A152649 -id:A152649 - cp's OEIS frontend

A007429 Inverse Moebius transform applied twice to natural numbers.

1, 4, 5, 11, 7, 20, 9, 26, 18, 28, 13, 55, 15, 36, 35, 57, 19, 72, 21, 77, 45, 52, 25, 130, 38, 60, 58, 99, 31, 140, 33, 120, 65, 76, 63, 198, 39, 84, 75, 182, 43, 180, 45, 143, 126, 100, 49, 285, 66, 152, 95, 165, 55, 232, 91, 234, 105, 124, 61, 385, 63, 132, 162, 247, 105, 260

Offset: 1

N. J. A. Sloane

Sum of the divisors d1 from the ordered pairs of divisors of n, (d1,d2) with d1<=d2, such that d1|d2. - Wesley Ivan Hurt, Mar 22 2022

a(n) is the sum of the sum-of-divisors of the divisors of n. - M. F. Hasler, Mar 29 2024

David M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston, MA, 1976, p. 120.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Travis Scholl, Table of n, a(n) for n = 1..100000 (terms 1 through 1000 were by T. D. Noe)
Olivier Bordelles, Mean values of generalized gcd-sum and lcm-sum functions, JIS, Vol. 10 (2007), Article 07.9.2, series g_4 (with an apparently wrong D.g.f. after equation 3).
N. J. A. Sloane, Transforms.

Cf. A000203, A007430, A134699, A152649, A280077.

[&+[SumOfDivisors(d): d in Divisors(n)]: n in [1..100]] // Jaroslav Krizek, Sep 24 2016

A007429 := proc(n)
    add(numtheory[sigma](d),d=numtheory[divisors](n)) ;
end proc:
seq(A007429(n),n=1..100) ; # R. J. Mathar, Aug 28 2015

f[n_] := Plus @@ DivisorSigma[1, Divisors@n]; Array[f, 52] (* Robert G. Wilson v, May 05 2010 *)
f[p_, e_] := (p*(p^(e + 1) - 1) - (p - 1)*(e + 1))/(p - 1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 10 2022 *)

A007429_upto(N)=vector(N,n, sumdiv(n,d, sigma(d))) \\ edited by M. F. Hasler, Mar 29 2024

a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)^2/(1-p*X))[n]) \\ Ralf Stephan

N=17; default(seriesprecision,N); x=z+O(z^(N+1))
c=sum(j=1,N,j*x^j);
t=1/prod(j=1,N, eta(x^(j))^(1/j))
t=log(t)
t=serconvol(t,c)
Vec(t)
/* Joerg Arndt, May 03 2008 */

a(n)=sumdiv(n,d, sumdiv(d,t, t ) );  /* Joerg Arndt, Oct 07 2012 */

from math import prod
from sympy import factorint
def A007429(n): return prod((p*(p**(e+1)-1)-(p-1)*(e+1))//(p-1)**2 for p,e in factorint(n).items()) # Chai Wah Wu, Mar 28 2024

def A(n): return sum(sigma(d) for d in n.divisors()) # Travis Scholl, Apr 14 2016

a(n) = Sum_{d|n} sigma(d), Dirichlet convolution of A000203 and A000012. - Jason Earls, Jul 07 2001
a(n) = Sum_{d|n} d*tau(n/d). - Vladeta Jovovic, Jul 31 2002
Multiplicative with a(p^e) = (p*(p^(e+1)-1)-(p-1)*(e+1))/(p-1)^2. - Vladeta Jovovic, Dec 25 2001
G.f.: Sum_{k>=1} sigma(k)*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
Moebius transform of A007430. - Benoit Cloitre, Mar 03 2004
Dirichlet g.f.: zeta(s-1)*zeta^2(s).
Equals A051731^2 * [1, 2, 3, ...]. Equals row sums of triangle A134577. - Gary W. Adamson, Nov 02 2007
Row sums of triangle A134699. - Gary W. Adamson, Nov 06 2007
a(n) = n * (Sum_{d|n} tau(d)/d) = n * (A276736(n) / A276737(n)). - Jaroslav Krizek, Sep 24 2016
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(sigma(k)/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 26 2018
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/72 = 1.352904... (A152649). - Amiram Eldar, Oct 22 2022

A098198 Decimal expansion of Pi^4/36 = zeta(2)^2.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Sep 21 2004

			2.70580808427784547879000924135291975693687737979... = 2*A152649 = A013661^2.

Ce Xu and Jianqiang Zhao, Sun's Three Conjectures on Apéry-like Sums Involving Harmonic Numbers, arXiv:2203.04184 [math.NT], 2022.
Index entries for transcendental numbers

Cf. A002088, A024916.

Mathematica
```
RealDigits[N[Pi^4/36, 256]]
```

zeta(2)^2 \\ Charles R Greathouse IV, Aug 08 2013

Decimal expansion of limit of q(n)= A024916(n)/A002088(n) = SummatorySigma / SummatoryTotient.
Equals Sum_{n>=1} A000005(n)/n^2. - R. J. Mathar, Dec 18 2010
Equals 10*Sum_{n>=2} (psi(n)+gamma)/n^3. - Jean-François Alcover, Feb 25 2013
Equals Zeta(4)*10/4 = A013662/0.4 = 1/A227929. - R. J. Mathar, Jul 20 2025
Equals 10 * zeta(3,1) = 10 * Sum_{n >= 1} 1/n Sum_{k >= n+1} 1/k^3 = 10 * Sum_{n >= 1} 1/n^3 * Sum_{k = 1..n-1} 1/k. - Peter Bala, Aug 07 2025

A233090 Decimal expansion of Sum_{n>=1} (-1)^(n-1)*H(n)/n^2, where H(n) is the n-th harmonic number.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Dec 04 2013, after the comment by Peter Bala about A233033

			0.7512855644747464283748363509446562442281164327128118011201697220886...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 43.

R. Barbieri, J. A. Mignaco, and E. Remiddi, Electron form factors up to fourth order. I., Il Nuovo Cim. 11A (4) (1972) 824-864, table II (13)
Philippe Flajolet and Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998), page 32.
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (C5.15)
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 561.

Cf. A002117 (zeta(3)), A197070 (3*zeta(3)/4), A233091 (7*zeta(3)/8), A076788 (alternating sum with denominator n), A152648 (non-alternating sum with denominator n^2), A152649 (non-alternating sum with denominator n^3), A233033 (alternating sum with denominator n^3).

RealDigits[ 5*Zeta[3]/8, 10, 100] // First

Equals 5*zeta(3)/8.
Equals -Integral_{x=0..1} (log(1+x)*log(1-x)/x)*dx. - Amiram Eldar, May 06 2023
Equals Sum_{m>=1} Sum_{n>=1} (-1)^(m-1)/(m*n*(m + n)) (see Finch). - Stefano Spezia, Nov 02 2024

A238168 Decimal expansion of sum_(n>=1) H(n)^2/n^5 where H(n) is the n-th harmonic number.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Feb 19 2014

			1.091882588664530085165782130699273873...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Philippe Flajolet, Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998) page 16.

Cf. A152648, A152649, A152651, A238166, A238167, A238169.

RealDigits[6*Zeta[7] -Zeta[2]*Zeta[5] -(5/2)*Zeta[3]*Zeta[4],10,100][[1]]

6*zeta(7) - zeta(2)*zeta(5) - (5/2)*zeta(3)*zeta(4) \\ G. C. Greubel, Dec 30 2017

Equals 6*zeta(7) - zeta(2)*zeta(5) - 5/2*zeta(3)*zeta(4).

A238181 Decimal expansion of sum_(n>=1) H(n)^2/n^3 where H(n) is the n-th harmonic number (Quadratic Euler Sum S(2,3)).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Feb 19 2014

			1.6519427927044986239626937611144940161...

Philippe Flajolet, Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998) page 24.

Cf. A152648, A152649, A152651, A218505, A238168, A238182, A238183.

7/2*Zeta[5] - Zeta[2]*Zeta[3] // RealDigits[#, 10, 100]& // First

7/2*zeta(5) - zeta(2)*zeta(3) \\ Stefano Spezia, May 22 2025

7/2*zeta(5) - zeta(2)*zeta(3).

A062822 Sum of divisors of the squarefree numbers: sigma(A005117(n)).

Original entry on oeis.org

1, 3, 4, 6, 12, 8, 18, 12, 14, 24, 24, 18, 20, 32, 36, 24, 42, 30, 72, 32, 48, 54, 48, 38, 60, 56, 42, 96, 44, 72, 48, 72, 54, 72, 80, 90, 60, 62, 96, 84, 144, 68, 96, 144, 72, 74, 114, 96, 168, 80, 126, 84, 108, 132, 120, 90, 112, 128, 144, 120, 98, 102, 216, 104, 192

Offset: 1

Jason Earls, Jul 20 2001

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000203, A005117, A002117, A152649, A265668, A001221.

a062822 1 = 1
a062822 n = product $ map (+ 1) $ a265668_row n
-- Reinhard Zumkeller, Dec 13 2015

DivisorSigma[1,#]&/@Select[Range[150],SquareFreeQ] (* Harvey P. Dale, May 18 2014 *)

j=[]; for(n=1,200, if(issquarefree(n),j=concat(j, sigma(n)))); j

from math import isqrt
from sympy import mobius, divisor_sigma
def A062822(n):
    def f(x): return n+x-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 divisor_sigma(m) # Chai Wah Wu, Aug 12 2024

a(n) = Product_{k=1..A001221(n)} (A265668(n,k) + 1). - Reinhard Zumkeller, Dec 13 2015
From Amiram Eldar, Nov 21 2022: (Start)
a(n) = A000203(A005117(n)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/(72*zeta(3)) = A152649 / A002117 = 1.1254908... . (End)

A238182 Decimal expansion of Sum_{n>=1} H(n)^2/n^4 where H(n) is the n-th harmonic number (Quadratic Euler Sum S(2,4)).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Feb 19 2014

No closed form of S(2,2q) is known to date, except for S(2,2) (A218505) and S(2,4) (this sequence).

			1.221879945319880138518806475290994812569...

Philippe Flajolet, Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998) page 24.

Cf. A001008/A002805, A152648, A152649, A152651, A218505, A238168, A238181, A238183.

97/24*Zeta[6] - 2*Zeta[3]^2 // RealDigits[#, 10, 100]& // First

97/24*zeta(6) - 2*zeta(3)^2.

A280025 Expansion of phi_{7, 4}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

Original entry on oeis.org

0, 1, 144, 2268, 18688, 78750, 326592, 825944, 2396160, 4966677, 11340000, 19501812, 42384384, 62777078, 118935936, 178605000, 306774016, 410422194, 715201488, 894002060, 1471680000, 1873240992, 2808260928, 3405105288, 5434490880, 6152734375, 9039899232

Offset: 0

Seiichi Manyama, Feb 22 2017

Multiplicative because A001158 is. - Andrew Howroyd, Jul 23 2018

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A280022 (phi_{5, 4}), this sequence (phi_{7, 4}).
Cf. A280024 (E_2^4*E_4), A282780 (E_2^3*E_6), A282752 (E_2^2*E_4^2), A282102 (E_2*E_4*E_6), A008411 (E_4^3), A280869 (E_6^2).
Cf. A001158 (sigma_3(n)), A281372 (n*sigma_3(n)), A282099 (n^2*sigma_3(n)), A282213 (n^3*sigma_3(n)), this sequence (n^4*sigma_3(n)).
Cf. A152649.

Table[n^4 * DivisorSigma[3, n], {n, 0, 30}] (* Amiram Eldar, Oct 31 2023 *)
nmax = 30; CoefficientList[Series[Sum[k^4*x^k*(1 + 120*x^k + 1191*x^(2*k) + 2416*x^(3*k) + 1191*x^(4*k) + 120*x^(5*k) + x^(6*k))/(1 - x^k)^8, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 02 2025 *)

a(n) = if(n < 1, 0, n^4 * sigma(n, 3)); \\ Andrew Howroyd, Jul 23 2018

a(n) = n^4*A001158(n) for n > 0.
a(n) = (7*(A280024(n) - 4*A282780(n) + 6*A282752(n) - 4*A282102(n)) + 3*A008411(n) + 4*A280869(n))/41472.
Sum_{k=1..n} a(k) ~ c * n^8, where c = Pi^4/720 = 0.1352904... (= A152649 / 10). - Amiram Eldar, Dec 08 2022
From Amiram Eldar, Oct 31 2023: (Start)
Multiplicative with a(p^e) = p^(4*e) * (p^(3*e+3)-1)/(p^3-1).
Dirichlet g.f.: zeta(s-4)*zeta(s-7). (End)
G.f.: Sum_{k>=1} k^4*x^k*(1 + 120*x^k + 1191*x^(2*k) + 2416*x^(3*k) + 1191*x^(4*k) + 120*x^(5*k) + x^(6*k))/(1 - x^k)^8. - Vaclav Kotesovec, Aug 02 2025

A360522 a(n) = Sum_{d|n} Max({d'; d'|n, gcd(d, d') = 1}).

Original entry on oeis.org

1, 3, 4, 6, 6, 12, 8, 11, 11, 18, 12, 24, 14, 24, 24, 20, 18, 33, 20, 36, 32, 36, 24, 44, 27, 42, 30, 48, 30, 72, 32, 37, 48, 54, 48, 66, 38, 60, 56, 66, 42, 96, 44, 72, 66, 72, 48, 80, 51, 81, 72, 84, 54, 90, 72, 88, 80, 90, 60, 144, 62, 96, 88, 70, 84, 144, 68

Offset: 1

Amiram Eldar, Feb 10 2023

a(n) is the sum of delta_d(n) over the divisors d of n, where delta_d(n) is the greatest divisor of n that is relatively prime to n.
Denoted by Sur(n) in Khan (2005).
Related sequences: A048691(n) = Sum_{d|n} #{d'; d' | n, gcd(d, d') = 1}, and A328485(n) = Sum_{d|n} Sum_{d' | n, gcd(d, d') = 1} d' (number and sum of divisors instead of maximal divisor, respectively).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Mizan R. Khan, Problem 10922, The American Mathematical Monthly, Vol. 109, No. 2 (2002), p. 201; Michael R. Avidon, The Sum of Divisors Won't Die, solution, ibid., Vol. 110, No. 10 (2003), p. 959.
Mizan R. Khan, A variant of the divisor functions sigma_a(n), JP Journal of Algebra, Number Theory and Applications, Vol. 5, No. 3 (2005), pp. 561-574.

Cf. A000203, A005117, A034448, A360523.
Cf. A065465, A072691, A152649, A330523.
Cf. A048691, A328485.

f[p_, e_] := p^e + e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

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

Multiplicative with a(p^e) = p^e + e.
Dirichlet g.f.: zeta(s-1)*zeta(s)^2 * Product_{p prime} (1 - 1/p^s - 1/p^(2*s-1) + 1/p^(2*s)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = A072691 * A065465 = A152649 * A330523 = 0.7250160726810604158... .
a(n) <= A000203(n) with equality if and only if n is squarefree (A005117).
limsup_{n->oo} sigma(n)/a(n) = oo, where sigma(n) is the sum of divisors of n (A000203) (Khan, 2002).
liminf_{n->oo} a(n)/usigma(n) = 1, where usigma(n) is the sum of unitary divisors of n (A034448) (Khan, 2005).
limsup_{n->oo} a(n)/usigma(n) = (55/54) * Product_{p prime} (1 + 1/(p^2+1)) = 1.4682298236... (Khan, 2005).

A238166 Decimal expansion of sum_(n>=1) H(n,2)/n^4 where H(n,2) = A007406(n)/A007407(n) is the n-th harmonic number of order 2.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Feb 19 2014

			1.1058264444388178540088457688766809845497962424196...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Philippe Flajolet, Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998) page 16.

Cf. A007406, A007407, A152648, A152649, A152651, A238167, A238168, A238169.

RealDigits[Zeta[3]^2 - 1/3*Zeta[6], 10, 100][[1]]

zeta(3)^2-Pi^6/2835 /* Michel Marcus, Jul 04 2014 */

Equals zeta(3)^2 - zeta(6)/3.

cp's OEIS Frontend

A007429 Inverse Moebius transform applied twice to natural numbers.

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A098198 Decimal expansion of Pi^4/36 = zeta(2)^2.

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A233090 Decimal expansion of Sum_{n>=1} (-1)^(n-1)*H(n)/n^2, where H(n) is the n-th harmonic number.

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A238168 Decimal expansion of sum_(n>=1) H(n)^2/n^5 where H(n) is the n-th harmonic number.

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A238181 Decimal expansion of sum_(n>=1) H(n)^2/n^3 where H(n) is the n-th harmonic number (Quadratic Euler Sum S(2,3)).

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A062822 Sum of divisors of the squarefree numbers: sigma(A005117(n)).

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A238182 Decimal expansion of Sum_{n>=1} H(n)^2/n^4 where H(n) is the n-th harmonic number (Quadratic Euler Sum S(2,4)).

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