A001017 -id:A001017 - cp's OEIS frontend

A253637 Second partial sums of ninth powers (A001017).

1, 514, 20710, 303050, 2538515, 14851676, 67518444, 254402940, 828707925, 2403012910, 6335265586, 15427298614, 35123831015, 75481410200, 154282348760, 301802764056, 567911055849, 1032378638010, 1819533917950, 3118689197890, 5212124524411, 8512829068724, 13614686274500, 21358351020500, 32916713032125, 49904578722726

Offset: 1

Luciano Ancora, Jan 07 2015

The formula for the second partial sums of m-th powers is: b(n,m) = (n+1)*F(m) - F(m+1), where F(m) are the m-th Faulhaber's formulas.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Luciano Ancora, Recurrence relation for the second partial sums of m-th powers
Luciano Ancora, Second partial sums of the m-th powers
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13.
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A001017.

List([1..30], n-> n*(n+1)*(n+2)*(n^2+n-1)*(n^2+3*n+1)*(6*n^4+24*n^3 +5*n^2-38*n+ 25)/660 ); # G. C. Greubel, Aug 28 2019

[n*(n+1)*(n+2)*(n^2+n-1)*(n^2+3*n+1)*(6*n^4+24*n^3+5*n^2-38*n+ 25)/660: n in [1..30]]; // G. C. Greubel, Aug 28 2019

seq(n*(n+1)*(n+2)*(n^2+n-1)*(n^2+3*n+1)*(6*n^4+24*n^3+5*n^2-38*n+ 25)/660, n=1..30); # G. C. Greubel, Aug 28 2019

CoefficientList[Series[(1 +502x +14608x^2 +88234x^3 +156190x^4 +88234x^5 +14608x^6 +502x^7 +x^8)/(1-x)^12, {x, 0, 30}], x] (* Vincenzo Librandi, Jan 19 2015 *)
Nest[Accumulate,Range[30]^9,2] (* Harvey P. Dale, Apr 18 2021 *)

a(n) = (6*n^11 + 66*n^10 + 275*n^9 + 495*n^8 + 198*n^7 - 462*n^6 - 330*n^5 + 330*n^4 + 231*n^3 - 99*n^2 - 50*n)/660; \\ Michel Marcus, Jan 08 2015

[n*(n+1)*(n+2)*(n^2+n-1)*(n^2+3*n+1)*(6*n^4+24*n^3+5*n^2-38*n+ 25)/660 for n in (1..30)] # G. C. Greubel, Aug 28 2019

a(n) = n*(n+1)*(n+2)*(n^2+n-1)*(n^2+3*n+1)*(6*n^4 + 24*n^3 + 5*n^2 - 38*n + 25)/660.
a(n) = 2*a(n-1) - a(n-2) + n^9.
G.f.: x*(1 + 502*x + 14608*x^2 + 88234*x^3 + 156190*x^4 + 88234*x^5 + 14608*x^6 + 502*x^7 + x^8)/(1-x)^12. - Vincenzo Librandi, Jan 19 2015

A254643 Third partial sums of ninth powers (A001017).

Original entry on oeis.org

1, 515, 21225, 324275, 2862790, 17714466, 85232910, 339635850, 1168343775, 3571356685, 9906622271, 25333920885, 60457751900, 135939162100, 290221510860, 592024274916, 1159935330765, 2192313968775, 4011847886725, 7130537084615

Offset: 1

Luciano Ancora, Feb 05 2015

			First differences:   1, 511, 19171, 242461, 1690981, ... (A022525)
------------------------------------------------------------------------
The ninth powers:    1, 512, 19683, 262144, 1953125, ... (A001017)
------------------------------------------------------------------------
First partial sums:  1, 513, 20196, 282340, 2235465, ... (A007487)
Second partial sums: 1, 514, 20710, 303050, 2538515, ... (A253637)
Third partial sums:  1, 515, 21225, 324275, 2862790, ... (this sequence)

Luciano Ancora, Table of n, a(n) for n = 1..1000
Luciano Ancora, Partial sums of m-th powers with Faulhaber polynomials
Luciano Ancora, Pascal’s triangle and recurrence relations for partial sums of m-th powers
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A001017, A007487, A022525, A253637.

List([1..30], n-> Binomial(n+3,4)*(2*n^8 +24*n^7 +98*n^6 +126*n^5 -97*n^4 -203*n^3 +127*n^2 +84*n -50)/110); # G. C. Greubel, Aug 28 2019

[Binomial(n+3,4)*(2*n^8 +24*n^7 +98*n^6 +126*n^5 -97*n^4 -203*n^3 +127*n^2 +84*n -50)/110: n in [1..30]]; // G. C. Greubel, Aug 28 2019

seq(binomial(n+3,4)*(2*n^8 +24*n^7 +98*n^6 +126*n^5 -97*n^4 -203*n^3 +127*n^2 +84*n -50)/110, n=1..30); # G. C. Greubel, Aug 28 2019

Table[n(1+n)(2+n)(3+n)(-50 +84n +127n^2 -204n^3 -97n^4 +126n^5 +98n^6 +24n^7 +2n^8)/2640, {n, 20}] (* or *)
CoefficientList[Series[(1 +502x +14608x^2 +88234x^3 +156190x^4 +88234x^5 +14608x^6 +502x^7 +x^8)/(1-x)^13, {x, 0, 19}], x] (* Ancora *)
Accumulate[Accumulate[Accumulate[Range[10]^9]]] (* Alonso del Arte, Feb 09 2015 *)

vector(30, n, m=n+3; binomial(m,4)*(2*(n*m)^4 -10*(n*m)^3 +11*(n*m)^2 +28*(n*m) -50)/110) \\ G. C. Greubel, Aug 28 2019

[binomial(n+3,4)*(2*n^8 +24*n^7 +98*n^6 +126*n^5 -97*n^4 -203*n^3 +127*n^2 +84*n -50)/110 for n in (1..30)] # G. C. Greubel, Aug 28 2019

G.f.: x*(1 +502*x +14608*x^2 +88234*x^3 +156190*x^4 +88234*x^5 +14608*x^6 +502*x^7 +x^8)/(1-x)^13.
a(n) = n*(1+n)*(2+n)*(3+n)*(-50 +84*n +127*n^2 -204*n^3 -97*n^4 +126*n^5 +98*n^6 +24*n^7 +2*n^8)/2640.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + n^9.

Edited by Alonso del Arte and Bruno Berselli, Feb 10 2015

A255179 Second differences of ninth powers (A001017).

Original entry on oeis.org

1, 510, 18660, 223290, 1448520, 6433590, 22151340, 63588210, 159338640, 359376750, 745368180, 1443884970, 2642886360, 4611828390, 7725765180, 12493804770, 19592282400, 29903014110, 44556993540, 64983894810, 92967744360, 130709124630, 180894272460

Offset: 0

Luciano Ancora, Feb 21 2015

			Second differences:  1, 510, 18660, 223290, 1448520, ... (this sequence)
First differences:   1, 511, 19171, 242461, 1690981, ... (A022525)
------------------------------------------------------------------------
The ninth powers:    1, 512, 19683, 262144, 1953125, ... (A001017)
------------------------------------------------------------------------
First partial sums:  1, 513, 20196, 282340, 2235465, ... (A007487)
Second partial sums: 1, 514, 20710, 303050, 2538515, ... (A253637)
Third partial sums:  1, 515, 21225, 324275, 2862790, ... (A254643)

Luciano Ancora, Table of n, a(n) for n = 0..1000
Luciano Ancora, Sums of powers of positive integers and their recurrence relations, section 0.5.
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A001017, A007487, A022525, A253637, A254643, A255177, A255178.

[1] cat [6*n*(3+28*n^2+42*n^4+12*n^6): n in [1..30]]; // Vincenzo Librandi, Mar 12 2015

Join[{1}, Table[6 n (3 + 28 n^2 + 42 n^4 + 12 n^6), {n, 1, 30}]]
Join[{1},Differences[Range[0,30]^9,2]] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,510,18660,223290,1448520,6433590,22151340,63588210,159338640},30] (* Harvey P. Dale, Jan 26 2019 *)

G.f.: (1 + 502*x + 14608*x^2 + 88234*x^3 + 156190*x^4 + 88234*x^5 + 14608*x^6 + 502*x^7 + x^8)/(1 - x)^8.

a(n) = 6*n*(3 + 28*n^2 + 42*n^4 + 12*n^6) for n>0, a(0)=1.

Corrected g.f. by Bruno Berselli, Feb 25 2015

Offset changed by Bruno Berselli, Mar 20 2015

A238170 Integer part of square root of A001017: a(n) = floor(n^(9/2)).

Original entry on oeis.org

0, 1, 22, 140, 512, 1397, 3174, 6352, 11585, 19683, 31622, 48558, 71831, 102978, 143739, 196069, 262144, 344365, 445375, 568056, 715541, 891223, 1098758, 1342070, 1625363, 1953125, 2330129, 2761448, 3252453, 3808824, 4436552, 5141947, 5931641, 6812597, 7792110

Offset: 0

Philippe Deléham, Feb 21 2014

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Integer part of square root of n^k: A000196 (k=1), A000093 (k=3), A155013 (k=5), A155014 (k=7), this sequence (k=9), A155015 (k=11), A155016 (k=13), A155018 (k=15), A155019 (k=17).

[Floor(n^(9/2)): n in [0..40]]; // Vincenzo Librandi, Feb 23 2014

Table[Floor[n^(9/2)], {n,0,30}] (* G. C. Greubel, Dec 30 2017 *)

a(n) = floor(n^(9/2)); \\ Joerg Arndt, Feb 23 2014

from math import isqrt
def A238170(n): return isqrt(n**9) # Chai Wah Wu, Jan 27 2023

a(n) = floor(n^(9/2)).
a(n) = A000196(A001017(n)).
a(n) = floor(n^4*sqrt(n)).

A279642 Exponential transform of the ninth powers A001017.

Original entry on oeis.org

1, 1, 513, 21220, 1130381, 108174916, 8543324917, 800980035472, 88064461381913, 9832425683734288, 1199454069536074601, 158528649288125900224, 21925314644323181005477, 3213026006947537325856832, 497390236613387084643144029, 80481275337746709959509939456

Offset: 0

Alois P. Heinz, Dec 16 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..348

Column k=9 of A279636.

Cf. A001017.

a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1)*j^9*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);

E.g.f.: exp(exp(x)*(x^9+36*x^8+462*x^7+2646*x^6+6951*x^5+7770*x^4 +3025*x^3 +255*x^2+x)).

A255183 Third differences of ninth powers (A001017).

Original entry on oeis.org

1, 509, 18150, 204630, 1225230, 4985070, 15717750, 41436870, 95750430, 200038110, 385991430, 698516790, 1199001390, 1968942030, 3113936790, 4768039590, 7098477630, 10310731710, 14653979430, 20426901270

Offset: 0

Luciano Ancora, Mar 18 2015

			Third differences:   1, 509, 18150, 204630, 1225230, ...  (this sequence)
Second differences:  1, 510, 18660, 223290, 1448520, ...  (A255179)
First differences:   1, 511, 19171, 242461, 1690981, ...  (A022525)
---------------------------------------------------------------------
The ninth powers:    1, 512, 19683, 262144, 1953125, ...  (A001017)
---------------------------------------------------------------------

Luciano Ancora, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A001017, A007487, A022525, A253637, A255179, A255182.

[1,509] cat [6*(84*n^6-252*n^5+630*n^4-840*n^3+756*n^2-378*n+85): n in [2..30]]; // Vincenzo Librandi, Mar 18 2015

Join[{1, 509}, Table[6 (84 n^6 - 252 n^5 + 630 n^4 - 840 n^3 + 756 n^2 - 378 n + 85), {n, 2, 30}]]
Join[{1,509},Differences[Range[0,20]^9,3]] (* Harvey P. Dale, Apr 24 2015 *)

G.f.: (1 + 502*x + 14608*x^2 + 88234*x^3 + 156190*x^4 + 88234*x^5 + 14608*x^6 + 502*x^7 + x^8)/(1 - x)^7.

a(n) = 6*(84*n^6 - 252*n^5 + 630*n^4 - 840*n^3 + 756*n^2 - 378*n + 85) for n>1, a(0)=1, a(1)=509.

Edited by Bruno Berselli, Mar 20 2015

A268335 Exponentially odd numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95, 96, 97

Offset: 1

Vladimir Shevelev, Feb 01 2016

nonn

The sequence is formed by 1 and the numbers whose prime power factorization contains only odd exponents.
The density of the sequence is the constant given by A065463.
Except for the first term the same as A002035. - R. J. Mathar, Feb 07 2016
Also numbers k all of whose divisors are bi-unitary divisors (i.e., A286324(k) = A000005(k)). - Amiram Eldar, Dec 19 2018
The term "exponentially odd integers" was apparently coined by Cohen (1960). These numbers were also called "unitarily 2-free", or "2-skew", by Cohen (1961). - Amiram Eldar, Jan 22 2024

Peter J. C. Moses, Table of n, a(n) for n = 1..2000
Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Mathematische Zeitschrift, Vol. 74 (1960), pp. 66-80.
Eckford Cohen, Some sets of integers related to the k-free integers, Acta Sci. Math. (Szeged), Vol. 22, No. 3-4 (1961), pp. 223-233.
Vladimir Shevelev, Exponentially S-numbers, arXiv:1510.05914 [math.NT], 2015.
Vladimir Shevelev, Set of all densities of exponentially S-numbers, arXiv preprint arXiv:1511.03860 [math.NT], 2015.
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175(2016), 385-395.
D. Suryanarayana and R. Sita Rama Chandra Rao, Distribution of unitarily k-free integers, Journal of the Australian Mathematical Society, Vol. 20 , No. 2 (1975), pp. 129-141.

Cf. A002035, A209061, A138302, A197680, A000578, A000584, A001014, A001017, A008456, A010803, A010805, A010806, A010808, A010811, A010812, A001221, A124010.

Select[Range@ 100, AllTrue[Last /@ FactorInteger@ #, OddQ] &] (* Version 10, or *)
Select[Range@ 100, Times @@ Boole[OddQ /@ Last /@ FactorInteger@ #] == 1 &] (* Michael De Vlieger, Feb 02 2016 *)

isok(n)=my(f = factor(n)); for (k=1, #f~, if (!(f[k,2] % 2), return (0))); 1; \\ Michel Marcus, Feb 02 2016

from itertools import count, islice
from sympy import factorint
def A268335_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(e&1 for e in factorint(n).values()),count(max(startvalue,1)))
A268335_list = list(islice(A268335_gen(),20)) # Chai Wah Wu, Jun 22 2023

Sum_{a(n)<=x} 1 = C*x + O(sqrt(x)*log x*e^(c*sqrt(log x)/(log(log x))), where c = 4*sqrt(2.4/log 2) = 7.44308... and C = Product_{prime p} (1 - 1/p*(p + 1)) = 0.7044422009991... (A065463).

Sum_{n>=1} 1/a(n)^s = zeta(2*s) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s)), s>1. - Amiram Eldar, Sep 26 2023

A001016 Eighth powers: a(n) = n^8.

Original entry on oeis.org

0, 1, 256, 6561, 65536, 390625, 1679616, 5764801, 16777216, 43046721, 100000000, 214358881, 429981696, 815730721, 1475789056, 2562890625, 4294967296, 6975757441, 11019960576, 16983563041, 25600000000, 37822859361, 54875873536, 78310985281, 110075314176

Offset: 0

N. J. A. Sloane

Besides the first term, this sequence lists the denominators in Pi^8/9450 = 1 + 1/256 + 1/6561 + 1/65536 + 1/390625 + 1/1679616 + ... - Mohammad K. Azarian, Nov 01 2011, edited by M. F. Hasler, Jul 03 2025
For n > 0, a(n) is the largest number k such that k + n^4 divides k^2 + n^4. - Derek Orr, Oct 01 2014
Fourth powers of squares and squares of 4th powers. Squares composed with themselves twice. - Wesley Ivan Hurt, Apr 01 2016

Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968), p. 982.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Cf. A000290 (squares), A000583 (fourth powers), A001014 - A001017 (6th - 9th powers), A008454 (10th powers), A010801 (13th powers).
Cf. A000542 (partial sums), A022524 (first differences), A013666 (zeta(8)).
Cf. A003380 - A003390 (sums of 2, ..., 12 eighth powers).

[n^8 : n in [0..50]]; // Wesley Ivan Hurt, Apr 01 2016

A001016:=n->n^8: seq(A001016(n), n=0..50); # Wesley Ivan Hurt, Apr 01 2016

Table[n^8, {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)

A001016(n):=n^8$
makelist(A001016(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

A001016(n)=n^8 \\ Charles R Greathouse IV, Sep 24 2015

A001016 = lambda n: n**8 # M. F. Hasler, Jul 03 2025

Multiplicative with a(p^e) = p^(8e). - David W. Wilson, Aug 01 2001
Totally multiplicative sequence with a(p) = p^8 for primes p. - Jaroslav Krizek, Nov 01 2009
G.f.: -x*(1+x)*(x^6+246*x^5+4047*x^4+11572*x^3+4047*x^2+246*x+1)/(x-1)^9. - R. J. Mathar, Jan 07 2011
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) + 40320. - Ant King, Sep 24 2013
From Wesley Ivan Hurt, Apr 01 2016: (Start)
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n > 8.
a(n) = A000290(n)^4 = A000290(A000290(A000290(n))).
a(n) = A000583(n)^2. (End)
From Amiram Eldar, Oct 08 2020: (Start)
Sum_{n>=1} 1/a(n) = zeta(8) = Pi^8/9450 (A013666).
Sum_{n>=1} (-1)^(n+1)/a(n) = 127*zeta(8)/128 = 127*Pi^8/1209600. (End)
E.g.f.: exp(x)*x*(1 + 127*x + 966*x^2 + 1701*x^3 + 1050*x^4 + 266*x^5 + 28*x^6 + x^7). - Stefano Spezia, Jul 29 2022

More terms from James Sellers, Sep 19 2000

A008455 11th powers: a(n) = n^11.

Original entry on oeis.org

0, 1, 2048, 177147, 4194304, 48828125, 362797056, 1977326743, 8589934592, 31381059609, 100000000000, 285311670611, 743008370688, 1792160394037, 4049565169664, 8649755859375, 17592186044416, 34271896307633, 64268410079232, 116490258898219, 204800000000000

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A000583, A000584, A001014, A001017, A008454, A013669.

Cf. A004813 - A004823 (sums of 2, ..., 12 positive eleventh powers).

[n^11: n in [0..40]]; // Vincenzo Librandi, Jul 05 2014

Table[n^11, {n, 0, 30}] (* Vincenzo Librandi, Jul 05 2014 *)

A008455(n):=n^11$ makelist(A008455(n),n,0,20); /* Martin Ettl, Dec 17 2012 */

A008455(n)=n^11 \\ M. F. Hasler, Jul 03 2025

A008455 = lambda n: n**11 # M. F. Hasler, Jul 03 2025

a(n) = A000584(n)*A001014(n).
Multiplicative with a(p^e) = p^(11*e). - David W. Wilson, Aug 01 2001
Totally multiplicative with a(p) = p^11 for primes p. - Jaroslav Krizek, Nov 01 2009
From Amiram Eldar, Oct 08 2020: (Start)
Sum_{n>=1} 1/a(n) = zeta(11) (A013669).
Sum_{n>=1} (-1)^(n+1)/a(n) = 1023*zeta(11)/1024. (End)

A003391 Numbers that are the sum of 2 positive 9th powers.

Original entry on oeis.org

2, 513, 1024, 19684, 20195, 39366, 262145, 262656, 281827, 524288, 1953126, 1953637, 1972808, 2215269, 3906250, 10077697, 10078208, 10097379, 10339840, 12030821, 20155392, 40353608, 40354119, 40373290, 40615751, 42306732, 50431303, 80707214, 134217729, 134218240, 134237411

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 03 2020: (Start)
4605366718202103 is in the sequence as 4605366718202103 = 8^9 + 55^9.
66540479494556160 is in the sequence as 66540479494556160 = 16^9 + 74^9.
208226326986883072 is in the sequence as 208226326986883072 = 28^9 + 84^9. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A003380 (8th), A004802 (10th).

Cf. A001017 (ninth powers).

cp's OEIS Frontend

A253637 Second partial sums of ninth powers (A001017).

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A254643 Third partial sums of ninth powers (A001017).

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A255179 Second differences of ninth powers (A001017).

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A238170 Integer part of square root of A001017: a(n) = floor(n^(9/2)).

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A279642 Exponential transform of the ninth powers A001017.

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A255183 Third differences of ninth powers (A001017).

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A268335 Exponentially odd numbers.

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