A271558 -id:A271558 - cp's OEIS frontend

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

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, ", "); ); }

A271562 a(n) = G_n(17), where G is the Goodstein function defined in A266201.

Original entry on oeis.org

17, 7625597484987, 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084095

Offset: 0

Natan Arie Consigli, Apr 13 2016

			G_1(17) = B_2(17)-1 = B_2(2^2^2+1)-1 = 3^3^3+1-1 = 7625597484987;
G_2(17) = B_3(3^3^3)-1 = 4^4^4-1 has 155 digits;
G_3(17) has 328 digits.

Nicholas Matteo, Table of n, a(n) for n = 0..4

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), A271561: G_n(14), A266201: G_n(n).

A271975 a(n) = G_n(18), where G is the Goodstein function defined in A266201.

Original entry on oeis.org

18, 7625597484989, 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097

Offset: 0

Natan Arie Consigli, Apr 24 2016

			G_1(18) = B_2(18)-1 = B_2(2^2^2+2)-1 = 3^3^3+3-1 = 7625597484989;
G_2(18) = B_3(3^3^3+2)-1 = 4^4^4+2-1 has 154 digits;
G_3(18) = B_4(4^4^4+1)-1 = 5^5^5 has 2184 digits;
G_4(18) = B_5(5^5^5)-1 = 6^6^6-1 = has 36305 digits.

Cf. A215409: G_n(3), A056193: G_n(4), 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), A271561: G_n(14), A222117: G_n(15), A059933: G_n(16), A271562: G_n(17), A211378: G_n(19), A266201: G_n(n).

A271976 a(n) = G_n(20), where G is the Goodstein function defined in A266201.

Original entry on oeis.org

20, 7625597485013, 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084137

Offset: 0

Natan Arie Consigli, Apr 24 2016

			G_1(20) = B_2(20)-1 = B_2(2^2^2+2^2)-1 = 3^3^3+3^3-1 = 7625597485013;
G_2(20) = B_3(3^3^3+2*3^2+2*3+2)-1 = 4^4^4+2*4^2+2*4+2-1  has 154 digits;
G_3(20) = B_4(4^4^4+2*4^2+2*4+1)-1 = 5^5^5+2*5^2+2*5+1-1 has 2184 digits.

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), A271561: G_n(14), A271562: G_n(17), A271975: G_n(18), A266201: G_n(n).

A296441 Array A(n, k) = G_k(n) where G_k(n) is the k-th term of the Goodstein sequence of n, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 0, 2, 3, 0, 0, 1, 3, 4, 0, 0, 0, 3, 26, 5, 0, 0, 0, 2, 41, 27, 6, 0, 0, 0, 1, 60, 255, 29, 7, 0, 0, 0, 0, 83, 467, 257, 30, 8, 0, 0, 0, 0, 109, 775, 3125, 259, 80, 9, 0, 0, 0, 0, 139, 1197, 46655, 3127, 553, 81, 10, 0, 0, 0, 0, 173, 1751, 98039, 46657, 6310, 1023, 83, 11

Offset: 0

Iain Fox, Dec 12 2017

G_0(n) = n. To get to the second term in the row, convert n to hereditary base 2 representation (see links), replace each 2 with a 3, and subtract 1. For the third term, convert the second term (G_1(n)) into hereditary base 3 notation, replace each 3 with a 4, and subtract one. This pattern continues until the sequence converges to 0, which, by Goodstein's Theorem, occurs for all n.

			| n\k |  0   1    2     3      4      5       6       7       8       9  ...
|-----|------------------------------------------------------------------------
|  0  |  0,  0,   0,    0,     0,     0,      0,      0,      0,      0, ...
|  1  |  1,  0,   0,    0,     0,     0,      0,      0,      0,      0, ...
|  2  |  2,  2,   1,    0,     0,     0,      0,      0,      0,      0, ...
|  3  |  3,  3,   3,    2,     1,     0,      0,      0,      0,      0, ...
|  4  |  4, 26,  41,   60,    83,   109,    139,    173,    211,    253, ...
|  5  |  5, 27, 255,  467,   775,  1197,   1751,   2454,   3325,   4382, ...
|  6  |  6, 29, 257, 3125, 46655, 98039, 187243, 332147, 555551, 885775, ...
| ... |

Iain Fox, Antidiagonals n = 0..20 of array, flattened
Eric Weisstein's World of Mathematics, Hereditary Representation
Eric Weisstein's World of Mathematics, Goodstein Sequence
Eric Weisstein's World of Mathematics, Goodstein's Theorem
Wikipedia, Hereditary base-n notation
Wikipedia, Goodstein sequence
Wikipedia, Goodstein's Theorem

n-th row: A000004 (n=0), A000007 (n=1), A215409 (n=3), A056193 (n=4), A266204 (n=5), A266205 (n=6), A271554 (n=7), A271555 (n=8), A271556 (n=9), A271557 (n=10), A271558 (n=11), A271559 (n=12), A271560 (n=13), A271561 (n=14), A222117 (n=15), A059933 (n=16), A271562 (n=17), A271975 (n=18) A211378 (n=19), A271976 (n=20).
k-th column: A001477 (k=0), A056004 (k=1), A057650 (k=2), A059934 (k=3), A059935 (k=4), A059936 (k=5), A271977 (k=6), A271978 (k=7), A271979 (k=8), A271985 (k=9), A271986 (k=10).
G_n(n) = A266201(n) (main diagonal of array).

B(n, b)=sum(i=1, #n=digits(n, b), n[i]*(b+1)^if(#n

cp's OEIS Frontend

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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A271562 a(n) = G_n(17), where G is the Goodstein function defined in A266201.

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A271975 a(n) = G_n(18), where G is the Goodstein function defined in A266201.

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A271976 a(n) = G_n(20), where G is the Goodstein function defined in A266201.

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A296441 Array A(n, k) = G_k(n) where G_k(n) is the k-th term of the Goodstein sequence of n, read by antidiagonals.

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