Tag Archives: Siehler numbers

3, 7, 15, 23, 39, 47, 71, 87, 111, 127, …


10 Siehler’s numbers

two consecutive reciprocal partial sums are

1/3+1/7+1/15+1/23+1/39+1/47+1/71+1/87+1/111+1/127+1/167+1/183+1/231+1/255+1/287+1/319+1/383+1/407+1/479+1/511+1/559+1/599+1/687+1/719+1/799+1/847+1/919+1/967+1/10794

\approx 0.722936

and

1/3+1/7+1/15+1/23+1/39+1/47+1/71+1/87+1/111+1/127+1/167+1/183+1/231+1/255+1/287+1/319+1/383+1/407+1/479+1/511+1/559+1/599+1/687+1/719+1/799+1/847+1/919+1/967+1/1079+

1/1111 \approx 0.723836

(04.nov.2011)

by the way: f(n)=\#{\mathbb{Z}_n}^* (cardinal of multiplicatively invertibles elements in the ring {\mathbb{Z}_n}) satisfy:

  • f(1)=\#{\mathbb{Z}_1}^*=0, since {\mathbb{Z}_1}=\{0\} and {\mathbb{Z}_1}^*=\varnothing
  • f(2)=\#{\mathbb{Z}_2}^*=1, since {\mathbb{Z}_2}=\{0,1\} and {\mathbb{Z}_2}^*=\{1\}
  • f(3)=\#{\mathbb{Z}_3}^*=2, since {\mathbb{Z}_3}=\{0,1,2\} and {\mathbb{Z}_3}^*=\{1,2\}
  • f(4)=\#{\mathbb{Z}_4}^*=2, since {\mathbb{Z}_4}=\{0,1,2,3\} and {\mathbb{Z}_4}^*=\{1,3\}
  • f(5)=\#{\mathbb{Z}_4}^*=4, since {\mathbb{Z}_5}=\{0,1,2,3,4\} and {\mathbb{Z}_4}^*=\{1,2,3,4\}

Hence

s(1)=3+4f(1)=3

s(2)=s(1)+4f(2)=7

s(3)=s(2)+4f(3)=15

s(4)=s(3)+4f(4)=23

s(5)=s(4)+4f(5)=39

s(n+1)=s(n)+4f(n+1)

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another reciprocal sum problem


ok we already know how to calculate how many rationals, \mathbb{Q}, exists in terms of the height function

h(a/b)=\max (|a|,|b|)

gcd(a,b)=1

It turns out that the A171503  entry of the OEIS tells more…

http://192.20.225.10/~njas/sequences/A171503

meanwhile it is natural to ask for convergence of

1/3 + 1/7 + 1/15 + 1/23 + 1/31 + 1/47 + 1/71 + 1/87 + 1/111 + 1/127+...

and for (non-) existence of a generating function…

The crescent sequence could be called Siehler’s numbers

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