|
%I #17 Apr 12 2023 10:52:22
%S 1,-1,5,-15,-489,-2865,35685,-135135,-5897745,58437855,3061162125,
%T -39296062575,-2278224696825,-33411730777425,-1496722493140875,
%U -6190283353629375,-1563094742062478625,-17805713551427426625,1456700757237661060125,11729538718345143320625
%N a(n) = (n!)*2^(n-1)*mu_h(n) where mu_h is the hypergeometric Moebius function associated to the Dirichlet character modulo 4 h={1,0,-1,0,1,...} (see comment).
%C Let h be a sequence and define the function g_h(x) = Sum_{1<=k<=1/x} h(k)/k*(k*x+1)/2 on the half-open interval on ]0,1]. Then the hypergeometric Moebius function associated to h is defined by the recursion Sum_{k=1..n} (mu_h(k)/k)*g_h(k/n) = 1/n. Here h(n)=1,0,-1,0,1,0,-1,...
%C a(n) allows us to characterize primes and primes modulo 4:
%C a(n)==0 (mod n) iff n is an odd composite number.
%C a(n)==1 (mod n) iff n is a power of 2 or an odd prime of the form 4k+1.
%C a(n)==n-1 (mod n) iff n is 1, 2, or an odd prime of the form 4k+3.
%D B. Cloitre, The Riemann hypothesis is a topological property, in preparation (2018-....)
%p a_list := proc(len) local s, gh, b: s := x -> select(k->modp(k,2)=1, [$1..1/x]);
%p gh := x -> add((-1)^iquo(k,2)/k*(k*x+1), k in s(x)): b := proc(n) option remember;
%p if n<=0 then 0 else 1/n - add(b(k)*gh(k/n), k=1..n-1)/2 fi end;
%p seq(2^(k-1)*k!*b(k), k=1..len) end: a_list(20); # _Peter Luschny_, Jan 27 2018
%t aList[len_] := Module[{s, gh, b}, s [x_] := Select[Range[1/x], OddQ]; gh[x_] := Sum[(-1)^Quotient[k, 2]/k*(k*x+1), {k, s[x]}]; b[n_] := b[n] = If[n <= 0, 0, 1/n - Sum[b[k]*gh[k/n], {k, 1, n-1}]/2]; Table[2^(k-1)* k!*b[k], {k, 1, len}]]; aList[20] (* _Jean-François Alcover_, Jul 21 2019, after _Peter Luschny_ *)
%o (PARI) v=vector(1000); b(n)=if(n<0,0,v[n]); v[1]=1; gh(x)=sum(k=1,1/x, if(k%2,(-1)^(k\2),0)/k*(k*x+1)/2); for(n=2,#v, v[n]=1/n-sum(k=1,n-1,b(k)*gh(k/n))); a(n)=b(n)*n!*2^(n-1);
%K sign
%O 1,3
%A _Benoit Cloitre_, Jan 20 2018
|
