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A298510 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). 0
1, -1, 5, -15, -489, -2865, 35685, -135135, -5897745, 58437855, 3061162125, -39296062575, -2278224696825, -33411730777425, -1496722493140875, -6190283353629375, -1563094742062478625, -17805713551427426625, 1456700757237661060125, 11729538718345143320625 (list; graph; refs; listen; history; text; internal format)



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,...

a(n) allows us to characterize primes and primes modulo 4:

a(n)==0 (mod n) iff n is an odd composite number.

a(n)==1 (mod n) iff n is a power of 2 or an odd prime of the form 4k+1.

a(n)==n-1 (mod n) iff n is 1, 2, or an odd prime the of form 4k+3.


B. Cloitre, The Riemann hypothesis is a topological property, in preparation (2018-....)


Table of n, a(n) for n=1..20.


a_list := proc(len) local s, gh, b: s := x -> select(k->modp(k, 2)=1, [$1..1/x]);

gh := x -> add((-1)^iquo(k, 2)/k*(k*x+1), k in s(x)): b := proc(n) option remember;

if n<=0 then 0 else 1/n - add(b(k)*gh(k/n), k=1..n-1)/2 fi end;

seq(2^(k-1)*k!*b(k), k=1..len) end: a_list(20); # Peter Luschny, Jan 27 2018


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-Fran├žois Alcover, Jul 21 2019, after Peter Luschny *)


(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);


Sequence in context: A216463 A213147 A112273 * A318898 A247882 A215901

Adjacent sequences:  A298507 A298508 A298509 * A298511 A298512 A298513




Benoit Cloitre, Jan 20 2018



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Last modified January 22 16:37 EST 2020. Contains 331152 sequences. (Running on oeis4.)