A060640 -id:A060640 - cp's OEIS frontend

A064944 a(n) = Sum_{i|n, j|n, j >= i} j.

1, 5, 7, 17, 11, 38, 15, 49, 34, 60, 23, 132, 27, 82, 82, 129, 35, 191, 39, 207, 112, 126, 47, 384, 86, 148, 142, 283, 59, 469, 63, 321, 172, 192, 172, 666, 75, 214, 202, 597, 83, 640, 87, 435, 403, 258, 95, 1016, 162, 485, 262, 511, 107, 812, 264, 813, 292, 324

Offset: 1

Vladeta Jovovic, Oct 28 2001

nonn

			a(6) = max(1,1)+max(1,2)+max(1,3)+max(1,6)+max(2,2)+max(2,3)+max(2,6)+max(3,3)+max(3,6)+max(6,6)=38, or a(6) = dot_product(1,2,3,4)*(1,2,3,6)=1*1+2*2+3*3+4*6=38.

Paolo Xausa, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A060640, A064945, A064946, A064947, A064948, A064949.

Cf. A027750, A135539, A000005, A000203, A064840, A337360, A337297.

a064944 = sum . zipWith (*) [1..] . a027750_row'
-- Reinhard Zumkeller, Jul 14 2015

with(numtheory): seq(add(i*sort(convert(divisors(n),'list'))[i],i=1..tau(n)), n=1..200);

A064944[n_] := #.Range[Length[#]] & [Divisors[n]];
Array[A064944, 100] (* Paolo Xausa, Aug 07 2025 *)

a(n) = my(d=divisors(n)); sum(i=1, length(d), i*d[i]); \\ Harry J. Smith, Sep 30 2009

from sympy import divisors
def A064944(n): return sum(a*b for a, b in enumerate(divisors(n),1)) # Chai Wah Wu, Aug 07 2025

a(n) = Sum_{i=1..tau(n)} i*d_i, where {d_i}, i=1..tau(n) is the increasing sequence of divisors of n.
a(n) = Sum_{i=1..A000005(n)} i*A027750(n, i). - Michel Marcus, Jun 10 2015
From Ridouane Oudra, Aug 01 2025: (Start)
a(n) = Sum_{d|n} (n/d)*A135539(n,d).
a(n) = A064946(n) + A000203(n).
a(n) = (A064948(n) + A000203(n))/2.
a(n) = A337360(n) - A064945(n).
a(n) = A064948(n) - A064946(n).
a(n) = A064840(n) - A064947(n). (End)

A263825 Total number c_{pi_1(B_1)}(n) of n-coverings over the first amphicosm.

Original entry on oeis.org

1, 7, 5, 23, 7, 39, 9, 65, 18, 61, 13, 143, 15, 87, 35, 183, 19, 182, 21, 245, 45, 151, 25, 465, 38, 189, 58, 375, 31, 429, 33, 549, 65, 277, 63, 806, 39, 327, 75, 875, 43, 663, 45, 719, 126, 439, 49, 1535, 66, 650, 95, 933, 55, 982, 91, 1425, 105, 637, 61, 2093

Offset: 1

N. J. A. Sloane, Oct 28 2015

nonn

Gheorghe Coserea, Table of n, a(n) for n = 1..20000
G. Chelnokov, M. Deryagina, A. Mednykh, On the Coverings of Amphicosms; Revised title: On the coverings of Euclidian manifolds B_1 and B_2, arXiv preprint arXiv:1502.01528 [math.AT], 2015.
G. Chelnokov, M. Deryagina and A. Mednykh, On the coverings of Euclidean manifolds B_1 and B_2, Communications in Algebra, Vol. 45, No. 4 (2017), 1558-1576.

Cf. A263825-A263830, A263832.

A263825 := proc(n)
    local a,l,m,s1,s2,s3,s4 ;
    # Theorem 2
    a := 0 ;
    for l in numtheory[divisors](n) do
        m := n/l ;
        s1 := 0 ;
        for twok in numtheory[divisors](m) do
            if type(twok,'even') then
                k := twok/2 ;
                s1 := s1+numtheory[sigma](k)*k ;
            end if;
        end do:
        s2 := 0 ;
        for d in numtheory[divisors](l) do
            s2 := s2+numtheory[mobius](l/d)*d^2*igcd(2,d) ;
        end do:
        s3 := 0 ;
        for k in numtheory[divisors](m) do
            s3 := s3+numtheory[sigma](m/k)*k ;
            if modp(m,2*k) = 0 then
                s3 := s3-numtheory[sigma](m/2/k)*k ;
            end if;
        end do:
        s4 := 0 ;
        for twok in numtheory[divisors](m) do
            if type(twok,'even') then
                s4 := s4+numtheory[sigma](m/twok)*twok ;
                if modp(m,2*twok) = 0 then
                    s4 := s4-numtheory[sigma](m/2/twok)*twok ;
                end if;
            end if;
        end do:
        a := a+A059376(l)*s1 + s2*s3 + A007434(l)*s4 ;
    end do:
    a/n ;
end proc: # R. J. Mathar, Nov 03 2015

A007434[n_] := Sum[ MoebiusMu[n/d] * d^2, {d, Divisors[n]}];
A059376[n_] := Sum[ MoebiusMu[n/d] * d^3, {d, Divisors[n]}];
A263825[n_] := Module[{a, l, m, s1, s2, s3, s4},
a = 0;
Do[m = n/l;
s1 = 0; Do[If[EvenQ[twok], k = twok/2; s1 = s1 + DivisorSigma[1, k]*k], {twok, Divisors[m]}];
s2 = 0; Do[s2 = s2 + MoebiusMu[l/d]*d^2*GCD[2, d], {d, Divisors[l]}];
s3 = 0; Do[s3 = s3 + DivisorSigma[1, m/k]*k ; If[Mod[m, 2*k] == 0, s3 = s3 - DivisorSigma[1, m/2/k]*k], {k, Divisors[m]}];
s4 = 0; Do[If[EvenQ[twok], s4 = s4 + DivisorSigma[1, m/twok]*twok; If[ Mod[m, 2*twok] == 0, s4 = s4 - DivisorSigma[1, m/2/twok]*twok]], {twok, Divisors[m]}]; a = a + A059376[l]*s1 + s2*s3 + A007434[l]*s4,
{l, Divisors[n]}]; a/n
];
Array[A263825, 60] (* Jean-François Alcover, Nov 21 2017, after R. J. Mathar *)

A001001(n) = sumdiv(n, d, sigma(d) * d);
A007434(n) = sumdiv(n, d, moebius(n\d) * d^2);
A059376(n) = sumdiv(n, d, moebius(n\d) * d^3);
A060640(n) = sumdiv(n, d, sigma(n\d) * d);
EpiPcZn(n) = sumdiv(n, d, moebius(n\d) * d^2 * gcd(d,2));
S1(n)      = if (n%2, 0, A001001(n\2));
S11(n)     = A060640(n) - if(n%2, 0, A060640(n\2));
S21(n)     = if (n%2, 0, 2*A060640(n\2)) - if (n%4, 0, 2*A060640(n\4));
a(n) = { 1/n * sumdiv(n, d,
  A059376(d) * S1(n\d) + EpiPcZn(d) * S11(n\d) + A007434(d) * S21(n\d));
};
vector(60, n, a(n))  \\ Gheorghe Coserea, May 04 2016

A349143 a(n) = Sum_{d|n} A038040(d) * A348507(n/d), where A038040(n) = n*tau(n), A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).

Original entry on oeis.org

0, 1, 1, 9, 1, 16, 1, 51, 13, 22, 1, 114, 1, 28, 25, 233, 1, 145, 1, 168, 31, 40, 1, 590, 21, 46, 106, 222, 1, 310, 1, 939, 43, 58, 37, 915, 1, 64, 49, 896, 1, 406, 1, 330, 262, 76, 1, 2570, 29, 297, 61, 384, 1, 1012, 49, 1202, 67, 94, 1, 2040, 1, 100, 340, 3489, 55, 598, 1, 492, 79, 574, 1, 4457, 1, 118, 360, 546, 55

Offset: 1

Antti Karttunen, Nov 08 2021

nonn

Dirichlet convolution of A348507 with A038040, which is the Dirichlet convolution of the identity function (A000027) with itself.
Dirichlet convolution of the identity function (A000027) with A349140.
Dirichlet convolution of sigma (A000203) with A349141.
Dirichlet convolution of A060640 with A348971.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000005, A000027, A000203, A003959, A038040, A060640, A348507, A348971, A349140, A349141, A349142.

Cf. also A349123, A348983.

f[p_, e_] := (p + 1)^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, #*DivisorSigma[0, #]*(s[n/#] - n/#) &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)

A038040(n) = (n*numdiv(n));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348507(n) = (A003959(n) - n);
A349143(n) = sumdiv(n,d,A038040(d)*A348507(n/d));

a(n) = Sum_{d|n} A038040(n/d) * A348507(d).
a(n) = Sum_{d|n} d * A349140(n/d).
a(n) = Sum_{d|n} A000203(d) * A349141(n/d).
a(n) = Sum_{d|n} A060640(d) * A348971(n/d).
For all n >= 1, a(n) >= A349123(n) >= A348983(n).

A068986 Numbers k such that Sum_{d|k} sigma(d)/d is an integer.

Original entry on oeis.org

1, 21, 105, 1050, 1155, 24921, 26565, 45150, 49842, 59455, 249210, 274131, 548262, 1129645, 1151325, 1248555, 1309350, 2302650, 2741310, 3388935, 4605300, 6305013, 12610026, 13417131, 14685385, 23722545, 25750550, 26834262, 30154410, 33388425, 42997750, 44056155

Offset: 1

Benoit Cloitre, Apr 06 2002

nonn

Also numbers k that divide A060640(k). - Seiichi Manyama, May 08 2021

Amiram Eldar, Table of n, a(n) for n = 1..100 (terms 1..64 from Michel Marcus)

Cf. A000203, A060640, A017665, A017666.

f[p_, e_] := ((e + 1)*p^(e + 2) - (e + 2)*p^(e + 1) + 1)/(p - 1)^2; q[1] = True; q[n_] := Divisible[Times @@ f @@@ FactorInteger[n], n]; Select[Range[10^6], q] (* Amiram Eldar, Dec 29 2024 *)

for(n=1,10^7,if(frac(sumdiv(n,d, sigma(d)/d))==0,print1(n,",")))

is(k) = {my(f = factor(k)); !(prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2];  ((e + 1)*p^(e + 2) - (e + 2)*p^(e + 1) + 1)/(p - 1)^2) % k);} \\ Amiram Eldar, Dec 29 2024

More terms from Jason Earls, Apr 09 2002
a(22)-a(25) from Jinyuan Wang, Apr 06 2020
a(26)-a(32) from Michel Marcus, Apr 06 2020

A061258 a(n) = Sum_{d|n} d*psi(d), where psi(d) is reduced totient function, cf. A002322.

Original entry on oeis.org

1, 3, 7, 11, 21, 21, 43, 27, 61, 63, 111, 53, 157, 129, 87, 91, 273, 183, 343, 151, 175, 333, 507, 117, 521, 471, 547, 305, 813, 261, 931, 347, 447, 819, 483, 431, 1333, 1029, 631, 327, 1641, 525, 1807, 781, 681, 1521, 2163, 373, 2101, 1563, 1095, 1103, 2757

Offset: 1

Vladeta Jovovic, Apr 21 2001

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

Cf. A002322, A057660, A001001, A060640, A061259.

Cf. A000005, A027750, A141258.

a061258 n = sum $ zipWith (*) ds $ map a002322 ds
            where ds = a027750_row n
-- Reinhard Zumkeller, Sep 02 2014

a[n_] := DivisorSum[n, # * CarmichaelLambda[#] &]; Array[a, 100] (* Amiram Eldar, Apr 13 2024 *)

a(n) = sumdiv(n, d, d * lcm(znstar(d)[2])); \\ Amiram Eldar, Apr 13 2024

a(n) = Sum_{k = 1..A000005(n)} (A027750(n,k)*A002322(A027750(n,k))). - Reinhard Zumkeller, Sep 02 2014

A245211 a(n) = Sum_{(d

Original entry on oeis.org

0, 1, 1, 5, 1, 11, 1, 17, 7, 15, 1, 47, 1, 19, 17, 49, 1, 62, 1, 67, 21, 27, 1, 151, 11, 31, 34, 87, 1, 145, 1, 129, 29, 39, 25, 254, 1, 43, 33, 219, 1, 189, 1, 127, 104, 51, 1, 423, 15, 130, 41, 147, 1, 278, 33, 287, 45, 63, 1, 589, 1, 67, 132, 321, 37, 277

Offset: 1

Jaroslav Krizek, Jul 23 2014

nonn

If q are proper divisors of n then values of sequence a(n) are the bending moments at point 0 of static forces of sizes tau(q) operating in places q on the cantilever as the nonnegative number axis of length n with support at point 0 by the schema: a(n) = Sum_{q | n} (q * tau(q)).
Number n = 144 is the smallest number n such that a(n) > n * tau(n) (see A245212 and A245214).
Conjecture: 21 is only number such that a(n) = n.

			For n = 21 with proper divisors [1, 3, 7] we have: a(21) = 7 * tau(7) + 3 * tau(3) + 1 * tau(1) = 7*2 + 3*2 + 1*1 = 21.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A000005, A100484, A245212, A245213, A245214.

[(&+[d*#([e: e in Divisors(d)]): d in Divisors(n)])-(n*(#[d: d in Divisors(n)])): n in [1..1000]];

a(n) = sumdiv(n, d, (dJens Kruse Andersen, Aug 13 2014

a(n) = A060640(n) - A038040(n) = Sum_{d | n} (d * tau(d)) - n*tau(n).
a(n) = A038040(n) - A245212(n).
a(n) = 1 for n = primes.
a(n) = n + 5 for even semiprimes q = 2p > 4 (see A100484) where p = odd prime.

A318491 a(n) is the numerator of Sum_{d|n} Sum_{j|d} 1/j.

Original entry on oeis.org

1, 5, 7, 17, 11, 35, 15, 49, 34, 11, 23, 119, 27, 75, 77, 129, 35, 85, 39, 187, 5, 115, 47, 343, 86, 135, 142, 255, 59, 77, 63, 321, 161, 175, 33, 289, 75, 195, 63, 539, 83, 25, 87, 391, 374, 235, 95, 301, 162, 43, 245, 459, 107, 355, 23, 105, 91, 295, 119, 1309, 123, 315, 170, 769, 297

Offset: 1

Ilya Gutkovskiy, Aug 27 2018

			1, 5/2, 7/3, 17/4, 11/5, 35/6, 15/7, 49/8, 34/9, 11/2, 23/11, 119/12, 27/13, 75/14, 77/15, 129/16, ...

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

Cf. A000005, A000203, A006171, A007429, A017665, A017666, A060640, A318492 (denominators).

Cf. A001620, A013661, A306016.

f:= proc(n) local d;
   numer(add(numtheory:-sigma(d)/d, d = numtheory:-divisors(n))) end proc:
map(f, [$1..65]); # Robert Israel, Jan 13 2025

Numerator[Table[Sum[DivisorSigma[-1, d], {d, Divisors[n]}], {n, 65}]]
Numerator[Table[Sum[DivisorSigma[1, d]/d, {d, Divisors[n]}], {n, 65}]]
Numerator[Table[Sum[d DivisorSigma[0, d], {d, Divisors[n]}]/n, {n, 65}]]
nmax = 65; Rest[Numerator[CoefficientList[Series[Sum[DivisorSigma[1, k] x^k/(k (1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x]]]
nmax = 65; Rest[Numerator[CoefficientList[Series[-Log[Product[(1 - x^k)^DivisorSigma[0, k], {k, 1, nmax}]], {x, 0, nmax}], x]]]

a(n) = numerator(sumdiv(n, d, sumdiv(d, j, 1/j))); \\ Michel Marcus, Aug 28 2018

Numerators of coefficients in expansion of Sum_{k>=1} sigma(k)*x^k/(k*(1 - x^k)), where sigma(k) = sum of divisors of k (A000203).
Numerators of coefficients in expansion of -log(Product_{k>=1} (1 - x^k)^tau(k)), where tau(k) = number of divisors of k (A000005).
a(n) = numerator of Sum_{d|n} sigma(d)/d.
a(n) = numerator of (1/n)*Sum_{d|n} d*tau(d).
If p is a prime, a(p) = 2*p + 1.
Sum_{k=1..n} a(k)/A318492(k) ~ zeta(2) * n * (log(n) + 2*gamma - 1 + zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 25 2024

A127099 Triangle T(n,m) = A126988 *A127093 read by rows.

Original entry on oeis.org

1, 3, 2, 4, 0, 3, 7, 6, 0, 4, 6, 0, 0, 0, 5, 12, 8, 9, 0, 0, 6, 8, 0, 0, 0, 0, 0, 7, 15, 14, 0, 12, 0, 0, 0, 8, 13, 0, 12, 0, 0, 0, 0, 0, 9, 18, 12, 0, 0, 15, 0, 0, 0, 0, 10, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 28, 24, 21, 16, 0, 18, 0, 0, 0, 0, 0, 12, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 24, 16, 0, 0

Offset: 1

Gary W. Adamson, Jan 05 2007

Multiply the infinite lower triangular matrices A126988 and A127093.

			First few rows of the triangle are:
1;
3, 2;
4, 0, 3;
7, 6, 0, 4;
6, 0, 0, 0, 5;
12, 8, 9, 0, 0, 6;
8, 0, 0, 0, 0, 0, 7;
15, 14, 0, 12, 0, 0, 0, 8;
13, 0, 12, 0, 0, 0, 0, 0, 9;
18, 12, 0, 0, 15, 0, 0, 0, 0, 10;
...

Cf. A060640 (row sums), A000203, A127093, A127094, A123229, A127096, A127097, A127098, A127013, A127057.

T(n,m) = sum_{j=m..n} A126988(n,j)*A127093(j,m).

T(n,1) = A000203(n).

Extended by R. J. Mathar, Aug 18 2009

A245212 a(n) = n * tau(n) - Sum_{(d

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 13, 15, 20, 25, 21, 25, 25, 37, 43, 31, 33, 46, 37, 53, 63, 61, 45, 41, 64, 73, 74, 81, 57, 95, 61, 63, 103, 97, 115, 70, 73, 109, 123, 101, 81, 147, 85, 137, 166, 133, 93, 57, 132, 170, 163, 165, 105, 154, 187, 161, 183, 169, 117, 131, 121

Offset: 1

Jaroslav Krizek, Jul 23 2014

sign

If d are divisors of n then values of sequence a(n) are the bending moments at point 0 of static forces of sizes tau(d) operating in places d on the cantilever as the nonnegative number axis of length n with support at point 0 by the schema: a(n) = (n * tau(n)) - Sum_{(d

If a(n) = 0 then n must be > 10^7.

Conjecture: a(n) = sigma(n) iff n is a power of 2 (A000079).

Number n = 72 is the smallest number n such that a(n) < n (see A245213).

Number n = 144 is the smallest number n such that a(n) < 0 (see A245214).

			For n = 6 with divisors [1, 2, 3, 6] we have: a(6) = 6 * tau(6) - (3 * tau(3) + 2 * tau(2) + 1 * tau(1)) = 6*4 - (3*2+2*2+1*1) = 13.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A000005, A100484, A245211, A245213, A245214.

[(2*(n*(#[d: d in Divisors(n)]))-(&+[d*#([e: e in Divisors(d)]): d in Divisors(n)])): n in [1..1000]];

a(n) = sumdiv(n, d, (-1)^(dJens Kruse Andersen, Aug 13 2014

a(n) = A038040(n) - A245211(n).

a(n) = 2 * A038040(n) - A060640(n) = 2 * (n * tau(n)) - Sum_{d | n} (d * tau(d)).

A318492 a(n) is the denominator of Sum_{d|n} Sum_{j|d} 1/j.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 11, 12, 13, 14, 15, 16, 17, 9, 19, 20, 1, 22, 23, 24, 25, 26, 27, 28, 29, 6, 31, 32, 33, 34, 7, 18, 37, 38, 13, 40, 41, 2, 43, 44, 45, 46, 47, 16, 49, 5, 51, 52, 53, 27, 5, 8, 19, 58, 59, 60, 61, 62, 21, 64, 65, 66, 67, 4, 69, 14, 71, 36, 73, 74, 75

Offset: 1

Ilya Gutkovskiy, Aug 27 2018

			1, 5/2, 7/3, 17/4, 11/5, 35/6, 15/7, 49/8, 34/9, 11/2, 23/11, 119/12, 27/13, 75/14, 77/15, 129/16, ...

Cf. A000005, A000203, A006171, A007429, A017665, A017666, A060640, A068986 (positions of 1's), A318491 (numerators).

Denominator[Table[Sum[DivisorSigma[-1, d], {d, Divisors[n]}], {n, 75}]]
Denominator[Table[Sum[DivisorSigma[1, d]/d, {d, Divisors[n]}], {n, 75}]]
Denominator[Table[Sum[d DivisorSigma[0, d], {d, Divisors[n]}]/n, {n, 75}]]
nmax = 75; Rest[Denominator[CoefficientList[Series[Sum[DivisorSigma[1, k] x^k/(k (1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x]]]
nmax = 75; Rest[Denominator[CoefficientList[Series[-Log[Product[(1 - x^k)^DivisorSigma[0, k], {k, 1, nmax}]], {x, 0, nmax}], x]]]

a(n) = denominator(sumdiv(n, d, sumdiv(d, j, 1/j))); \\ Michel Marcus, Aug 28 2018

Denominators of coefficients in expansion of Sum_{k>=1} sigma(k)*x^k/(k*(1 - x^k)), where sigma(k) = sum of divisors of k (A000203).
Denominators of coefficients in expansion of -log(Product_{k>=1} (1 - x^k)^tau(k)), where tau(k) = number of divisors of k (A000005).
a(n) = denominator of Sum_{d|n} sigma(d)/d.
a(n) = denominator of (1/n)*Sum_{d|n} d*tau(d).

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A064944 a(n) = Sum_{i|n, j|n, j >= i} j.

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A263825 Total number c_{pi_1(B_1)}(n) of n-coverings over the first amphicosm.

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A349143 a(n) = Sum_{d|n} A038040(d) * A348507(n/d), where A038040(n) = n*tau(n), A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).

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A068986 Numbers k such that Sum_{d|k} sigma(d)/d is an integer.

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A061258 a(n) = Sum_{d|n} d*psi(d), where psi(d) is reduced totient function, cf. A002322.

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A245211 a(n) = Sum_{(d

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A318491 a(n) is the numerator of Sum_{d|n} Sum_{j|d} 1/j.

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