A027423 -id:A027423 - cp's OEIS frontend

A335407 Number of anti-run permutations of the prime indices of n!.

1, 1, 1, 2, 0, 2, 3, 54, 0, 30, 105, 6090, 1512, 133056, 816480, 127209600, 0, 10090080, 562161600, 69864795000, 49989139200, 29593652088000, 382147120555200, 41810689605484800, 4359985823793600, 3025062801079038720, 49052072750637116160, 25835971971637227375360

Offset: 0

Gus Wiseman, Jul 01 2020

nonn

An anti-run is a sequence with no adjacent equal parts.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Conjecture: Only vanishes at n = 4 and n = 8.
a(16) = 0. Proof: 16! = 2^15 * m where bigomega(m) = A001222(m) = 13. We can't separate 15 1's with 13 other numbers. - David A. Corneth, Jul 04 2020

			The a(0) = 1 through a(6) = 3 anti-run permutations:
  ()  ()  (1)  (1,2)  .  (1,2,1,3,1)  (1,2,1,2,1,3,1)
               (2,1)     (1,3,1,2,1)  (1,2,1,3,1,2,1)
                                      (1,3,1,2,1,2,1)

Andrew Howroyd, Table of n, a(n) for n = 0..200

The version for Mersenne numbers is A335432.
Anti-run compositions are A003242.
Anti-run patterns are counted by A005649.
Permutations of prime indices are A008480.
Anti-runs are ranked by A333489.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Strict permutations of prime indices are A335489.
Cf. A056239, A106351, A112798, A114938, A278990, A292884, A325535, A335125.
Factorial numbers: A000142, A001222, A002982, A007489, A022559, A027423, A054991, A108731, A181819, A181821, A325272, A325273, A325617.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n!]],!MatchQ[#,{_,x_,x_,_}]&]],{n,0,10}]

\\ See A335452 for count.
a(n)={count(factor(n!)[,2])} \\ Andrew Howroyd, Feb 03 2021

a(n) = A335452(A000142(n)). - Andrew Howroyd, Feb 03 2021

Terms a(14) and beyond from Andrew Howroyd, Feb 03 2021

A337071 Number of strict chains of divisors starting with n!.

Original entry on oeis.org

1, 1, 2, 6, 40, 264, 3776, 40256, 1168000, 34204032, 1107791872, 23233380352, 1486675898368, 38934372315136, 1999103691427840, 132874800979423232, 20506322412604129280, 776179999255323115520, 107455579038104865996800, 4651534843901106606571520, 731092060557632280262082560

Offset: 0

Gus Wiseman, Aug 16 2020

nonn

			The a(1) = 1 through a(3) = 6 chains:
  1  2    6
     2/1  6/1
          6/2
          6/3
          6/2/1
          6/3/1
The a(4) = 40 chains:
  24  24/1   24/2/1   24/4/2/1   24/8/4/2/1
      24/2   24/3/1   24/6/2/1   24/12/4/2/1
      24/3   24/4/1   24/6/3/1   24/12/6/2/1
      24/4   24/4/2   24/8/2/1   24/12/6/3/1
      24/6   24/6/1   24/8/4/1
      24/8   24/6/2   24/8/4/2
      24/12  24/6/3   24/12/2/1
             24/8/1   24/12/3/1
             24/8/2   24/12/4/1
             24/8/4   24/12/4/2
             24/12/1  24/12/6/1
             24/12/2  24/12/6/2
             24/12/3  24/12/6/3
             24/12/4
             24/12/6

A325617 is the maximal case.
A337070 is the version for superprimorials.
A337074 counts the case with distinct prime multiplicities.
A337105 is the case ending with one.
A000005 counts divisors.
A000142 lists factorial numbers.
A027423 counts divisors of factorial numbers.
A067824 counts chains of divisors starting with n.
A074206 counts chains of divisors from n to 1.
A076716 counts factorizations of factorial numbers.
A253249 counts chains of divisors.
Cf. A001013, A001055, A002033, A022559, A124010, A167865, A336941, A336942.

chnsc[n_]:=Prepend[Join@@Table[Prepend[#,n]&/@chnsc[d],{d,Most[Divisors[n]]}],{n}];
Table[Length[chnsc[n!]],{n,0,5}]

a(n) = 2*A337105(n) for n > 1.

a(n) = A067824(n!).

a(19)-a(20) from Alois P. Heinz, Aug 23 2020

A095149 Triangle read by rows: Aitken's array (A011971) but with a leading diagonal before it given by the Bell numbers (A000110), 1, 1, 2, 5, 15, 52, ...

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 5, 2, 3, 5, 15, 5, 7, 10, 15, 52, 15, 20, 27, 37, 52, 203, 52, 67, 87, 114, 151, 203, 877, 203, 255, 322, 409, 523, 674, 877, 4140, 877, 1080, 1335, 1657, 2066, 2589, 3263, 4140, 21147, 4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147

Offset: 0

Gary W. Adamson, May 30 2004

Or, prefix Aitken's array (A011971) with a leading diagonal of 0's and take the differences of each row to get the new triangle.
With offset 1, triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} in which k is the largest entry in the block containing 1 (1 <= k <= n). - Emeric Deutsch, Oct 29 2006
Row term sums = the Bell numbers starting with A000110(1): 1, 2, 5, 15, ...
The k-th term in the n-th row is the number of permutations of length n starting with k and avoiding the dashed pattern 23-1. Equivalently, the number of permutations of length n ending with k and avoiding 1-32. - Andrew Baxter, Jun 13 2011
From Gus Wiseman, Aug 11 2020: (Start)
Conjecture: Also the number of divisors d with distinct prime multiplicities of the superprimorial A006939(n) that are of the form d = m * 2^k where m is odd. For example, row n = 4 counts the following divisors:
  1     2     4    8     16
  3     18    12   24    48
  5     50    20   40    80
  7     54    28   56    112
  9     1350  108  72    144
  25          540  200   400
  27          756  360   432
  45               504   720
  63               600   1008
  75               1400  1200
  135                    2160
  175                    2800
  189                    3024
  675                    10800
  4725                   75600
Equivalently, T(n,k) is the number of length-n vectors 0 <= v_i <= i whose nonzero values are distinct and such that v_n = k.
Crossrefs:
A008278 is the version counted by omega A001221.
A336420 is the version counted by Omega A001222.
A006939 lists superprimorials or Chernoff numbers.
A008302 counts divisors of superprimorials by Omega.
A022915 counts permutations of prime indices of superprimorials.
A098859 counts partitions with distinct multiplicities.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
Cf. A000005, A000142, A027423, A076954, A124010, A146291, A181818, A336417, A336419, A336421, A336499, A336942.
(End)

			Triangle starts:
   1;
   1,  1;
   2,  1,  2;
   5,  2,  3,  5;
  15,  5,  7, 10, 15;
  52, 15, 20, 27, 37, 52;
From _Gus Wiseman_, Aug 11 2020: (Start)
Row n = 3 counts the following set partitions (described in Emeric Deutsch's comment above):
  {1}{234}      {12}{34}    {123}{4}    {1234}
  {1}{2}{34}    {12}{3}{4}  {13}{24}    {124}{3}
  {1}{23}{4}                {13}{2}{4}  {134}{2}
  {1}{24}{3}                            {14}{23}
  {1}{2}{3}{4}                          {14}{2}{3}
(End)

Alois P. Heinz, Rows n = 0..150, flattened (first 51 rows from Chai Wah Wu)
Andrew M. Baxter and Lara K. Pudwell, Enumeration schemes for dashed patterns, arXiv:1108.2642 [math.CO], 2011.
Anders Claesson, Generalized pattern avoidance, Europ. J. Combin., 22 7 (2001), 961-971. See Proposition 3.
A. Bernini, M. Bouvel and L. Ferrari, Some statistics on permutations avoiding generalized patterns, PU.M.A. Vol. 18 (2007), No. 3-4, pp. 223-237 (see transposed array p. 227).

Cf. A000110, A005493, A008278, A011971, A188919, A271466.

T(2n,n) gives A020556.

with(combinat): T:=proc(n,k) if k=1 then bell(n-1) elif k=2 and n>=2 then bell(n-2) elif k<=n then add(binomial(k-2,i)*bell(n-2-i),i=0..k-2) else 0 fi end: matrix(8,8,T): for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
Q[1]:=t*s: for n from 2 to 11 do Q[n]:=expand(t^n*subs(t=1,Q[n-1])+s*diff(Q[n-1],s)-Q[n-1]+s*Q[n-1]) od: for n from 1 to 11 do P[n]:=sort(subs(s=1,Q[n])) od: for n from 1 to 11 do seq(coeff(P[n],t,k),k=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Oct 29 2006
A011971 := proc(n,k) option remember ; if k = 0 then if n=0 then 1; else A011971(n-1,n-1) ; fi ; else A011971(n,k-1)+A011971(n-1,k-1) ; fi ; end: A000110 := proc(n) option remember; if n<=1 then 1 ; else add( binomial(n-1,i)*A000110(n-1-i),i=0..n-1) ; fi ; end: A095149 := proc(n,k) option remember ; if k = 0 then A000110(n) ; else A011971(n-1,k-1) ; fi ; end: for n from 0 to 11 do for k from 0 to n do printf("%d, ",A095149(n,k)) ; od ; od ; # R. J. Mathar, Feb 05 2007
# alternative Maple program:
b:= proc(n, m, k) option remember; `if`(n=0, 1, add(
      b(n-1, max(j, m), max(k-1, -1)), j=`if`(k=0, m+1, 1..m+1)))
    end:
T:= (n, k)-> b(n, 0, n-k):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Dec 20 2018

nmax = 10; t[n_, 1] = t[n_, n_] = BellB[n-1]; t[n_, 2] = BellB[n-2]; t[n_, k_] /; n >= k >= 3 := t[n, k] = t[n, k-1] + t[n-1, k-1]; Flatten[ Table[ t[n, k], {n, 1, nmax}, {k, 1, n}]] (* Jean-François Alcover, Nov 15 2011, from formula with offset 1 *)

# requires Python 3.2 or higher.
from itertools import accumulate
A095149_list, blist = [1,1,1], [1]
for _ in range(2*10**2):
    b = blist[-1]
    blist = list(accumulate([b]+blist))
    A095149_list += [blist[-1]]+ blist
# Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 20 2014

With offset 1, T(n,1) = T(n,n) = T(n+1,2) = B(n-1) = A000110(n-1) (the Bell numbers). T(n,k) = T(n,k-1) + T(n-1,k-1) for n >= k >= 3. T(n,n-1) = B(n-1) - B(n-2) = A005493(n-3) for n >= 3 (B(q) are the Bell numbers A000110). T(n,k) = A011971(n-2,k-2) for n >= k >= 2. In other words, deleting the first row and first column we obtain Aitken's array A011971 (also called Bell or Pierce triangle; offset in A011971 is 0). - Emeric Deutsch, Oct 29 2006

T(n,1) = B(n-1); T(n,2) = B(n-2) for n >= 2; T(n,k) = Sum_{i=0..k-2} binomial(k-2,i)*B(n-2-i) for 3 <= k <= n, where B(q) are the Bell numbers (A000110). Generating polynomial of row n is P[n](t) = Q[n](t,1), where Q[n](t,s) = t^n*Q[n-1](1,s) + s*dQ[n-1](t,s)/ds + (s-1) Q[n-1](t,s); Q[1](t,s) = ts. - Emeric Deutsch, Oct 29 2006

Corrected and extended by R. J. Mathar, Feb 05 2007

Entry revised by N. J. A. Sloane, Jun 01 2005, Jun 16 2007

A336421 Number of ways to choose a divisor of a divisor, both having distinct prime exponents, of the n-th superprimorial number A006939(n).

Original entry on oeis.org

1, 3, 13, 76, 571, 5309, 59341, 780149

Offset: 0

Gus Wiseman, Jul 25 2020

A number has distinct prime exponents iff its prime signature is strict.

The n-th superprimorial or Chernoff number is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1).

			The a(2) = 13 ways:
  12/1/1  12/2/1  12/3/1  12/4/1  12/12/1
          12/2/2  12/3/3  12/4/2  12/12/2
                          12/4/4  12/12/3
                                  12/12/4
                                  12/12/12

A000258 shifted once to the left is dominated by this sequence.
A336422 is the generalization to non-superprimorials.
A000110 counts divisors of superprimorials with distinct prime exponents.
A006939 lists superprimorials or Chernoff numbers.
A008302 counts divisors of superprimorials by bigomega.
A022915 counts permutations of prime indices of superprimorials.
A076954 can be used instead of A006939.
A130091 lists numbers with distinct prime exponents.
A181796 counts divisors with distinct prime exponents.
A181818 gives products of superprimorials.
A317829 counts factorizations of superprimorials.
Cf. A000005, A008278, A027423, A071625, A118914, A124010, A327498, A336417, A336419, A336420, A336426, A336500, A336568.

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
strsig[n_]:=UnsameQ@@Last/@FactorInteger[n];
Table[Total[Cases[Divisors[chern[n]],d_?strsig:>Count[Divisors[d],e_?strsig]]],{n,0,5}]

A079202 Number of isomorphism classes of non-associative non-commutative non-anti-associative non-anti-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 0, 0, 0, 32, 2155, 0, 0, 0, 12, 60, 184, 34544, 147032271

Offset: 1

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

A079202(n)+A079203(n)+A079204(n)+A079205(n)+A079197(n)+A079207(n)+A079208(n)+A079209(n)+A063524(n)=A079171(n)
Elements per row: 1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)
First four rows: 0; 0,0; 0,0,32,2155; 0,0,0,12,60,184,34544,147032271
A079230(x) is equal to the sum of the products of each element in row x of this sequence and the corresponding element of A079210.
The sum of each row x of this sequence is given by A079231(x).

C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to groupoids

Cf. A079203, A079204, A079205, A079197, A079207, A079208, A079209, A079230, A079231.

A079203 Number of isomorphism classes of non-associative non-commutative non-anti-associative anti-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 2, 0, 6, 34, 952, 0, 1, 12, 6, 181, 283, 13333, 31839187

Offset: 1

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

A079202(n)+A079203(n)+A079204(n)+A079205(n)+A079197(n)+A079207(n)+A079208(n)+A079209(n)+A063524(n)=A079171(n)
Elements per row: 1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)
First four rows: 0; 0,2; 0,6,34,952; 0,1,12,6,181,283,13333,31839187
A079232(x) is equal to the sum of the products of each element in row x of this sequence and the corresponding element of A079210.
The sum of each row x of this sequence is given by A079233(x).

C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to groupoids

Cf. A079202, A079204, A079205, A079197, A079207, A079208, A079209, A079232, A079233.

A079204 Number of isomorphism classes of non-associative non-commutative anti-associative non-anti-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 146, 12992

Offset: 1

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

A079202(n)+A079203(n)+A079204(n)+A079205(n)+A079197(n)+A079207(n)+A079208(n)+A079209(n)+A063524(n)=A079171(n)
Elements per row: 1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)
First four rows: 0; 0,0; 0,0,0,8; 0,0,0,0,0,0,146,12992
A079234(x) is equal to the sum of the products of each element in row x of this sequence and the corresponding element of A079210.
The sum of each row x of this sequence is given by A079235(x).

C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to groupoids

Cf. A079202, A079203, A079205, A079197, A079207, A079208, A079209, A079234, A079235.

A079205 Number of isomorphism classes of non-associative non-commutative anti-associative anti-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 29, 0, 237, 4374

Offset: 1

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

A079202(n)+A079203(n)+A079204(n)+A079205(n)+A079197(n)+A079207(n)+A079208(n)+A079209(n)+A063524(n)=A079171(n)
Elements per row: 1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)
First four rows: 0; 2,0; 0,2,0,0; 0,0,2,0,29,0,237,4374
A079236(x) is equal to the sum of the products of each element in row x of this sequence and the corresponding element of A079210.
The sum of each row x of this sequence is given by A079237(x).

C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to groupoids

Cf. A079202, A079203, A079204, A079197, A079207, A079208, A079209, A079236, A079237.

A079207 Number of isomorphism classes of associative non-commutative non-anti-associative non-anti-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 6, 0, 0, 0, 4, 4, 0, 46, 73, 0, 0, 0, 0, 4, 0, 0, 8, 0, 2, 36, 0, 43, 2, 473, 1020, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 8, 0, 0, 4, 0, 36, 0, 0, 0, 0, 84, 0, 0, 38, 415, 0, 758, 32, 6682, 18426, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8

Offset: 0

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Elements per row: 1,1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)

			Triangle T(n,k) begins:
  0;
  0;
  0, 0;
  0, 0, 4, 6;
  0, 0, 0, 4, 4, 0, 46, 73;
  0, 0, 0, 0, 4, 0, 0, 8, 0, 2, 36, 0, 43, 2, 473, 1020;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..217 (rows 0..8)
C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to semigroups

Row sums give A079241.

Cf. A027423 (row lengths), A079175, A079201, A079202, A079203, A079204, A079205, A079197, A079208, A079209, A079240.

A079202(n,k) + A079203(n,k) + A079204(n,k) + A079205(n,k) + A079197(n,k) + A079208(n,k) + T(n,k) + A079201(n,k) = A079171(n,k).
A079240(n) = Sum_{k>=1} T(n,k)*A079210(n,k).
T(n,k) = A079175(n,k) - A079201(n,k) - A079208(n,k). - Andrew Howroyd, Jan 27 2022

a(0)=0 prepended and terms a(16) and beyond from Andrew Howroyd, Jan 27 2022

A079208 Number of isomorphism classes of associative non-commutative non-anti-associative anti-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Elements per row: 1,1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)

The only closed binary operations that are both commutative and anti-commutative are those on sets of size <= 1. The significance of non-commutative (and non-anti-associative) in the name is that it excludes this possibility. Otherwise, the first two terms would be 1. - Andrew Howroyd, Jan 26 2022

			Triangle T(n,k) begins:
  0;
  0;
  2, 0;
  2, 0, 0, 0;
  2, 0, 0, 0, 1, 0, 0, 0;
  2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..217 (rows 0..8)
C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to semigroups

Row sums give A079243.

Cf. A027423 (row lengths), A079171, A079201, A079202, A079203, A079204, A079205, A079197, A079207, A079209, A079242.

A079202(n,k) + A079203(n,k) + A079204(n,k) + A079205(n,k) + A079197(n,k) + A079207(n,k) + T(n,k) + A079201(n,k) = A079171(n,k).

A079242(n,k) = Sum_{k>=1} T(n,k)*A079210(n,k).

a(0)=0 prepended and terms a(16) and beyond from Andrew Howroyd, Jan 26 2022

cp's OEIS Frontend

A335407 Number of anti-run permutations of the prime indices of n!.

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A337071 Number of strict chains of divisors starting with n!.

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A095149 Triangle read by rows: Aitken's array (A011971) but with a leading diagonal before it given by the Bell numbers (A000110), 1, 1, 2, 5, 15, 52, ...

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A336421 Number of ways to choose a divisor of a divisor, both having distinct prime exponents, of the n-th superprimorial number A006939(n).

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A079202 Number of isomorphism classes of non-associative non-commutative non-anti-associative non-anti-commutative closed binary operations on a set of order n, listed by class size.

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A079203 Number of isomorphism classes of non-associative non-commutative non-anti-associative anti-commutative closed binary operations on a set of order n, listed by class size.

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A079204 Number of isomorphism classes of non-associative non-commutative anti-associative non-anti-commutative closed binary operations on a set of order n, listed by class size.

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Author

Keywords

Comments

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A079205 Number of isomorphism classes of non-associative non-commutative anti-associative anti-commutative closed binary operations on a set of order n, listed by class size.

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A079207 Number of isomorphism classes of associative non-commutative non-anti-associative non-anti-commutative closed binary operations on a set of order n, listed by class size.

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