{n->infinity} 1/n * Sum_{k=1..n} k^2/A057660(k).

A348061 a(n) = Sum_{k=1..n, gcd(n,k) = 1} n / gcd(n,k-1).

Original entry on oeis.org

1, 1, 4, 3, 16, 4, 36, 11, 34, 16, 100, 12, 144, 36, 64, 43, 256, 34, 324, 48, 144, 100, 484, 44, 396, 144, 304, 108, 784, 64, 900, 171, 400, 256, 576, 102, 1296, 324, 576, 176, 1600, 144, 1764, 300, 544, 484, 2116, 172, 1758, 396, 1024, 432, 2704, 304, 1600, 396, 1296, 784, 3364, 192

Offset: 1

Ilya Gutkovskiy, Sep 26 2021

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A057660, A062355, A348060.

Table[Sum[If[GCD[n, k] == 1, n/GCD[n, k - 1], 0], {k, n}], {n, 60}]
f[p_, e_] := (p^(2 e + 1) - (p + 1) p^(2 e - 1) + 1)/(p + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60]

a(n) = sum(k=1, n, if (gcd(n, k)==1, n/gcd(n, k-1))); \\ Michel Marcus, Sep 27 2021

Multiplicative with a(p^e) = (p^(2*e+1) - (p + 1) * p^(2*e-1) + 1) / (p + 1).

Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} (1 - 1/p^2 - 1/(1 + p + p^2)) = 0.1381393084... . - Amiram Eldar, Nov 18 2022

A372965 a(n) = Sum_{k = 1..n} ( n/gcd(k, n) )^4.

Original entry on oeis.org

1, 17, 163, 529, 2501, 2771, 14407, 16913, 39529, 42517, 146411, 86227, 342733, 244919, 407663, 541201, 1336337, 671993, 2345779, 1323029, 2348341, 2488987, 6156503, 2756819, 7815001, 5826461, 9605467, 7621303, 19803869, 6930271, 27705631, 17318417, 23864993

Offset: 1

Seiichi Manyama, May 18 2024

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A057660, A068963, A068970.
Cf. A001160, A008683.
Cf. A372966.
Cf. A013661, A013664, A157292.

f[p_, e_] := (p^(5*e+5) - p^(5*e+4) + p^4 - 1)/(p^5-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)

a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^4*sigma(d, 5));

PARI
```
a(n) = sumdiv(n, d, eulerphi(d^5));
```

a(n) = Sum_{d|n} mu(n/d) * (n/d)^4 * sigma_5(d).
a(n) = Sum_{d|n} d^(5-m) * phi(d^m) for m > 0.
G.f.: Sum_{k>=1} k^(5-m) * phi(k^m) * x^k/(1 - x^k) for m > 0.
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (p^(5*e+5) - p^(5*e+4) + p^4 - 1)/(p^5-1).
Dirichlet g.f.: zeta(s)*zeta(s-5)/zeta(s-4).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(6)/zeta(2) = 2*Pi^4/315 = 0.6184704192... (1/A157292). (End)

A060367 Average order of an element in a cyclic group of order n rounded down.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 6, 5, 6, 6, 10, 6, 12, 9, 9, 10, 16, 10, 18, 11, 14, 15, 22, 12, 20, 18, 20, 16, 28, 14, 30, 21, 23, 24, 25, 18, 36, 27, 28, 22, 40, 21, 42, 27, 28, 33, 46, 24, 42, 31, 37, 33, 52, 30, 42, 33, 42, 42, 58, 26, 60, 45, 41, 42, 50, 35, 66, 44, 51

Offset: 1

Avi Peretz (njk(AT)netvision.net.il), Apr 01 2001

Danny Rorabaugh, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A001157, A018804, A057660, A065764, A065827, A253905.

seq(floor(numtheory:-sigma[2](n^2)/numtheory:-sigma(n^2)/n), n=1..1000); # Robert Israel, Mar 24 2015

f[n_] := Block[{i, j, k}, Reap@ For[j = 1, j <= n, j++, Sow[Floor[Sum[1/GCD[j, k], {k, 1, j}]]]]] // Flatten // Rest; f@ 49 (* Michael De Vlieger, Mar 24 2015 *)
a[n_] := Floor[DivisorSigma[2, n^2]/DivisorSigma[1, n^2]/n]; Array[a, 100] (* Amiram Eldar, Jul 25 2025 *)

a(n) = {my(f = factor(n^2)); floor(sigma(f, 2)/(n * sigma(f)));} \\ Amiram Eldar, Jul 25 2025

[floor(sum([1/gcd(n,k) for k in range(1,n+1)])) for n in range(1,50)] # Danny Rorabaugh, Mar 24 2015

Sequence A057660 gives the sum of the orders of the elements in a cyclic group with n elements so a(n) = floor(A057660(n) / n) = floor(Sum_{k=1..n} 1/GCD(n, k)) = floor(Sum of 1/d times phi(n/d)) for all divisors d of n, where phi is Euler's phi function. This sum may also be expressed as the product of (p^(2*e(p)+1)+1)/((p+1)*p^e(p)) over all prime divisors p of n where the canonical factorization of n is the product of p^e(p), the e(p) being the exponents of the power of p in the factorization.
From Amiram Eldar, Jul 25 2025: (Start)
a(n) = floor(sigma_2(n^2)/(n*sigma(n))) = floor(A001157(n^2)/(n*A000203(n^2))) = floor(A065827(n)/(n*A065764(n))).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(3)/zeta(2) (A253905). (End)

Offset corrected and terms a(18)-a(50) added by Danny Rorabaugh, Mar 24 2015

cp's OEIS Frontend

A345295 Decimal expansion of Product_{p primes} (1 - 1/p)*(1 + (1 - 1/p^2)*Sum_{k>=1} 1/(p^k + p^(-k-1))).

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A070999 Numbers n such that the denominator of Sum_{k=1..n} 1/gcd(n,k) is not equal to n.

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A108223 a(n) = sigma_{2n}(n^2)/sigma_n(n^2), where sigma_n(m) = Sum_{d|m} d^n.

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A127477 Triangle T(n,k) read by rows: matrix product A054522 * A054523.

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Maple

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A130211 Triangle read by rows: matrix product A054522 * A000012.

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A226512 a(n) = minimum of sum of orders of all elements of G, where G is a group of order n.

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A345294 Decimal expansion of Product_{p primes} (1 - 1/p)*(1 + (1 + 1/p)*Sum_{k>=1} 1/(p^k + p^(-k-1))).

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A348061 a(n) = Sum_{k=1..n, gcd(n,k) = 1} n / gcd(n,k-1).

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A345295 Decimal expansion of Product_{p primes} (1 - 1/p)(1 + (1 - 1/p^2)Sum_{k>=1} 1/(p^k + p^(-k-1))).

A345294 Decimal expansion of Product_{p primes} (1 - 1/p)(1 + (1 + 1/p)Sum_{k>=1} 1/(p^k + p^(-k-1))).