This site is supported by donations to The OEIS Foundation.

 Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 14 19:27 EST 2019. Contains 329987 sequences. (Running on oeis4.)