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 A266201 Goodstein numbers: a(n) = G_n(n), where G is the Goodstein function. 33
 0, 0, 1, 2, 83, 1197, 187243, 37665879, 20000000211, 855935016215, 44580503598539, 2120126221988686, 155568095557812625, 6568408355712901455, 295147905179358418247, 14063084452070776884879 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS To write an integer n in base-k hereditary representation, write n in ordinary base-k representation, and then do the same recursively for all exponents which are greater than k. For example, the hereditary representation of 132132 in base-2 is: 132132 = 2^17 + 2^10 + 2^5 + 2^2 = 2^(2^4 + 1) + 2^(2^3 + 2) + 2^(2^2 + 1) + 2^2 = 2^(2^(2^2) + 1) + 2^(2^(2+1) + 2) + 2^(2^2 + 1) + 2^2. Define B_k(n) to be the function that substitutes k+1 for all the bases of the base-k hereditary representation of n. E.g., B_2(101) = B_2(2^(2^2 + 2) + 2^(2^2 + 1) + 2^2 + 1) = 3^(3^3 + 3) + 3^(3^3 + 1) + 3^3 + 1 = 228767924549638. (Sometimes B_k(n) is referred to as n "bumped" from base k.) The Goodstein function is defined as: G_k(n) = B_{k+1}(G_{k-1}(n)) - 1 with G_0(n) = n, i.e., iteration of bumping the number to the next larger base and subtracting one; see example section for instances. Goodstein's theorem says that for any nonnegative n, the sequence G_k(n) eventually stabilizes and then decreases by 1 in each step until it reaches 0. (The subsequent values of G_k(n) < 0 are not part of the sequence.) LINKS Eric Weisstein's World of Mathematics, Goodstein sequences EXAMPLE Compute a(5) = G_5(5): G_0(5) = 5; G_1(5) = B_2(G_0(5))-1 = B_2(2^2+1)-1 = (3^3+1)-1 = 27 = 3^3; G_2(5) = B_3(G_1(5))-1 = B_3(3^3)-1 = 4^4-1 = 255 = 3*4^3+3*4^2+3*4+3; G_3(5) = B_4(G_2(5))-1 = B_4(3*4^3+3*4^2+3*4+3)-1 = 467; G_4(5) = B_5(G_3(5))-1 = B_5(3*5^3+3*5^2+3*5+2)-1 = 775; G_5(5) = B_6(G_4(5))-1 = B_6(3*6^3+3*6^2+3*6+1)-1 = 1197. PROG (PARI) (B(n, b)=sum(i=1, #n=digits(n, b), n[i]*(b+1)^if(#n

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Last modified November 27 17:16 EST 2020. Contains 338683 sequences. (Running on oeis4.)