A289813 -id:A289813 - cp's OEIS frontend

A293226 Restricted growth sequence transform of A293225, a filter combining two products, the other formed from the 1-digits (A293221) and the other from the 2-digits (A293222) present in the ternary expansions of proper divisors of n.

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 4, 17, 18, 19, 2, 20, 2, 21, 22, 23, 24, 25, 2, 26, 27, 28, 2, 29, 2, 30, 31, 32, 2, 33, 34, 35, 12, 36, 2, 37, 38, 39, 40, 41, 2, 42, 2, 43, 44, 45, 46, 47, 2, 48, 49, 50, 2, 51, 2, 52, 53, 54, 55, 56, 2, 57, 58, 59, 2, 60, 61, 62, 63, 64, 2, 65, 66, 67, 68, 69, 70, 71, 2, 72

Offset: 1

Antti Karttunen, Oct 03 2017

nonn

For all i, j: a(i) = a(j) => A001065(i) = A001065(j).

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A001065, A293221, A293222, A293223, A293224, A293225.

Cf. also A290093, A290094, A293215, A293217, A293232.

rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A289813(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); };
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); };
A293221(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289813(d)))); m; };
A293222(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289814(d)))); m; };
Anot_submitted(n) = (1/2)*(2 + ((A293222(n) + A293221(n))^2) - A293222(n) - 3*A293221(n)); \\ Eq.class-wise equal to A293225.
write_to_bfile(1,rgs_transform(vector(19683,n,Anot_submitted(n))),"b293226.txt");

A290093 Compound filter (for base-3 digit runlengths): a(n) = P(A290091(n), A290092(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 3, 2, 3, 10, 5, 2, 5, 7, 3, 21, 5, 10, 36, 14, 5, 27, 12, 2, 5, 16, 5, 14, 23, 7, 12, 29, 3, 21, 5, 21, 78, 27, 5, 27, 12, 10, 78, 14, 36, 136, 44, 14, 90, 25, 5, 27, 23, 27, 90, 61, 12, 42, 38, 2, 5, 16, 5, 14, 23, 16, 23, 67, 5, 27, 23, 14, 44, 40, 23, 61, 80, 7, 12, 67, 12, 25, 80, 29, 38, 121, 3, 21, 5, 21, 78, 27, 5, 27, 12, 21, 465, 27, 78, 300, 90, 27

Offset: 0

Antti Karttunen, Jul 25 2017

nonn

For all i, j: a(i) = a(j) => A006047(i) = A006047(j) => A053735(i) = A053735(j).

Antti Karttunen, Table of n, a(n) for n = 0..19683

Cf. A000027, A278222, A289813, A289814, A290091, A290092.

Cf. A006047, A053735, A290079 (some of the matched sequences).

(define (A290093 n) (* (/ 1 2) (+ (expt (+ (A290091 n) (A290092 n)) 2) (- (A290091 n)) (- (* 3 (A290092 n))) 2)))

a(n) = (1/2)*(2 + ((A290091(n)+A290092(n))^2) - A290091(n) - 3*A290092(n)).

A304759 Binary encoding of 1-digits in ternary representation of A048673(n).

Original entry on oeis.org

1, 0, 2, 2, 3, 0, 0, 6, 7, 4, 1, 2, 4, 4, 0, 14, 5, 12, 6, 10, 9, 0, 4, 6, 1, 0, 4, 10, 5, 8, 1, 30, 8, 8, 14, 26, 2, 8, 13, 22, 3, 16, 0, 2, 17, 12, 8, 14, 1, 0, 10, 2, 10, 0, 9, 22, 3, 8, 11, 18, 9, 0, 18, 62, 0, 20, 12, 18, 1, 24, 13, 54, 15, 0, 28, 18, 0, 24, 12, 46, 37, 4, 8, 34, 7, 4, 0, 6, 11, 32, 23, 26, 22, 0

Offset: 1

Antti Karttunen, May 30 2018

Compare the logarithmic scatterplot to those of A291759, A292250 and A304760.

Antti Karttunen, Table of n, a(n) for n = 1..16383
Index entries for sequences related to binary expansion of n

Cf. A048673, A289813, A304758 (rgs-transform), A340381.
Cf. also A291759, A292250, A304760, A305295.
Cf. A340376 (positions of zeros), A340378 (binary weight).

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A048673(n) = (A003961(n)+1)/2;
A289813(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From A289813
A304759(n) = A289813(A048673(n));

a(n) = A289813(A048673(n)).

A291770 A binary encoding of the zeros in ternary representation of n.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 3, 2, 2, 1, 0, 0, 1, 0, 0, 3, 2, 2, 1, 0, 0, 1, 0, 0, 7, 6, 6, 5, 4, 4, 5, 4, 4, 3, 2, 2, 1, 0, 0, 1, 0, 0, 3, 2, 2, 1, 0, 0, 1, 0, 0, 7, 6, 6, 5, 4, 4, 5, 4, 4, 3, 2, 2, 1, 0, 0, 1, 0, 0, 3, 2, 2, 1, 0, 0, 1, 0, 0, 15, 14, 14, 13, 12, 12, 13, 12, 12, 11, 10, 10, 9, 8, 8, 9, 8, 8, 11, 10, 10, 9, 8, 8, 9, 8, 8, 7, 6, 6

Offset: 1

Antti Karttunen, Sep 11 2017

The ones in the binary representation of a(n) correspond to the nonleading zeros in the ternary representation of n. For example: ternary(33) = 1020 and binary(a(33)) = 101 (a(33) = 5).

			   n      a(n)    ternary(n)  binary(a(n))
                  A007089(n)  A007088(a(n))
  --      ----    ----------  ------------
   1        0            1           0
   2        0            2           0
   3        1           10           1
   4        0           11           0
   5        0           12           0
   6        1           20           1
   7        0           21           0
   8        0           22           0
   9        3          100          11
  10        2          101          10
  11        2          102          10
  12        1          110           1
  13        0          111           0
  14        0          112           0
  15        1          120           1
  16        0          121           0
  17        0          122           0
  18        3          200          11
  19        2          201          10
  20        2          202          10
  21        1          210           1
  22        0          211           0
  23        0          212           0
  24        1          220           1
  25        0          221           0
  26        0          222           0
  27        7         1000         111
  28        6         1001         110
  29        6         1002         110
  30        5         1010         101

Antti Karttunen, Table of n, a(n) for n = 1..59049

Cf. A007088, A007089, A077267, A289813, A289814, A291771.

Table[FromDigits[IntegerDigits[n, 3] /. k_ /; k < 3 :> If[k == 0, 1, 0], 2], {n, 110}] (* Michael De Vlieger, Sep 11 2017 *)

from sympy.ntheory.factor_ import digits
def a(n):
    k=digits(n, 3)[1:]
    return int("".join('1' if i==0 else '0' for i in k), 2)
print([a(n) for n in range(1, 111)]) # Indranil Ghosh, Sep 21 2017

(define (A291770 n) (if (zero? n) n (let loop ((n n) (b 1) (s 0)) (if (< n 3) s (let ((d (modulo n 3))) (if (zero? d) (loop (/ n 3) (+ b b) (+ s b)) (loop (/ (- n d) 3) (+ b b) s)))))))

For all n >= 0, a(A000244(n)) = A000225(n), that is, a(3^n) = (2^n) - 1. [The records in the sequence].
For all n >= 1, A000120(a(n)) = A077267(n).
For all n >= 1, A278222(a(n)) = A291771(n).

A293223 Restricted growth sequence transform of A293221, a product formed from the 1-digits present in the ternary expansion of proper divisors of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 4, 3, 3, 2, 5, 2, 6, 7, 4, 2, 8, 2, 9, 4, 10, 2, 11, 3, 12, 8, 9, 2, 13, 2, 14, 8, 10, 4, 15, 2, 6, 16, 9, 2, 11, 2, 9, 17, 3, 2, 18, 6, 14, 8, 9, 2, 19, 8, 20, 4, 21, 2, 22, 2, 23, 16, 24, 16, 25, 2, 26, 7, 27, 2, 28, 2, 29, 16, 26, 30, 31, 2, 32, 19, 19, 2, 33, 8, 29, 34, 27, 2, 35, 14, 36, 37, 21, 4, 38, 2, 24, 39, 40, 2, 41, 2, 20, 42

Offset: 1

Antti Karttunen, Oct 03 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A293221.

Cf. also A290091, A293224, A293225, A293226, A293215, A293217, A293232.

rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A289813(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From Remy Sigrist
A293221(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289813(d)))); m; };
write_to_bfile(1,rgs_transform(vector(19683,n,A293221(n))),"b293223.txt");

A244042 In ternary representation of n, replace 2's with 0's.

Original entry on oeis.org

0, 1, 0, 3, 4, 3, 0, 1, 0, 9, 10, 9, 12, 13, 12, 9, 10, 9, 0, 1, 0, 3, 4, 3, 0, 1, 0, 27, 28, 27, 30, 31, 30, 27, 28, 27, 36, 37, 36, 39, 40, 39, 36, 37, 36, 27, 28, 27, 30, 31, 30, 27, 28, 27, 0, 1, 0, 3, 4, 3, 0, 1, 0, 9, 10, 9, 12, 13, 12, 9, 10, 9

Offset: 0

Joonas Pohjonen, Jun 17 2014

			16 = 121_3, replacing 2 with 0 gives 101_3 = 10, so a(16) = 10.

Alois P. Heinz, Table of n, a(n) for n = 0..6560

Cf. A004488, A117966, A005836, A289813, A289814.

a:= proc(n) local t, r, i; t, r:= n, 0;
      for i from 0 while t>0 do
        r:= r+3^i*(d-> `if`(d=2, 0, d))(irem(t, 3, 't'))
      od; r
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Jun 17 2014

Array[FromDigits[IntegerDigits[#, 3] /. 2 -> 0, 3] &, 72, 0] (* Michael De Vlieger, Mar 17 2018 *)

a(n) = my(d=digits(n, 3)); fromdigits(apply(x->(if (x==2, 0, x)), d), 3); \\ Michel Marcus, Jun 10 2017

from sympy.ntheory.factor_ import digits
def a(n):return int("".join(map(str, digits(n, 3)[1:])).replace('2', '0'), 3) # Indranil Ghosh, Jun 10 2017

a(n) = n - 2 * A005836(A289814(n) + 1) = A005836(A289813(n) + 1). - Andrey Zabolotskiy, Nov 11 2019

A292245 Base-2 expansion of a(n) encodes the steps where numbers of the form 3k+1 are encountered when map x -> A253889(x) is iterated down to 1, starting from x=n.

Original entry on oeis.org

1, 2, 2, 5, 4, 4, 11, 4, 8, 17, 10, 18, 9, 8, 22, 17, 8, 8, 17, 22, 36, 41, 8, 42, 17, 16, 44, 21, 34, 32, 35, 20, 32, 33, 36, 64, 69, 18, 34, 73, 16, 74, 37, 44, 82, 33, 34, 34, 89, 16, 64, 69, 16, 68, 65, 34, 64, 33, 44, 64, 33, 72, 16, 65, 82, 68, 85, 16, 128, 137, 84, 72, 69, 34, 138, 145, 32, 84, 145, 88, 88, 149, 42, 162, 65, 68, 164, 45, 64

Offset: 1

Antti Karttunen, Sep 15 2017

			For n=1 (the termination value of the iteration), 1 is of the form 3k+1, thus a(1) = 1*(2^0) = 1.
For n=2, 2 is not of the form 3k+1, while A253889(2) = 1 is, thus a(2) = 0*(2^0) + 1*2(^1) = 2.
For n=4, 4 is of the form 3k+1, while A253889(4) = 2 is not, but then A253889(2) = 1 again is, thus a(4) = 1*(2^0) + 0*(2^1) + 1*(2^2) = 5.

Cf. A048673, A064216, A289813, A292243, A292244, A292246, A292248.

f[n_] := Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1]; g[n_] := (Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n; Map[FromDigits[#, 2] &[IntegerDigits[#, 3] /. 2 -> 0] &, Array[a, 98]] (* Michael De Vlieger, Sep 16 2017 *)

a(1) = 1; for n > 1, a(n) = 2*a(A253889(n)) + [n ≡ 1 (mod 3)], where the last part of the formula is Iverson bracket, giving 1 only if n is of the form 3k+1, and 0 otherwise.
a(n) = A289813(A292243(n)).
Other identities. For all n >= 1:
a(A048673(n)) = A292248(n).
a(n) + A292244(n) = A064216(n).
a(n) AND A292244(n) = a(n) AND A292246(n) = 0, where AND is a bitwise-AND (A004198).

A304740 Restricted growth sequence transform of A304760(n), formed from 1-digits in ternary representation of A254103(n).

Original entry on oeis.org

1, 2, 1, 3, 3, 4, 1, 1, 5, 6, 6, 2, 3, 7, 6, 8, 9, 4, 1, 6, 10, 1, 1, 5, 5, 11, 11, 3, 10, 12, 12, 8, 13, 6, 6, 14, 3, 15, 11, 1, 16, 17, 12, 14, 3, 17, 12, 2, 9, 18, 19, 2, 20, 4, 1, 12, 16, 4, 1, 11, 21, 12, 12, 2, 22, 4, 1, 23, 10, 19, 19, 14, 5, 24, 24, 2, 20, 7, 6, 25, 26, 23, 23, 3, 21, 19, 19, 25, 5, 23, 23, 10, 21, 1, 1, 10, 13, 27, 27, 21, 28, 1, 1

Offset: 0

Antti Karttunen, May 29 2018

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A254103, A289813, A304760.

Cf. also A304746.

A254103(n) = if(!n,n,if(!(n%2),(3*A254103(n/2))-1,(3*(1+A254103((n-1)/2)))\2));
A289813(n) = { my (d=digits(n, 3)); from digits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From A289813
A304760(n) = A289813(A254103(n));
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
v304740 = rgs_transform(vector(65538,n,A304760(n-1)));
A304740(n) = v304740[1+n];

A292371 A binary encoding of 1-digits in the base-4 representation of n.

Original entry on oeis.org

0, 1, 0, 0, 2, 3, 2, 2, 0, 1, 0, 0, 0, 1, 0, 0, 4, 5, 4, 4, 6, 7, 6, 6, 4, 5, 4, 4, 4, 5, 4, 4, 0, 1, 0, 0, 2, 3, 2, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 3, 2, 2, 0, 1, 0, 0, 0, 1, 0, 0, 8, 9, 8, 8, 10, 11, 10, 10, 8, 9, 8, 8, 8, 9, 8, 8, 12, 13, 12, 12, 14, 15, 14, 14, 12, 13, 12, 12, 12, 13, 12, 12, 8, 9, 8, 8, 10, 11, 10, 10, 8, 9, 8, 8, 8, 9, 8, 8, 8

Offset: 0

Antti Karttunen, Sep 15 2017

			   n      a(n)     base-4(n)  binary(a(n))
                  A007090(n)  A007088(a(n))
  --      ----    ----------  ------------
   1        1          1           1
   2        0          2           0
   3        0          3           0
   4        2         10          10
   5        3         11          11
   6        2         12          10
   7        2         13          10
   8        0         20           0
   9        1         21           1
  10        0         22           0
  11        0         23           0
  12        0         30           0
  13        1         31           1
  14        0         32           0
  15        0         33           0
  16        4        100         100
  17        5        101         101
  18        4        102         100

Antti Karttunen, Table of n, a(n) for n = 0..65536
Rémy Sigrist, Interactive scatterplot of (a(n), A292372(n), A292373(n)) for n=0..4^8-1 [provided your web browser supports the Plotly library, you should see icons on the top right corner of the page: if you choose "Orbital rotation", then you will be able to rotate the plot alongside three axes, the 3D plot here corresponds to a Sierpiński triangle-based pyramid]
Index entries for sequences related to binary expansion of n

Cf. A003188, A004198, A007088, A007090, A059905, A160381, A292272, A292370, A292372, A292373.

Cf. A289813 (analogous sequence for base 3).

Table[FromDigits[IntegerDigits[n, 4] /. k_ /; IntegerQ@ k :> If[k == 1, 1, 0], 2], {n, 0, 112}] (* Michael De Vlieger, Sep 21 2017 *)

from sympy.ntheory.factor_ import digits
def a(n):
    k=digits(n, 4)[1:]
    return 0 if n==0 else int("".join('1' if i==1 else '0' for i in k), 2)
print([a(n) for n in range(116)]) # Indranil Ghosh, Sep 21 2017

def A292371(n): return int(bin(n&~(n>>1))[:1:-2][::-1],2) # Chai Wah Wu, Jun 30 2022

a(n) = A059905(A292272(n)) = A059905(n AND A003188(n)), where AND is bitwise-AND (A004198).

For all n >= 0, A000120(a(n)) = A160381(n).

A293225 Compound filter: a(n) = P(A293224(n), A293223(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 2, 2, 5, 2, 8, 2, 12, 4, 13, 2, 32, 2, 40, 30, 33, 2, 59, 2, 58, 42, 69, 2, 143, 8, 80, 29, 83, 2, 178, 2, 197, 38, 96, 25, 239, 2, 100, 121, 163, 2, 221, 2, 202, 194, 103, 2, 448, 61, 365, 59, 245, 2, 333, 48, 576, 187, 256, 2, 720, 2, 278, 546, 718, 138, 606, 2, 503, 114, 1009, 2, 1101, 2, 437, 651, 678, 532, 831, 2, 1400, 172, 213, 2, 1508, 71, 500, 597

Offset: 1

Antti Karttunen, Oct 03 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A000027, A019565, A293221, A293222, A293223, A293224, A293226 (rgs-version of this filter).

Cf. also A289813, A289814, A290093, A290094.

rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A289813(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); };
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); };
A293221(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289813(d)))); m; };
A293222(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289814(d)))); m; };
v293223 = rgs_transform(vector(19683,n,A293221(n)));
A293223(n) = v293223[n];
v293224 = rgs_transform(vector(19683,n,A293222(n)));
A293224(n) = v293224[n];
A293225(n) = (1/2)*(2 + ((A293224(n) + A293223(n))^2) - A293224(n) - 3*A293223(n));

(define (A293225 n) (* 1/2 (+ (expt (+ (A293224 n) (A293223 n)) 2) (- (A293224 n)) (- (* 3 (A293223 n))) 2)))

a(n) = (1/2)*(2 + ((A293224(n) + A293223(n))^2) - A293224(n) - 3*A293223(n)).

cp's OEIS Frontend

A293226 Restricted growth sequence transform of A293225, a filter combining two products, the other formed from the 1-digits (A293221) and the other from the 2-digits (A293222) present in the ternary expansions of proper divisors of n.

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A290093 Compound filter (for base-3 digit runlengths): a(n) = P(A290091(n), A290092(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A304759 Binary encoding of 1-digits in ternary representation of A048673(n).

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A291770 A binary encoding of the zeros in ternary representation of n.

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A293223 Restricted growth sequence transform of A293221, a product formed from the 1-digits present in the ternary expansion of proper divisors of n.

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A244042 In ternary representation of n, replace 2's with 0's.

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Formula

A292245 Base-2 expansion of a(n) encodes the steps where numbers of the form 3k+1 are encountered when map x -> A253889(x) is iterated down to 1, starting from x=n.

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Programs

Mathematica

Formula

A304740 Restricted growth sequence transform of A304760(n), formed from 1-digits in ternary representation of A254103(n).

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