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).

43, 7, 89, 17, 89, 97, 34, 89, 17, 214, 17, 89, 23, 43, 19, 17, 31, 97, 139, 83, 239, 51, 151, 149, 107, 13, 191, 37, 17, 79, 13, 269, 19, 359, 7, 79, 7, 89, 13, 107, 13, 419, 23, 127, 83, 34, 79, 83, 214, 37, 127, 37, 158, 31, 239

Offset: 1

Author

Hugo Pfoertner, Aug 15 2004

Keywords

• nonn
• tabl

Comments

The rectangular table (Table 1, page 35) in Ibstedt's book gives the position of the first noninteger term for parameters x1 and m:

m\x1: 2 3 4 5 6 7 8 9 10 11

1 43 7 17 34 17 17 51 17 7 34

2 89 89 89 89 31 151 79 89 79 601

3 97 17 23 97 149 13 13 83 23 13

4 214 43 139 107 269 107 214 139 251 107

5 19 83 13 19 13 37 13 37 347 19

6 239 191 359 419 127 127 239 191 239 461

7 37 7 23 37 23 37 17 23 7 37

8 79 127 158 79 103 103 163 103 163 79

9 83 31 41 83 71 83 71 23 41 31

10 239 389 169 137 239 239 239 239 239 389

Examples

			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.
		

References

• R. K. Guy, Unsolved Problems in Number Theory, E15.
• 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.

Links

• Hibiki Gima, Toshiki Matsusaka, Taichi Miyazaki, and Shunta Yara, On integrality and asymptotic behavior of the (k,l)-Göbel sequences, arXiv:2402.09064 [math.NT], 2024. See p. 2.
• H. Ibstedt, Some sequences of large integers, Fibonacci Quart. 28 (1990), 200-203.
• Alex Stone, The Astonishing Behavior of Recursive Sequences, Quanta Magazine, Nov 16 2023, 13 pages.
• Eric Weisstein's World of Mathematics: Göbel's Sequence

Crossrefs

Cf. A003504 for more references and links, A005166, A005167.

Extensions

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