cp's OEIS Frontend

A097398 Matrix T(m,x(1)), m>=1, x(1)>=2, read by antidiagonals, where T(m,x(1)) gives the position of the first noninteger term in the sequence defined by x(n)=(x(n-1)*(x(n-1)^m+n-1))/n for n>=2 with exponent m and the given starting value x(1).

%I A097398 #21 Feb 16 2025 08:32:54
%S A097398 43,7,89,17,89,97,34,89,17,214,17,89,23,43,19,17,31,97,139,83,239,51,
%T A097398 151,149,107,13,191,37,17,79,13,269,19,359,7,79,7,89,13,107,13,419,23,
%U A097398 127,83,34,79,83,214,37,127,37,158,31,239
%N A097398 Matrix T(m,x(1)), m>=1, x(1)>=2, read by antidiagonals, where T(m,x(1)) gives the position of the first noninteger term in the sequence defined by x(n)=(x(n-1)*(x(n-1)^m+n-1))/n for n>=2 with exponent m and the given starting value x(1).
%C A097398 The rectangular table (Table 1, page 35) in Ibstedt's book gives the position of the first noninteger term for parameters x1 and m:
%C A097398   m\x1: 2   3   4   5   6   7   8   9  10  11
%C A097398    1   43   7  17  34  17  17  51  17   7  34
%C A097398    2   89  89  89  89  31 151  79  89  79 601
%C A097398    3   97  17  23  97 149  13  13  83  23  13
%C A097398    4  214  43 139 107 269 107 214 139 251 107
%C A097398    5   19  83  13  19  13  37  13  37 347  19
%C A097398    6  239 191 359 419 127 127 239 191 239 461
%C A097398    7   37   7  23  37  23  37  17  23   7  37
%C A097398    8   79 127 158  79 103 103 163 103 163  79
%C A097398    9   83  31  41  83  71  83  71  23  41  31
%C A097398   10  239 389 169 137 239 239 239 239 239 389
%D A097398 R. K. Guy, Unsolved Problems in Number Theory, E15.
%D A097398 Henry Ibstedt, Mainly natural numbers - a few elementary studies on Smarandache sequences and other number problems, Henry Ibstedt. - Martinsville, Ind.: Bookman, 2003. Chapter IV, Some Sequences of Large Integers, pp. 32-37.
%H A097398 Hibiki Gima, Toshiki Matsusaka, Taichi Miyazaki, and Shunta Yara, <a href="https://arxiv.org/abs/2402.09064">On integrality and asymptotic behavior of the (k,l)-Göbel sequences</a>, arXiv:2402.09064 [math.NT], 2024. See p. 2.
%H A097398 H. Ibstedt, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/28-3/ibstedt.pdf">Some sequences of large integers</a>, Fibonacci Quart. 28 (1990), 200-203.
%H A097398 Alex Stone, <a href="https://www.quantamagazine.org/the-astonishing-behavior-of-recursive-sequences-20231116/">The Astonishing Behavior of Recursive Sequences</a>, Quanta Magazine, Nov 16 2023, 13 pages.
%H A097398 Eric Weisstein's World of Mathematics: <a href="https://mathworld.wolfram.com/GoebelsSequence.html">Göbel's Sequence</a>
%e A097398 T(1,3)=a(2)=7: x(1)=3, x(2)=x(1)*(x(1)^1+2-1)/n=3*(3+2-1)/2=6, x(3)=6*(6+3-1)/3=16, x(4)=16*(16+4-1)/4=76, x(5)=76*(76+5-1)/5=1216, x(6)=1216*(1216+6-1)/6=247456, x(7)=247456*(247456+7-1)/7=8747993810+2/7; i.e., x(7) is the first noninteger term in the sequence x(n) = x(n-1)*(x(n-1)^1+n-1)/n, n>=2, x(1)=3.
%Y A097398 Cf. A003504 for more references and links, A005166, A005167.
%K A097398 nonn,tabl
%O A097398 1,1
%A A097398 _Hugo Pfoertner_, Aug 15 2004
%E A097398 m=10 row corrected by _Don Reble_, Dec 07 2004, who remarks that the versions in the books of Ibstedt and Guy are both wrong