A142150 -id:A142150 - cp's OEIS frontend

A242483 Numbers n such that A242481(n) = ((n*(n+1)/2) mod n + sigma(n) mod n + antisigma(n) mod n) / n = 2.

4, 8, 10, 14, 16, 22, 26, 32, 34, 36, 38, 44, 46, 48, 50, 52, 58, 60, 62, 64, 68, 72, 74, 76, 82, 84, 86, 90, 92, 94, 96, 98, 106, 108, 110, 116, 118, 122, 124, 128, 130, 132, 134, 136, 142, 144, 146, 148, 152, 154, 156, 158, 164, 166, 168, 170, 172, 178, 182

Offset: 1

Jaroslav Krizek, May 16 2014

nonn

Numbers n such that A242480(n) = (n*(n+1)/2) mod n + sigma(n) mod n + antisigma(n) mod n = (A142150(n) + A054024(n) + A229110(n)) = ((A000217(n) mod n) + (A000203(n) mod n) + (A024816(n) mod n)) = 2n. Numbers n such that A242481(n) = (A142150(n) + A054024(n) + A229110(n)) / n = ((A000217(n) mod n) + (A000203(n) mod n) + (A024816(n) mod n)) / n = 2.

Conjecture: with number 1 complement of A242482.

			8 is in sequence because [(8*(8+1)/2) mod 8 + sigma(8) mod 8 + antisigma(8) mod 8] / 8 = (36 mod 8 + 15 mod 8 + 21 mod 8) / 8 = (4 + 7 + 5 ) / 8 = 2.

Jaroslav Krizek, Table of n, a(n) for n = 1..5000

Cf. A242480, A242481, A242482, A242484, A242485, A242486.

[n: n in [1..1000] | 2 eq (((n*(n+1)div 2 mod n + SumOfDivisors(n) mod n + (n*(n+1)div 2-SumOfDivisors(n)) mod n)))div n]

A276457 a(n) is the number of times that a(n-1) appears in the concatenation of all numbers from a(0) to a(n-2), with a(0) = 0.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, 0, 8, 0, 9, 0, 10, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 2, 10, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 3, 10, 4, 4, 5, 4, 6, 4, 7, 4, 8, 4, 9, 4, 10, 5, 5, 6, 5, 7, 5, 8, 5, 9, 5, 10, 6, 6, 7, 6, 8, 6, 9, 6, 10, 7, 7, 8, 7, 9, 7, 10

Offset: 0

Yuriy Sibirmovsky, Sep 03 2016

138, 185 and 199 are three smallest numbers that do not appear among the first 5000 terms of the sequence. They first appear at n = 8776, 5117 and 10580 respectively.
1187 and 1190 are two smallest numbers that do not appear among the first 100000 terms.
Question: will every natural number eventually appear in the sequence?
The sequence can be started with any number a(0). The terms will be different, but for larger n behavior will be similar for all a(0).

			From a(0) to a(0), a(1) appears once, thus a(2) = 1.
From a(0) to a(1), a(2) appears 0 times, thus a(3) = 0.
...
From a(0) to a(19), a(20) = 10 appears once, in the form of '1,0'. Thus a(21) = 1.

Yuriy Sibirmovsky, Table of n, a(n) for n = 0..100000
Yuriy Sibirmovsky, Plot for n=0..4999
Yuriy Sibirmovsky, Plot for n=0..59999

Cf. A142150, A171181, A248034.

Similar in spirit to van Eck's A181391.

Nm=100;
A=Table[0,{j,1,Nm}];
A[[3]]=1;
Do[B=Table[IntegerDigits[A[[l]]],{l,1,j-1}];
A[[j+1]]=SequenceCount[Flatten[B],IntegerDigits[A[[j]]]],{j,3,Nm-1}];
A

a(n) = A142150(n) = A171181(n), if 0<=n<=20.

a(n) = A248034(n-19), if 21<=n<=120. - Omar E. Pol, Sep 03 2016

A086099 a(n) = OR(k AND (n-k): 0<=k<=n), AND and OR bitwise.

Original entry on oeis.org

0, 0, 1, 0, 3, 2, 3, 0, 7, 6, 7, 4, 7, 6, 7, 0, 15, 14, 15, 12, 15, 14, 15, 8, 15, 14, 15, 12, 15, 14, 15, 0, 31, 30, 31, 28, 31, 30, 31, 24, 31, 30, 31, 28, 31, 30, 31, 16, 31, 30, 31, 28, 31, 30, 31, 24, 31, 30, 31, 28, 31, 30, 31, 0, 63, 62, 63, 60, 63, 62, 63, 56, 63, 62

Offset: 0

Reinhard Zumkeller, Jul 09 2003

a(2^n - 1) = 0, a(3*2^n - 1) = 2^n;

A086100(n) = A007088(a(n)).

			a(4) = (0 AND 4) OR (1 AND 3) OR (2 AND 2) OR (3 AND 1) OR (4 AND 0) -> (000 AND 100) OR (001 AND 011) OR (010 AND 010) OR (011 AND 001) OR (111 AND 000) = 000 OR 011 OR 010 OR 011 OR 000 = 011 -> a(4)=3.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, AND
Eric Weisstein's World of Mathematics, OR
Reinhard Zumkeller, Logical Convolutions

Cf. A003817 (even bisection), A062383.
Cf. A086100 (in binary), A007088.
Cf. A053644, A006519.
Cf. A000004, A142149, A142150, A142151, A001477.

import Data.Bits ((.&.), (.|.))
a086099 n = foldl1 (.|.) $ zipWith (.&.) [0..] $ reverse [0..n] :: Integer
-- Reinhard Zumkeller, Jun 04 2012

a[n_] := BitOr @@ Table[BitAnd[k, n - k], {k, 0, n}]; Table[a[n], {n, 0, 73}] (* Jean-François Alcover, Jun 19 2012 *)

PARI
```
a(n) = n++; 1<Kevin Ryde, Apr 11 2023
```

a(2*n) = 2*2^floor(log_2(n)) - 1 = A003817(n).
a(2*n+1) = 2*a(n).
a(n) = A053644(n+1) - A006519(n+1). - Ridouane Oudra, Apr 09 2023

A115514 Triangle read by rows: row n >= 1 lists first n positive terms of A004526 (integers repeated) in decreasing order.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 1, 3, 3, 2, 2, 1, 1, 4, 3, 3, 2, 2, 1, 1, 4, 4, 3, 3, 2, 2, 1, 1, 5, 4, 4, 3, 3, 2, 2, 1, 1, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 6, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 7, 6, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 7, 7, 6, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1

Offset: 1

Roger L. Bagula, Mar 07 2006

T(n,k) = number of 2-element subsets of {1,2,...,n+2} such that the absolute difference of the elements is k+1, where 1 <= k < = n. E.g., T(7,3) = 3, the subsets are {1,5}, {2,6}, and {3,7}. - Christian Barrientos, Jun 27 2022

			Triangle begins as, for n >= 1, 1 <= k <= n,
  1;
  1, 1;
  2, 1, 1;
  2, 2, 1, 1;
  3, 2, 2, 1, 1;
  3, 3, 2, 2, 1, 1;
  4, 3, 3, 2, 2, 1, 1;
  ...

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A000012, A000073, A004526, A008805, A008967.

Cf. A002620 (row sums), A008805 (diagonal sums), A142150 (alternating row sums)

[Floor((n-k+2)/2): k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 14 2024

# Assuming offset 0:
Even := n -> (1 + (-1)^n)/2: # Iverson's even.
p := n -> add(add(Even(k)*x^j, j = 0..n-k), k = 0..n):
for n from 0 to 9 do seq(coeff(p(n), x, k), k=0..n) od; # Peter Luschny, Jun 03 2021

Table[Floor[(n-k+2)/2], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 14 2024 *)

flatten([[(n-k+2)//2 for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 14 2024

Sum_{k=1..n} T(n, k) = A002620(n+1) (row sums). - Gary W. Adamson, Oct 25 2007
T(n, k) = [x^k] p(n), where p(n) are partial Gaussian polynomials (A008967) defined by p(n) = Sum_{k=0..n} Sum_{j=0..n-k} even(k)*x^j, and even(k) = 1 if k is even and otherwise 0. We assume offset 0. - Peter Luschny, Jun 03 2021
T(n, k) = floor((n+2-k)/2). - Christian Barrientos, Jun 27 2022
From G. C. Greubel, Mar 14 2024: (Start)
T(n, k) = A128623(n, k)/n.
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A142150(n+1).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A008805(n-1).
Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = A002265(n+3). (End)

Edited by N. J. A. Sloane, Mar 23 2008 and Dec 15 2017

A142149 a(n) = XOR{k OR (n-k): 0<=k<=n}.

Original entry on oeis.org

0, 1, 3, 3, 6, 5, 5, 7, 12, 9, 15, 11, 10, 13, 9, 15, 24, 17, 27, 19, 30, 21, 29, 23, 20, 25, 23, 27, 18, 29, 17, 31, 48, 33, 51, 35, 54, 37, 53, 39, 60, 41, 63, 43, 58, 45, 57, 47, 40, 49, 43, 51, 46, 53, 45, 55, 36, 57, 39, 59, 34, 61, 33, 63, 96, 65, 99, 67, 102, 69, 101, 71

Offset: 0

Reinhard Zumkeller, Jul 15 2008

a(n) = XOR{k AND (n-k): 0<=k<=n}.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Reinhard Zumkeller, Logical Convolutions

Cf. A003817, A000004, A086099, A142150, A142151, A001477.

import Data.Bits (xor, (.|.))
a142149 :: Integer -> Integer
a142149 = foldl xor 0 . zipWith (.|.) [0..] . reverse . enumFromTo 1
-- Reinhard Zumkeller, Mar 31 2015

a(n)=if(n%2, n, bitxor(n, n/2)) \\ Charles R Greathouse IV, Jul 01 2022

def A142149(n): return n if n&1 else (n^ n>>1) # Chai Wah Wu, Jun 29 2022

a(2*n) = A048724(n) and a(2*n+1) = A005408(n).

A180714 Sum of the x- and y-coordinates of a point moving in a clockwise spiral.

Original entry on oeis.org

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

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jan 21 2011

sign

A spiral on the simple square grid is constructed starting at (0,0) and walking in the closest self-avoiding clockwise loop: up 1 unit, right 1 unit, down 2 units, left 2 units, up 3 units etc. The step widths in the x-coordinate are 0, 1, 0, -2, 0, 3, ... a signed version of A142150; the step widths in the y-coordinate are 1, 0, -2, 0, 3, ... The x-coordinate after n steps (n>=0) is a signed variant of A002265(n+3), namely 0, 0, 1, 1, -1, -1, 2, 2, -2, -2, 3, ...; the y-coordinate after n steps is 0, 1, 1, -1, -1, 2, 2, ... (n >= 0). The sum of the x- and y-coordinates after n steps (at corners of the spiral) is c(n) = 0, 1, 2, 0, -2, 1, 4, 0, -4, 1, 6, 0, -6, 1, 8, 0, ..., with g.f. -x*(1+x)/( (x-1)*(x^2+1)^2). The current sequence is obtained by recording the sum of the two coordinates at all intermediate positions walking with a stride of 1 along the edges of the spiral, equivalent to showing all interpolating integers between two values of c(n). The first differences a(n+1)-a(n) are two 1's, four -1's, six 1's, eight -1's etc., blocks of +1 and -1 with run lengths increasing by 2. - R. J. Mathar, Jan 22 2011

			Spiral begins at x=0, y=0, then moves up-right-down-left-up-right-...
a(0)=0+0=0, a(1)=1+0=1, a(2)=1+1=2, a(3)=0+1=1, a(4)=-1+1=0, a(5)=-1+0=-1, ...

Peter Kagey, Table of n, a(n) for n = 0..10000
Image of the spiral

Cf. A065358, A174344.

A191967 n * (numbers that are not divisible by 3).

Original entry on oeis.org

0, 1, 4, 12, 20, 35, 48, 70, 88, 117, 140, 176, 204, 247, 280, 330, 368, 425, 468, 532, 580, 651, 704, 782, 840, 925, 988, 1080, 1148, 1247, 1320, 1426, 1504, 1617, 1700, 1820, 1908, 2035, 2128, 2262, 2360, 2501, 2604, 2752, 2860, 3015, 3128, 3290, 3408

Offset: 0

Reinhard Zumkeller, Jul 07 2012

A033579 and A033570 interleaved.

Eric Weisstein's World of Mathematics, Pentagonal Number.
Wikipedia, Pentagonal number.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000326, A001651, A033570, A033579, A142150, A182079.

Haskell
```
a191967 n = n * a001651 n
```

A001651:=func; [n*A001651(n): n in [0..48]]; // Bruno Berselli, Jul 09 2012

Table[n (6 n - 3 - (-1)^n)/4, {n, 0, 48}] (* Bruno Berselli, Jul 09 2012 *)

a(n)=n\2*3*n+if(n%2,n,-n) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = n * A001651(n).
a(n) = A000326(n) - A142150(n).
a(2*n) = A033579(n) = 4 * A000326(n);
a(2*n+1) = A033570(n) = A000326(2*n+1).
G.f.: x*(1+3*x+6*x^2+2*x^3)/((1+x)^2*(1-x)^3). - Bruno Berselli, Jul 09 2012
a(n) = A182079(3n). - Bruno Berselli, Jul 09 2012
From Amiram Eldar, Feb 18 2022: (Start)
Sum_{n>=1} 1/a(n) = Pi/(4*sqrt(3)) + 9*log(3)/4 - 2*log(2).
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/4 + 3*log(3)/4 - 2*log(2). (End)

A242485 Possible values of A242480(n) in increasing order.

Original entry on oeis.org

0, 2, 3, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 35, 37, 39, 40, 41, 42, 43, 44, 45, 47, 49, 51, 52, 53, 54, 55, 56, 57, 59, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 83

Offset: 1

Jaroslav Krizek, May 27 2014

nonn

A242480(n) = (n*(n+1)/2) mod n + sigma(n) mod n + antisigma(n) mod n = A142150(n) + A054024(n) + A229110(n) = (A000217(n) mod n) + (A000203(n) mod n) + (A024816(n) mod n).

Supersequence of odd numbers > 1. Complement of A242486.

			16 is in the sequence because there is a number m such that A242480(m) = 16; m = 8.

Cf. A242480, A242481, A242482, A242483, A242484, A242486.

A242486 Numbers n such that A242480(x) = n has no solution.

Original entry on oeis.org

1, 4, 10, 14, 22, 26, 34, 36, 38, 46, 48, 50, 58, 60, 62, 74, 82, 84, 86, 90, 94, 98, 106, 108, 110, 118, 122, 130, 132, 134, 142, 146, 154, 156, 158, 166, 170, 178, 182, 190, 194, 202, 206, 210, 214, 218, 226, 230, 238, 242, 250, 252, 254, 262, 266, 270, 274

Offset: 1

Jaroslav Krizek, May 27 2014

nonn

A242480(n) = (n*(n+1)/2) mod n + sigma(n) mod n + antisigma(n) mod n = A142150(n) + A054024(n) + A229110(n) = (A000217(n) mod n) + (A000203(n) mod n) + (A024816(n) mod n).

All values of a(n) are even for n > 1. Complement of A242485.

			14 is in the sequence because there is no x whose A242480(x) = 14.

Cf. A242480, A242481, A242482, A242483, A242484, A242485.

A267120 Triangle of coefficients of Gaussian polynomials [2n+3,3]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where k = 0,1,...,g with g=3n.

Original entry on oeis.org

1, 0, -1, 1, 1, -1, 0, 5, -2, -4, 1, 1, 0, 2, -2, -15, 7, 17, -5, -7, 1, 1, 1, 0, -15, 6, 53, -23, -67, 22, 38, -8, -10, 1, 1, 0, -3, 3, 55, -28, -189, 81, 261, -90, -182, 46, 68, -11, -13, 1, 1, -1, 0, 30, -12, -229, 106, 691, -292, -1010, 359, 817, -229, -387, 79, 107, -14, -16, 1, 1

Offset: 0

Stephen O'Sullivan, Jan 10 2016

The entry a(n,k), n >= 0, k = 0,1,...,g, where g=3n, of this irregular triangle is the coefficient of (1+q^2)^k*q^(g-k) in the representation of the Gaussian polynomial [2n+3,3]q = Sum{k=0..g) a(n,k)*(1+q^2)^k*q^(g-k).
Row n is of length 3n+1.
The sequence arises in the formal derivation of the stability polynomial B(x) = Sum_{i=0..N} d_i T(iM,x) of rank N, and degree L, where T(iM,x) denotes the Chebyshev polynomial of the first kind of degree iM (A053120). The coefficients d_i are determined by order conditions on the stability polynomial.
Conjecture: More generally, the Gaussian polynomial [2*n+m+1-(m mod 2),m]q = Sum{k=0..g(m;n)} a(m;n,k)*(1+q^2)^k*q^(g(m;n)-k), for m >= 0, n >= 0, where g(m;n) = m*n if m is odd and (2*n+1)*m/2 if m is even, and the tabf array entries a(m;n,k) are the coefficients of the g.f. for the row n polynomials G(m;n,x) = (d^m/dt^m)G(m;n,t,x)/m!|{t=0}, with G(m;n,t,x) = (1+t)*Product{k=1..n+(m - m (mod 2))/2}(1 + t^2 + 2*t*T(k,x/2) (Chebyshev's T-polynomials). Hence a(m;n,k) = [x^k]G(m;n,x), for k=0..g(m;n). The present entry is the instance m = 3. (Thanks to Wolfdieter Lang for clarifying the text on the general prescription of a(m;n,k).)
From Robert Israel, Jan 15 2016: (Start)
a(n,0) = A056594(n).
a(n,1) = (-1)^((n+1)/2) * A142150(n+1).
a(2n,2) = 5*(-1)^(n+1)*A000217(n), a(2n+1,2) =(-1)^n*(n+1).
It appears that Sum_{j=0..k+1} C(k+1,j)*a(n+2*j,k) = 0.
(End)

			The irregular triangle a(n, k) begins:
n/k 0  1   2   3   4    5   6   7   8   9  10 11 12
0:  1
1:  0 -1   1   1
2: -1  0   5  -2  -4   1    1
3:  0  2  -2 -15   7   17  -5  -7   1   1
4:  1  0 -15   6  53  -23 -67  22  38  -8 -10  1  1
...
Row n=5: 0 -3   3  55 -28 -189  81 261 -90 -182  46 68 -11 -13 1 1;
Row n=6: -1  0 30 -12 -229  106 691 -292 -1010 359 817 -229 -387 79 107 -14 -16 1 1.
Row n=7: 0 4 -4 -134 70 896 -416 -2561 1073 3903 -1415 -3529 1057 1991 -467 -709 121 155 -17 -19 1 1.
... Reformatted and extended. - _Wolfdieter Lang_, Feb 13 2016

Stephen O'Sullivan, Table of n, a(n) for n = 0..1425
S. O'Sullivan, A class of high-order Runge-Kutta-Chebyshev stability polynomials, Journal of Computational Physics, 300 (2015), 665-678.
Wikipedia, Gaussian binomial coefficients.

Cf. A000217. A056594, A142150, A267482, A267483, A267484, A267485, A267486.

A267120 := proc (n, k) local y: y := expand(subs(t = 0, diff((1+t)*product(1+t^2+2*t*ChebyshevT(i, x/2), i = 1 .. n+1),t$3)/3!)): if k = 0 then subs(x = 0, y) else subs(x = 0, diff(y, x$k)/k!) end if: end proc: seq(seq(A267120(n, k), k = 0 .. 3*n), n = 0 .. 20);
# More efficient:
N:= 20: # to get rows 0 to N
P[0]:= (1+t)*(t^2 + t*x + 1):
B[0]:= 1:
for n from 1 to N do
  P[n]:= expand(series(P[n-1]*(1+t^2+2*t*orthopoly[T](n+1,x/2)),t,4));
  B[n]:= coeff(P[n],t,3);
od:
seq(seq(coeff(B[n],x,j),j=0..3*n),n=0..N); # Robert Israel, Jan 15 2016

row[n_] := 1/3! D[(1+t)*Product[1+t^2+2*t*ChebyshevT[i, x/2], {i, 1, n+1}], {t, 3}] /. t -> 0 // CoefficientList[#, x]&; Table[row[n], {n, 0, 6}] // Flatten (* Jean-François Alcover, Jan 16 2016 *)

G.f. for row polynomial: G(n,x) = (d^3/dt^3)((1+t)*Product_{i=1..n+1}(1+t^2+2t*T(i,x/2))/3!)|_{t=0}.

cp's OEIS Frontend

A242483 Numbers n such that A242481(n) = ((n*(n+1)/2) mod n + sigma(n) mod n + antisigma(n) mod n) / n = 2.

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A276457 a(n) is the number of times that a(n-1) appears in the concatenation of all numbers from a(0) to a(n-2), with a(0) = 0.

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A086099 a(n) = OR(k AND (n-k): 0<=k<=n), AND and OR bitwise.

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A115514 Triangle read by rows: row n >= 1 lists first n positive terms of A004526 (integers repeated) in decreasing order.

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A142149 a(n) = XOR{k OR (n-k): 0<=k<=n}.

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A180714 Sum of the x- and y-coordinates of a point moving in a clockwise spiral.

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A191967 n * (numbers that are not divisible by 3).

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