A306016 -id:A306016 - cp's OEIS frontend

A072156 Numerator of Sum_{k=1..n} phi(k)/k^2.

1, 5, 53, 115, 3163, 3263, 170687, 352399, 1096397, 223513, 28103473, 28459213, 4963286677, 5029541437, 25532475569, 51741301813, 15299527769557, 15415359085157, 5677532668504877, 1144538596366201, 1156827116999161, 1166157760248361, 626832724103131129

Offset: 1

N. J. A. Sloane, Jun 28 2002

			1, 5/4, 53/36, 115/72, 3163/1800, 3263/1800, 170687/88200, ...

Vincenzo Librandi, Table of n, a(n) for n = 1..300

Cf. A000010, A072157.

Cf. A001620, A013661, A306016.

List([1..25], n-> NumeratorRat( Sum([1..n], k-> Phi(k)/k^2) ) ); # G. C. Greubel, Aug 25 2019

[Numerator( &+[EulerPhi(k)/k^2: k in [1..n]] ): n in [1..25]]; // G. C. Greubel, Aug 25 2019

with(numtheory); seq(numer(add(phi(k)/k^2, k = 1..n)), n = 1..25); # G. C. Greubel, Aug 25 2019

Numerator[Table[Sum[EulerPhi[k]/k^2,{k,n}],{n,30}]] (* Vincenzo Librandi, Nov 15 2011 *)
Numerator[Accumulate[Table[EulerPhi[k]/k^2, {k, 1, 30}]]] (* Amiram Eldar, Dec 28 2024 *)

a(n) = numerator( sum(k=1,n, eulerphi(k)/k^2));
vector(25, n, a(n)) \\ G. C. Greubel, Aug 25 2019

[numerator( sum(euler_phi(k)/k^2 for k in (1..n)) ) for n in (1..25)] # G. C. Greubel, Aug 25 2019

a(n)/A072157(n) ~ (log(n) + gamma - zeta'(2)/zeta(2)) / zeta(2), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 28 2024

A087050 Square root of the largest square >1 dividing the n-th nonsquarefree number.

Original entry on oeis.org

2, 2, 3, 2, 4, 3, 2, 2, 5, 3, 2, 4, 6, 2, 2, 3, 4, 7, 5, 2, 3, 2, 2, 3, 8, 2, 6, 5, 2, 4, 9, 2, 2, 3, 2, 4, 7, 3, 10, 2, 6, 4, 2, 3, 2, 11, 2, 5, 3, 8, 2, 3, 2, 2, 12, 7, 2, 5, 2, 3, 2, 4, 9, 2, 2, 13, 3, 2, 5, 4, 6, 2, 2, 3, 8, 14, 3, 10, 2, 3, 4, 2, 6, 2, 4, 15, 2, 2, 3, 2, 4, 11, 9, 2, 7, 2, 5, 6, 16, 2, 3

Offset: 1

Wolfdieter Lang, Sep 08 2003

			n=10, A013929(10) = 27, a(10)^2 = 3^2 = 9. 27 = 9*3.
n=39, A013929(39) = 100, a(39)^2 = 10^2 = 100.

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

Cf. A087049, A013929, A000188, A000290.

Cf. A001610, A013661, A306016.

f[p_, e_] := p^Floor[e/2]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; s /@ Select[Range[300], !SquareFreeQ[#] &] (* Amiram Eldar, Feb 11 2021 *)

from math import isqrt, prod
from sympy import mobius, factorint
def A087050(n):
    def f(x): return n+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return prod(p**(e>>1) for p, e in factorint(m).items() if e>1) # Chai Wah Wu, Jul 22 2024

a(n)^2 is the largest square factor (from A000290) of the nonsquarefree number A013929(n), n>=1.
a(n) = A000188(A013929(n)). - Amiram Eldar, Feb 11 2021
Sum_{k=1..n} a(k) ~ (n/(2*(zeta(2)-1))) * (log(n) + 3*gamma - 3 - 2*zeta'(2)/zeta(2) - log(1-1/zeta(2))), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 14 2024

A143304 Decimal expansion of Norton's constant.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Aug 05 2008

The average number of divisions required by the Euclidean algorithm, for a pair of independently and uniformly chosen numbers in the range [1, N] is (12*log(2)/Pi^2) * log(N) + c + O(N^(e-1/6)), for any e>0, where c is this constant (Norton, 1990). - Amiram Eldar, Aug 27 2020

			0.06535142592303732137...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 157.

Graham H. Norton, On the asymptotic analysis of the Euclidean algorithm, J. Symbolic Comput., Vol. 10 (1990), pp. 53-58.
Eric Weisstein's World of Mathematics, Norton's Constant.

Cf. A001620, A074962, A086237, A306016.

RealDigits[-((Pi^2 - 6*Log[2]*(24 * Log[Glaisher] + 2*EulerGamma + Log[2] - 2 * Log[Pi] - 3))/Pi^2), 10, 100][[1]] (* Amiram Eldar, Aug 27 2020 *)

Equals -((Pi^2 - 6*log(2)*(-3 + 2*EulerGamma + log(2) + 24*log(Glaisher) - 2*log(Pi)))/Pi^2).

Equals (12*log(2)/Pi^2) * (zeta'(2)/zeta(2) - 1/2) + A086237 - 1/2. - Amiram Eldar, Aug 27 2020

A268732 Sum of the numbers of divisors of gcd(x,y) with x*y <= n.

Original entry on oeis.org

1, 3, 5, 9, 11, 15, 17, 23, 27, 31, 33, 41, 43, 47, 51, 60, 62, 70, 72, 80, 84, 88, 90, 102, 106, 110, 116, 124, 126, 134, 136, 148, 152, 156, 160, 176, 178, 182, 186, 198, 200, 208, 210, 218, 226, 230, 232, 250, 254, 262, 266, 274, 276, 288, 292, 304, 308, 312, 314, 330

Offset: 1

Michel Marcus, Feb 12 2016

nonn

Partial sums of A124315.

Daniel Suteu, Table of n, a(n) for n = 1..10000
Masum Billal, Asymptotic Result of A Generalization of A GCD-Sum, arXiv:2206.05023 [math.NT], 2022.
Adrian W. Dudek, On the Success of Mishandling Euclid's Lemma, arXiv:1602.03555 [math.HO], 2016. See Remark 1 p. 3.
Adrian W. Dudek, On the Success of Mishandling Euclid's Lemma, The American Mathematical Monthly, Vol. 123, No. 9 (2016), 924-927.
Randell Heyman, A summation involving the number of divisors function and the GCD function, arXiv:2003.13937 [math.NT], 2020.

Cf. A000005, A124315.

Cf. A001620, A013661, A306016.

Table[Total@ Flatten@ Map[Function[k, DivisorSigma[0, GCD[#, k]] & /@ Select[Range@ n, # k <= n &]], Range@ n], {n, 60}] (* Michael De Vlieger, Feb 12 2016 *)

a(n) = sum(k=1, n, sumdiv(k, d, numdiv(gcd(d, k/d))));

a(n) = sum(k=1, sqrtint(n), 2*sum(j=1, sqrtint(n\(k*k)), n\(j*k*k))-sqrtint(n\(k*k))^2); \\ Daniel Suteu, Jan 08 2019

a(n)=sum(k=1,n,sum(j=1,sqrt(n/k),floor(n/k/j^2))); \\ Benoit Cloitre, Oct 02 2022

a(n) = Sum_{k=1..floor(sqrt(n))} (2*Sum_{j=1..floor(sqrt(n/k^2))} floor(n/(j*k^2)) - floor(sqrt(n/k^2))^2). - Daniel Suteu, Jan 08 2019
a(n) = n*zeta(2)*(log(n) + 2*gamma - 1 + 2*zeta'(2)/zeta(2)) + O(sqrt(n)*log(n)), where gamma is the Euler-Mascheroni constant A001620. - Daniel Suteu, Jan 11 2019
a(n) = Sum_{i=1..n} Sum_{j=1..n} floor(sqrt(n/(i*j))). - Ridouane Oudra, Apr 13 2025

A356472 Numerator of the average of gcd(i,n) for i = 1..n.

Original entry on oeis.org

1, 3, 5, 2, 9, 5, 13, 5, 7, 27, 21, 10, 25, 39, 3, 3, 33, 7, 37, 18, 65, 63, 45, 25, 13, 75, 3, 26, 57, 9, 61, 7, 35, 99, 117, 14, 73, 111, 125, 9, 81, 65, 85, 42, 21, 135, 93, 5, 19, 39, 55, 50, 105, 9, 189, 65, 185, 171, 117, 6, 121, 183, 13, 4, 45, 105, 133, 66, 75, 351, 141, 35, 145, 219, 13, 74, 39, 125, 157

Offset: 1

Matthias Kaak, Aug 08 2022

			For n = 3, the average of the gcd's is (gcd(1,3) + gcd(2,3) + gcd(3,3))/3 = (1 + 1 + 3)/3 = 5/3 and its numerator is a(3)=5.

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

Cf. A356473 (denominators), A018804, A057661 (LCM).

Cf. A001620, A013661, A306016.

map numerator (map (\i -> sum (map (\j -> gcd i j) [1..i]) % i) [1..])

Table[Numerator[Sum[GCD[I, j], {j, 1, I}]/I], {I, 100}]
f[p_, e_] := e*(p - 1)/p + 1; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Apr 28 2023 *)

a(n) = numerator(sum(i=1, n, gcd(i, n))/n); \\ Michel Marcus, Aug 08 2022

a(n,f=factor(n))=my(k=prod(i=1, #f~, (f[i, 2]*(f[i, 1]-1)/f[i, 1] + 1)*f[i, 1]^f[i, 2])); k/gcd(k,n) \\ Charles R Greathouse IV, Sep 08 2022

from math import prod, gcd
from sympy import factorint
def A356472(n):
    f = factorint(n)
    return (m:=prod((p-1)*e+p for p, e in f.items()))//gcd(prod(f),m) # Chai Wah Wu, Sep 08 2022

a(n) = numerator(A018804(n)/n).
a(n) << n^(1+e) for any e > 0. a(n) > 1 for all n > 1. - Charles R Greathouse IV, Sep 08 2022
Sum_{k=1..n} a(k)/A356473(k) ~ (n/zeta(2)) * (log(n) + 2*gamma - 1 - zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 25 2024

A360164 a(n) is the sum of the square roots of the unitary divisors of n that are odd squares.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 8, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 6, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jan 29 2023

First differs from A336649 at n = 27.

The unitary analog of A360163.

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

Cf. A001620, A056624, A069290, A108269, A306016, A336649, A358347, A360163.

f[p_, e_] := If[OddQ[e], 1, p^(e/2) + 1]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1] == 2, 1, if(f[i, 2]%2, 1, f[i, 1]^(f[i, 2]/2) + 1))); }

a(n) = Sum_{d|n, gcd(d, n/d)=1, d odd square} sqrt(d).
a(n) = A360162(n) if n is not of the form (2*m - 1)*4^k where m >= 1, k >= 1 (A108269).
Multiplicative with a(2^e) = 1, and for p > 2, a(p^e) = p^(e/2) + 1 if e is even and 1 if e is odd.
Dirichlet g.f.: (zeta(s)*zeta(2*s-1)/zeta(3*s-1))*(2^(3*s)-2^(s+1))/(2^(3*s)-2).
Sum_{k=1..n} a(k) ~ (2*n/Pi^2)*(log(n) + 3*gamma - 1 + log(2) - 3*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620).

A063754 Dirichlet convolution of totient and cototient.

Original entry on oeis.org

0, 1, 1, 3, 1, 7, 1, 8, 5, 11, 1, 20, 1, 15, 13, 20, 1, 31, 1, 32, 17, 23, 1, 52, 9, 27, 21, 44, 1, 71, 1, 48, 25, 35, 21, 88, 1, 39, 29, 84, 1, 99, 1, 68, 61, 47, 1, 128, 13, 83, 37, 80, 1, 123, 29, 116, 41, 59, 1, 200, 1, 63, 81, 112, 33, 155, 1, 104, 49, 159, 1, 228, 1, 75, 101

Offset: 1

Labos Elemer, Aug 14 2001

a(n) = 1 if and only if n is prime. - Robert Israel, Feb 04 2018

a(n) = n+1 if and only if n = 2*p with p an odd prime (A100484 \ {4}). - Bernard Schott, Jun 19 2023

			n = 24: divisors = {1, 2, 3, 4, 6, 8, 12, 24}, d-phi(d) = {0, 1, 1, 2, 4, 4, 8, 16}, phi(n/d) = {8, 4, 4, 2, 2, 2, 1, 1}, products = {0, 4, 4, 4, 8, 8, 8, 16}, a(24) = 52.

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

Cf. A000010, A018804, A029935, A051953, A100484.

Cf. A001620, A013661, A306016.

f:= n -> add(numtheory:-phi(d)*(n/d - numtheory:-phi(n/d)), d=numtheory:-divisors(n)):
map(f, [$1..100]); # Robert Israel, Feb 04 2018

f1[p_, e_] := (e*(p - 1)/p + 1)*p^e; f2[p_, e_] := (e+1)*(p^e - p^(e-1)) - (e-1)*(p^(e-1) - p^(e-2)); a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct; a[1] = 0; Array[a, 100] (* Amiram Eldar, Apr 28 2023 *)

a(n) = sumdiv(n, d, eulerphi(d)*(n/d - eulerphi(n/d))); \\ Michel Marcus, Feb 05 2018

a(n) = Sum_{d|n} A000010(d)*A051953(n/d).
From Richard L. Ollerton, May 06 2021: (Start)
a(n) = Sum_{k=1..n} A051953(gcd(n,k)).
a(n) = Sum_{k=1..n} A051953(n/gcd(n,k))*A000010(gcd(n,k))/A000010(n/gcd(n,k)).
a(n) = A018804(n) - A029935(n). (End)
Sum_{k=1..n} a(k) ~ (1/(2*zeta(2)))*(1 - 1/zeta(2)) * n^2 * (log(n) + 2*gamma - 1/2 - ((zeta(2)-2)/(zeta(2)-1))*(zeta'(2)/zeta(2))), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 11 2024

Offset corrected by Robert Israel, Feb 04 2018

A068074 a(n) = Sum_{d|n} (-1)^d*2^omega(n/d) where omega(x) is the number of distinct prime factors in the factorization of x.

Original entry on oeis.org

-1, -1, -3, 1, -3, -3, -3, 3, -5, -3, -3, 3, -3, -3, -9, 5, -3, -5, -3, 3, -9, -3, -3, 9, -5, -3, -7, 3, -3, -9, -3, 7, -9, -3, -9, 5, -3, -3, -9, 9, -3, -9, -3, 3, -15, -3, -3, 15, -5, -5, -9, 3, -3, -7, -9, 9, -9, -3, -3, 9, -3, -3, -15, 9, -9, -9, -3, 3, -9, -9, -3, 15, -3, -3, -15, 3, -9, -9, -3, 15, -9, -3, -3, 9, -9, -3, -9, 9, -3

Offset: 1

Benoit Cloitre, Apr 14 2002

Gérald Tenenbaum and Jie Wu, Cours spécialisés No. 2: "Exercices corrigés de théorie analytique et probabiliste des nombres", Collection SMF, chapter II.7.1, p. 105.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005, A001221, A048691.

Cf. A013661, A001620, A306016.

a068074 n | odd n     = - a048691 n
          | otherwise = 2 * a048691 (n `div` 2) - a048691 n
-- Reinhard Zumkeller, Jul 12 2012

a[n_?OddQ] := -DivisorSigma[0, n^2]; a[n_?EvenQ] := 2*DivisorSigma[0, n^2/4] - DivisorSigma[0, n^2]; Table[a[n], {n, 1, 89}] (* Jean-François Alcover, Nov 15 2011, after Vladeta Jovovic *)
f[p_, e_] := 2*e + 1; f[2, e_] := 3-2*e; a[1] = -1; a[n_] := -Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = sumdiv(n, d, (-1)^d*2^omega(n/d)); \\ Michel Marcus, Oct 08 2017

a(n) = -numdiv(n^2) + if(!(n%2), 2*numdiv(n^2/4)); \\ Amiram Eldar, Apr 24 2025

Asymptotic formula: Sum_{k=1..n} a(k)/k ~ -C*log(n)^2 with C = 3*log(2)/Pi^2.
a(n) = -tau(n^2) for odd n and 2*tau(n^2/4) - tau(n^2) for even n. b(n) = abs(a(n)) is multiplicative with b(2^e) = abs(2*e-3) and b(p^e) = 2*e+1 for an odd prime p. - Vladeta Jovovic, Apr 25 2002
a(n) = if n odd then -A048691(n) else 2*A048691(n/2) - A048691(n). - Reinhard Zumkeller, Jul 12 2012
From Amiram Eldar, Apr 24 2025: (Start)
More precisely, b(n) = -a(n) is multiplicative with b(2^e) = 3-2*e and b(p^e) = 2*e+1 for an odd prime p.
Dirichlet g.f.: -(zeta(s)^3/zeta(2*s)) * (1-1/2^(s-1)).
Sum_{k=1..n} a(k) ~ -(log(2)/zeta(2)) * n * (log(n) + 3*gamma - 1 - log(2)/2 - 2*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). (End)

A309153 a(n) = A000203(n)*A001227(n).

Original entry on oeis.org

1, 3, 8, 7, 12, 24, 16, 15, 39, 36, 24, 56, 28, 48, 96, 31, 36, 117, 40, 84, 128, 72, 48, 120, 93, 84, 160, 112, 60, 288, 64, 63, 192, 108, 192, 273, 76, 120, 224, 180, 84, 384, 88, 168, 468, 144, 96, 248, 171, 279, 288, 196, 108, 480, 288, 240, 320, 180, 120, 672, 124, 192, 624, 127, 336, 576, 136, 252, 384

Offset: 1

Omar E. Pol, Jul 14 2019

A001227(n) is denoted by Delta_0(n) in Glaisher 1907.

a(n) = A000203(n) iff n is a power of 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Cf. A000079, A000203, A001227, A062354, A062355, A064840.

Cf. A001620, A002117, A244115, A306016

Array[DivisorSum[#, 1 &, OddQ] DivisorSigma[1, #] &, 69] (* Michael De Vlieger, Nov 22 2019 *)
f[p_, e_] := (e+1)*(p^(e+1)-1)/(p-1); f[2, e_] := 2^(e+1) - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 01 2022 *)

a(n) = sigma(n)*delta(n).
Multiplicative with a(2^e) = 2^(e+1) - 1 and a(p^e) = (e+1)*(p^(e+1)-1)/(p-1) for p > 2. - Amiram Eldar, Nov 01 2022
From Amiram Eldar, Dec 04 2023: (Start)
Dirichlet g.f.: (4^s - 3*2^s + 2)/(4^s - 2) * (zeta(s)*zeta(s-1))^2/zeta(2*s-1).
Sum_{k=1..n} a(k) ~ (Pi^4/(168*zeta(3))) * n^2 * (log(n) + 2*gamma - 1/2 + 22*log(2)/21 + 2*zeta'(2)/zeta(2) - 2*zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620). (End)

A345308 Decimal expansion of Sum_{p primes} log(p) / (p-1)^2.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Jun 13 2021

			1.226968805653470005965662568745762562988257454901426311714794620109...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 156 (constant C10 + 1).
Eric Weisstein's World of Mathematics, Prime Factor, formula (13)-(14), (constant U).

Cf. A138312, A306016, A345364.

ratfun = 1/((p - 1)^2); zetas = 0; ratab = Table[konfun = Simplify[ratfun + c/(p^power - 1)] // Together; coefs = CoefficientList[Numerator[konfun], p]; sol = Solve[Last[coefs] == 0, c][[1]]; zetas = zetas + c*Zeta'[power]/Zeta[power] /. sol; ratfun = konfun /. sol, {power, 2, 25}]; Do[Print[N[Sum[Log[p]*ratfun /. p -> Prime[k], {k, 1, m}] + zetas, 110]], {m, 1000, 5000, 1000}]

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A072156 Numerator of Sum_{k=1..n} phi(k)/k^2.

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A087050 Square root of the largest square >1 dividing the n-th nonsquarefree number.

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A143304 Decimal expansion of Norton's constant.

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A268732 Sum of the numbers of divisors of gcd(x,y) with x*y <= n.

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A356472 Numerator of the average of gcd(i,n) for i = 1..n.

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A360164 a(n) is the sum of the square roots of the unitary divisors of n that are odd squares.

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A063754 Dirichlet convolution of totient and cototient.

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