A222117 -id:A222117 - cp's OEIS frontend

A271555 a(n) = G_n(8), where G is the Goodstein function defined in A266201.

8, 80, 553, 6310, 93395, 1647195, 33554571, 774841151, 20000000211, 570623341475, 17832200896811, 605750213184854, 22224013651116433, 875787780761719208, 36893488147419103751, 1654480523772673528938, 78692816150593075151501, 3956839311320627178248684

Offset: 0

Natan Arie Consigli, Apr 10 2016

			G_1(8) = B_2(8)-1 = B_2(2^(2+1))-1 = 3^(3+1)-1 = 80;
G_2(8) = B_3(2*3^3+2*3^2+2*3+2)-1 = 2*4^4+2*4^2+2*4+2-1 = 553;
G_3(8) = B_4(2*4^4+2*4^2+2*4+1)-1 = 2*5^5+2*5^2+2*5+1-1 = 6310;
G_4(8) = B_5(2*5^5+2*5^2+2*5)-1 = 2*6^6+2*6^2+2*6-1 = 93395;
G_5(8) = B_6(2*6^6+2*6^2+6+5)-1 = 2*7^7+2*7^2+7+5-1 = 1647195;
G_6(8) = B_7(2*7^7+2*7^2+7+4)-1 = 2*8^8+2*8^2+8+4-1 = 33554571;
G_7(8) = B_8(2*8^8+2*8^2+8+3)-1 = 2*9^9+2*9^2+9+3-1 = 774841151.

Nicholas Matteo, Table of n, a(n) for n = 0..384
R. L. Goodstein, On the Restricted Ordinal Theorem, The Journal of Symbolic Logic 9, no. 2 (1944), 33-41.
Wikipedia, Goodstein sequence

Cf. A056193: G_n(4), A059933: G_n(16), A211378: G_n(19), A215409: G_n(3), A222117: G_n(15), A266204: G_n(5), A266205: G_n(6), A271554: G_n(7), A266201: G_n(n).

lista(nn) = {print1(a = 8, ", "); for (n=2, nn, pd = Pol(digits(a, n)); q = sum(k=0, poldegree(pd), if (c=polcoeff(pd, k), c*x^subst(Pol(digits(k, n)), x, n+1), 0)); a = subst(q, x, n+1) - 1; print1(a, ", "); ); }

a(3) corrected by Nicholas Matteo, Aug 15 2019

A271556 a(n) = G_n(9), where G is the Goodstein function defined in A266201.

Original entry on oeis.org

9, 81, 1023, 9842, 140743, 2471826, 50333399, 1162263921, 30000003325, 855935016215, 26748301350411, 908625319783885, 33336020476682897, 1313681671142588955, 55340232221128667935, 2481720785659010308168, 118039224225889612744771, 5935258966980940767393628

Offset: 0

Natan Arie Consigli, Apr 10 2016

			G_1(9) = B_2(9)-1 = B_2(2^(2+1)+1)-1 = 3^(3+1) + 1-1 = 81;
G_2(9) = B_3(3^(3+1))-1 = 4^(4+1)-1 = 1023;
G_3(9) = B_4(3*4^4 + 3*4^3 + 3*4^2 + 3*4 + 3)-1 = 3*5^5 + 3*5^3 + 3*5^2 + 3*5 + 3-1 = 9842;
G_4(9) = B_5(3*5^5 + 3*5^3 + 3*5^2 + 3*5 + 2)-1 = 3*6^6 + 3*6^3 + 3*6^2 + 3*6 + 2-1 = 140743;
G_5(9) = B_6(3*6^6 + 3*6^3 + 3*6^2 + 3*6 + 1)-1 = 3*7^7 + 3*7^3 + 3*7^2 + 3*7 + 1-1 = 2471826;
G_6(9) = B_7(3*7^7 + 3*7^3 + 3*7^2 + 3*7)-1 = 3*8^8 + 3*8^3 + 3*8^2 + 3*8-1 = 50333399.

Nicholas Matteo, Table of n, a(n) for n = 0..384
R. L. Goodstein, On the Restricted Ordinal Theorem, The Journal of Symbolic Logic 9, no. 2 (1944), 33-41.

Cf. A056193: G_n(4), A059933: G_n(16), A211378: G_n(19), A215409: G_n(3), A222117: G_n(15), A266204: G_n(5), A266205: G_n(6), A271554: G_n(7), A271555: G_n(8), A266201: G_n(n).

lista(nn) = {print1(a = 9, ", "); for (n=2, nn, pd = Pol(digits(a, n)); q = sum(k=0, poldegree(pd), if (c=polcoeff(pd, k), c*x^subst(Pol(digits(k, n)), x, n+1), 0)); a = subst(q, x, n+1) - 1; print1(a, ", "); ); }

A271557 a(n) = G_n(10), where G is the Goodstein function defined in A266201.

Original entry on oeis.org

10, 83, 1025, 15625, 279935, 4215754, 84073323, 1937434592, 50000555551, 1426559238830, 44580503598539, 1514375534972427, 55560034130686045, 2189469451908364943, 92233720368553350471, 4136201309431691363859, 196732040376482697880697, 9892098278301567958688175

Offset: 0

Natan Arie Consigli, Apr 11 2016

			G_1(10) = B_2(10)-1 = B_2(2^(2+1)+2)-1 = 3^(3+1)+3-1 = 83;
G_2(10) = B_3(3^(3+1)+2)-1 = 4^(4+1)+2-1 = 1025;
G_3(10) = B_4(4^(4+1)+1)-1 = 5^(5+1)+1-1 = 15625;
G_4(10) = B_5(5*5^(5+1))-1 = 6^(6+1)-1= 279935;
G_5(10) = B_6(5*6^6+5*6^5+5*6^4+5*6^3+5*6^2+5*6+5)-1 = 5*7^7+5*7^5+5*7^4+5*7^3+5*7^2+5*7+5-1 = 4215754;
G_6(10) = B_7(5*7^7+5*7^5+5*7^4+5*7^3+5*7^2+5*7+4)-1 = 5*8^8+5*8^5+5*8^4+5*8^3+5*8^2+5*8+4-1 = 84073323;
G_7(10) = B_8(5*8^8+5*8^5+5*8^4+5*8^3+5*8^2+5*8+3)-1 = 5*9^9+5*9^5+5*9^4+5*9^3+5*9^2+5*9+3-1 = 1937434592;
G_8(10) = B_9(5*9^9+5*9^5+5*9^4+5*9^3+5*9^2+5*9+2)-1 = 5*10^10+5*10^5+5*10^4+5*10^3+5*10^2+5*10+2-1 = 50000555551.

Nicholas Matteo, Table of n, a(n) for n = 0..384
R. L. Goodstein, On the Restricted Ordinal Theorem, The Journal of Symbolic Logic 9, no. 2 (1944), 33-41.

Cf. A056193: G_n(4), A059933: G_n(16), A211378: G_n(19), A215409: G_n(3), A222117: G_n(15), A266204: G_n(5), A266205: G_n(6), A271554: G_n(7), A271555: G_n(8), A271556: G_n(9), A266201: G_n(n).

lista(nn) = {print1(a = 10, ", "); for (n=2, nn, pd = Pol(digits(a, n)); q = sum(k=0, poldegree(pd), if (c=polcoeff(pd, k), c*x^subst(Pol(digits(k, n)), x, n+1), 0)); a = subst(q, x, n+1) - 1; print1(a, ", "); ); }

A056041 Value for which b(a(n))=0 when b(2)=n and b(k+1) is calculated by writing b(k) in base k, reading this as being written in base k+1 and then subtracting 1.

Original entry on oeis.org

2, 3, 5, 7, 23, 63, 383, 2047

Offset: 0

Henry Bottomley, Aug 04 2000

a(8)=3*2^(3*2^27+27)-1 which is more than 10^(10^8) and equal to the final base of the Goodstein sequence starting with g(2)=4; indeed, apart from the initial term, the sequence starting with b(2)=8 is identical to the Goodstein sequence starting with g(2)=4. The initial terms of a(n) [2, 3, 5 and 7] are equal to the initial terms of the equivalent final bases of Goodstein sequences starting at the same points. a(9)=2^(2^(2^70+70)+2^70+70)-1 which is more than 10^(10^(10^20)).
It appears that if n is even then a(n) is one less than three times a power of two, while if n is odd then a(n) is one less than a power of two.
Comment from John Tromp, Dec 02 2004: The sequence 2,3,5,7,3*2^402653211 - 1, ... gives the final base of the Goodstein sequence starting with n. This is an example of a very rapidly growing function that is total (i.e. defined on any input), although this fact is not provable in first-order Peano Arithmetic. See the links for definitions. This grows even faster than the Friedman sequence described in the Comments to A014221.
In fact there are two related sequences: (i) The Goodstein function l(n) = number of steps for the Goodstein sequence to reach 0 when started with initial term n >= 0: 0, 1, 3, 5, 3*2^402653211 - 3, ...; and (ii) the same sequence + 2: 2, 3, 5, 7, 3*2^402653211 - 1, ..., which is the final base reached. Both grow too rapidly to have their own entries in the database.
Related to the hereditary base sequences - see cross-reference lines.
This sequence gives the final base of the weak Goodstein sequence starting with n; compare A266203, the length of the weak Goodstein sequence. a(n) = A266203(n) + 2.

			a(3)=7 because starting with b(2)=3=11 base 2, we get b(3)=11-1 base 3=10 base 3=3, b(4)=10-1 base 4=3, b(5)=3-1 base 5=2, b(6)=2-1 base 6=1 and b(7)=1-1 base 7=0.

R. L. Goodstein, On the Restricted Ordinal Theorem, J. Symb. Logic 9, 33-41, 1944.
L. Kirby, and J. Paris, Accessible independence results for Peano arithmetic, Bull. London Mathematical Society, 14 (1982), 285-293.
J. Tromp, Programming Pearls
Eric Weisstein's World of Mathematics, Goodstein Sequence
Wikipedia, Goodstein's theorem

Cf. A266202, A268687, A268689, A268688.
Equals A266203 + 2.
Weak Goodstein sequences: A267647, A267648, A271987, A271988, A271989, A271990, A271991, A137411, A271992, A265034.
Steps of strong Goodstein sequences: A056004, A057650, A059934, A059935, A059936, A271977.
Strong Goodstein sequences: A215409, A056193, A266204, A222117, A059933.
Woodall numbers: A003261.

A222112 Initial step in Goodstein sequences: write n-1 in hereditary binary representation, then bump to base 3.

Original entry on oeis.org

0, 1, 3, 4, 27, 28, 30, 31, 81, 82, 84, 85, 108, 109, 111, 112, 7625597484987, 7625597484988, 7625597484990, 7625597484991, 7625597485014, 7625597485015, 7625597485017, 7625597485018, 7625597485068, 7625597485069, 7625597485071, 7625597485072, 7625597485095

Offset: 1

Reinhard Zumkeller, Feb 13 2013

nonn

See A056004 for an alternate version.

			n = 19: 19 - 1 = 18 = 2^4 + 2^1 = 2^2^2 + 2^1
-> a(19) = 3^3^3 + 3^1 = 7625597484990;
n = 20: 20 - 1 = 19 = 2^4 + 2^1 + 2^0 = 2^2^2 + 2^1 + 2^0
-> a(20) = 3^3^3 + 3^1 + 3^0 = 7625597484991;
n = 21: 21 - 1 = 20 = 2^4 + 2^2 = 2^2^2 + 2^2
-> a(21) = 3^3^3 + 3^3 = 7625597485014.

Helmut Schwichtenberg and Stanley S. Wainer, Proofs and Computations, Cambridge University Press, 2012; 4.4.1, page 148ff.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
R. L. Goodstein, On the Restricted Ordinal Theorem, The Journal of Symbolic Logic, Vol. 9, No. 2, Jun., 1944.
Wikipedia, Goodstein's Theorem
Reinhard Zumkeller, Haskell programs for Goodstein sequences

Cf. A056004: G_1(n), A057650 G_2(n), A056041; A266201: G_n(n);

Cf. A215409: G_n(3), A056193: G_n(4), A266204: G_n(5), A266205: G_n(6), A222117: G_n(15), A059933: G_n(16), A211378: G_n(19).

Haskell
```
-- See Link
```

A222112(n)=sum(i=1, #n=binary(n-1), if(n[i],3^if(#n-i<2, #n-i, A222112(#n-i+1)))) \\ See A266201 for more general code. - M. F. Hasler, Feb 13 2017, edited Feb 19 2017

A271558 a(n) = G_n(11), where G is the Goodstein function defined in A266201.

Original entry on oeis.org

11, 84, 1027, 15627, 279937, 5764801, 134217727, 2749609302, 70077777775, 1997331745490, 62412976762503, 2120126221988686, 77784048573561751, 3065257233947460930, 129127208517971179375, 5790681833207409243109, 275424856527080300658781, 13848937589622201728586799

Offset: 0

Natan Arie Consigli, Apr 11 2016

			G_1(11) = B_2(11)-1 = B_2(2^(2+1)+2+1)-1 = 3^(3+1)+3+1-1 = 84;
G_2(11) = B_3(3^(3+1)+3)-1 = 4^(4+1)+4-1 = 1027;
G_3(11) = B_4(4^(4+1)+3)-1 = 5^(5+1)+3-1 = 15627;
G_4(11) = B_5(5^(5+1)+2)-1 = 6^(6+1)+2-1 = 279937;
G_5(11) = B_6(6^(6+1)+1)-1 = 7^(7+1)+1-1 = 5764801;
G_6(11) = B_7(7^(7+1))-1 = 8^(8+1)-1 = 134217727.

Nicholas Matteo, Table of n, a(n) for n = 0..384
R. L. Goodstein, On the Restricted Ordinal Theorem, The Journal of Symbolic Logic 9, no. 2 (1944), 33-41.

Cf. A056193: G_n(4), A059933: G_n(16), A211378: G_n(19), A215409: G_n(3), A222117: G_n(15), A266204: G_n(5), A266205: G_n(6), A271554: G_n(7), A271555: G_n(8), A271556: G_n(9), A271557: G_n(10), A266201: G_n(n).

lista(nn) = {print1(a = 11, ", "); for (n=2, nn, pd = Pol(digits(a, n)); q = sum(k=0, poldegree(pd), if (c=polcoeff(pd, k), c*x^subst(Pol(digits(k, n)), x, n+1), 0)); a = subst(q, x, n+1) - 1; print1(a, ", "); ); }

a(9)-a(13) corrected by Nicholas Matteo, Aug 15 2019

a(14) onwards from Nicholas Matteo, Aug 28 2019

A271559 a(n) = G_n(12), where G is the Goodstein function defined in A266201.

Original entry on oeis.org

12, 107, 1065, 15685, 280019, 5764910, 134217867, 3486784574, 100000000211, 3138428376974, 106993205379371, 3937376385699637, 155568095557812625, 6568408355712891083, 295147905179352826375, 14063084452067724991593, 708235345355337676358285, 37589973457545958193356327

Offset: 0

Natan Arie Consigli, Apr 11 2016

Goodstein's theorem shows that such sequence converges to zero for any starting value.

			G_1(12) = B_2(12)-1 = B_2(2^(2+1)+2^2)-1 = 3^(3+1)+3^3-1 = 107;
G_2(12) = B_3(3^(3+1)+2*3^2+2*3+2)-1 = 4^(4+1)+2*4^2+2*4+2-1 = 1065;
G_3(12) = B_4(4^(4+1)+2*4^2+2*4+1)-1 = 5^(5+1)+2*5^2+2*5+1-1 = 15685;
G_4(12) = B_5(5^(5+1)+2*5^2+2*5)-1 = 6^(6+1)+2*6^2+2*6-1 = 280019;
G_5(12) = B_6(6^(6+1)+2*6^2+6+5)-1 = 7^(7+1)+2*7^2+7+5-1 = 5764910;
G_6(12) = B_7(7^(7+1)+2*7^2+7+4)-1 = 8^(8+1)+2*8^2+8+4-1 = 134217867;
G_7(12) = B_8(8^(8+1)+2*8^2+8+3)-1 = 9^(9+1)+2*9^2+9+3-1 = 3486784574.

Nicholas Matteo, Table of n, a(n) for n = 0..383
R. L. Goodstein, On the Restricted Ordinal Theorem, The Journal of Symbolic Logic 9, no. 2 (1944), 33-41.

Cf. A056193: G_n(4), A059933: G_n(16), A211378: G_n(19), A215409: G_n(3), A222117: G_n(15), A266204: G_n(5), A266205: G_n(6), A271554: G_n(7), A271555: G_n(8), A271556: G_n(9), A271557: G_n(10), A271558: G_n(11), A266201: G_n(n).

lista(nn) = {print1(a = 12, ", "); for (n=2, nn, pd = Pol(digits(a, n)); q = sum(k=0, poldegree(pd), if (c=polcoeff(pd, k), c*x^subst(Pol(digits(k, n)), x, n+1), 0)); a = subst(q, x, n+1) - 1; print1(a, ", "); ); }

A271560 a(n) = G_n(13), where G is the Goodstein function defined in A266201.

Original entry on oeis.org

13, 108, 1279, 16092, 280711, 5765998, 134219479, 3486786855, 100000003325, 3138428381103, 106993205384715, 3937376385706415, 155568095557821073, 6568408355712901455, 295147905179352838943, 14063084452067725006646, 708235345355337676376131, 37589973457545958193377292

Offset: 0

Natan Arie Consigli, Apr 11 2016

			G_1(13) = B_2(13)-1 = B_2(2^(2+1)+2^2+1)-1 = 3^(3+1)+3^3+1-1 = 108;
G_2(13) = B_3(3^(3+1)+3^3)-1 = 4^(4+1)+4^4-1 = 1279;
G_3(13) = B_4(4^(4+1)+3*4^3+3*4^2+3*4+3)-1 = 5^(5+1)+3*5^3+3*5^2+3*5+3-1 = 16092;
G_4(13) = B_5(5^(5+1)+3*5^3+3*5^2+3*5+2)-1 = 6^(6+1)+3*6^3+3*6^2+3*6+2-1 = 280711;
G_5(13) = B_6(6^(6+1)+3*6^3+3*6^2+3*6+1)-1 = 7^(7+1)+3*7^3+3*7^2+3*7+1-1 = 5765998;
G_6(13) = B_7(7^(7+1)+3*7^3+3*7^2+3*7)-1 = 8^(8+1)+3*8^3+3*8^2+3*8-1 = 134219479;
G_7(13) = B_8(8^(8+1)+3*8^3+3*8^2+2*8+7)-1 = 9^(9+1)+3*9^3+3*9^2+2*9+7-1 = 3486786855.

Nicholas Matteo, Table of n, a(n) for n = 0..383

Cf. A056193: G_n(4), A059933: G_n(16), A211378: G_n(19), A215409: G_n(3), A222117: G_n(15), A266204: G_n(5), A266205: G_n(6), A271554: G_n(7), A271555: G_n(8), A271556: G_n(9), A271557: G_n(10), A271558: G_n(11), A271559: G_n(12), A266201: G_n(n).

lista(nn) = {my(a=13); print1(a, ", "); for (n=2, nn, my(pd = Pol(digits(a, n)), q = sum(k=0, poldegree(pd), my(c=polcoeff(pd, k)); if (c, c*x^subst(Pol(digits(k, n)), x, n+1), 0))); a = subst(q, x, n+1) - 1; print1(a, ", "); ); }

A271561 a(n) = G_n(14), where G is the Goodstein function defined in A266201.

Original entry on oeis.org

14, 110, 1281, 18750, 326591, 5862840, 134404971, 3487116548, 100000555551, 3138429262496, 106993206736331, 3937376387710451, 155568095560708189, 6568408355716958693, 295147905179358418247, 14063084452067732533983, 708235345355337686361209, 37589973457545958206423881

Offset: 0

Natan Arie Consigli, Apr 13 2016

			G_1(14) = B_2(14)-1 = B_2(2^(2+1)+2^2+2)-1 = 3^(3+1)+3^3+3-1 = 110;
G_2(14) = B_3(3^(3+1)+3^3+2)-1 = 4^(4+1)+4^4+2-1 = 1281;
G_3(14) = B_4(4^(4+1)+4^4+1)-1 = 5^(5+1)+5^5+1-1 = 18750;
G_4(14) = B_5(5^(5+1)+5^5)-1 = 6^(6+1)+6^6-1 = 326591.

Nicholas Matteo, Table of n, a(n) for n = 0..383

Cf. A056193: G_n(4), A059933: G_n(16), A211378: G_n(19), A215409: G_n(3), A222117: G_n(15), A266204: G_n(5), A266205: G_n(6), A271554: G_n(7), A271555: G_n(8), A271556: G_n(9), A271557: G_n(10), A271558: G_n(11), A271559: G_n(12), A271560: G_n(13), A266201: G_n(n).

lista(nn) = {print1(a = 14, ", "); for (n=2, nn, pd = Pol(digits(a, n)); q = sum(k=0, poldegree(pd), if (c=polcoeff(pd, k), c*x^subst(Pol(digits(k, n)), x, n+1), 0)); a = subst(q, x, n+1) - 1; print1(a, ", "); ); }

A222113 Goodstein sequence starting with a(1) = 16: to calculate a(n) for n>1, subtract 1 from a(n-1) and write the result in the hereditary representation base n, then bump the base to n+1.

Original entry on oeis.org

16, 112, 1284, 18753, 326594, 6588345, 150994944, 3524450281, 100077777776, 3138578427935, 106993479003784, 3937376861542205, 155568096352467864, 6568408356994335931, 295147905181357143920, 14063084452070776884880, 708235345355342213988446

Offset: 1

Reinhard Zumkeller, Feb 13 2013

Compare to A222117: the underlying variants to define Goodstein sequences are equivalent.

			a(1) - 1 = 15 = 2^3 + 2^2 + 2^1 + 2^0 = 2^(2^1+1) + 2^2 + 2^1 + 2^0
-> a(2) = 3^(3^1+1) + 3^3 + 3^1 + 3^0 = 112;
a(2) - 1 = 111 = 3^(3^1+1) + 3^3 + 3^1
-> a(3) = 4^(4^1+1) + 4^4 + 4^1 = 1284;
a(3) - 1 = 1283 = 4^(4^1+1) + 4^4 + 3*4^0
-> a(4) = 5^(5^1+1) + 5^5 + 3*5^0 = 18753;
a(4) - 1 = 18752 = 5^(5^1+1) + 5^5 + 2*5^0
-> a(5) = 6^(6^1+1) + 6^6 + 2*6^0 = 326594;
a(5) - 1 = 326593 = 6^(6^1+1) + 6^6 + 6^0
-> a(6)  = 7^(7^1+1) + 7^7 + 7^0 = 6588345.

Helmut Schwichtenberg and Stanley S. Wainer, Proofs and Computations, Cambridge University Press, 2012; 4.4.1, page 148ff.

Reinhard Zumkeller, Table of n, a(n) for n = 1..250
R. L. Goodstein, On the Restricted Ordinal Theorem, The Journal of Symbolic Logic, Vol. 9, No. 2, Jun., 1944.
Wikipedia, Goodstein's Theorem
Reinhard Zumkeller, Haskell programs for Goodstein sequences

Cf. A222112.

Haskell
```
-- See Link
```

cp's OEIS Frontend

A271555 a(n) = G_n(8), where G is the Goodstein function defined in A266201.

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A271556 a(n) = G_n(9), where G is the Goodstein function defined in A266201.

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A271557 a(n) = G_n(10), where G is the Goodstein function defined in A266201.

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A056041 Value for which b(a(n))=0 when b(2)=n and b(k+1) is calculated by writing b(k) in base k, reading this as being written in base k+1 and then subtracting 1.

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A222112 Initial step in Goodstein sequences: write n-1 in hereditary binary representation, then bump to base 3.

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A271558 a(n) = G_n(11), where G is the Goodstein function defined in A266201.

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A271559 a(n) = G_n(12), where G is the Goodstein function defined in A266201.

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A271560 a(n) = G_n(13), where G is the Goodstein function defined in A266201.

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A271561 a(n) = G_n(14), where G is the Goodstein function defined in A266201.

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