A224892 - OEIS

%I #20 Jan 06 2020 18:13:28

%S 1,-2,-3,-4,-5,0,-7,0,-9,0,-11,12,-13,0,0,0,-17,18,-19,20,0,0,-23,24,

%T -25,0,0,28,-29,30,-31,32,0,0,0,36,-37,0,0,40,-41,42,-43,44,45,0,-47,

%U 48,-49,50,0,52,-53,54,0,56,0,0,-59,60,-61,0,63,0,0,66,-67,68,0,70,-71,72,-73,0,75,76,0,78,-79,80,0,0,-83,84,0,0,0,88,-89,90

%N Dirichlet g.f.: Product_{k>=2} (1 - 1/k^(s-1)).

%C Old name (which appeared to be incorrect) was "a(1)=1; for n>1, let n = p_1^e_1*p_2^e_2*...*p_k^e_k be the prime factorization of n; then a(n) = -n if k=1 and e_1 is 1 or 2; a(n) = +n if k=2 and e1, e_2 are not both 1; a(n) = -n if k >= 3; and a(n) = 0 otherwise."

%C Every factor (1 - 1/n^(s-1)) corresponds to an operator whose row sums are the numerators in the Dirichlet series that converges to log(n).

%H Michael De Vlieger, <a href="/A224892/b224892.txt">Table of n, a(n) for n = 1..512</a>

%F Dirichlet g.f.: Product_{k>=2} (1 - 1/k^(s-1)).

%t Clear[nn, logarithm, LOGPRODUCT, LOGi, n, k]; nn = 90; logarithm = 1; LOGPRODUCT = Table[Table[If[n/k == logarithm, n/k, If[n == k, 1, 0]], {k, 1, nn}], {n, 1, nn}]; Monitor[Do[logarithm = i; LOGi = Table[Table[If[n/k == logarithm, -n/k, If[n == k, 1, 0]], {k, 1, nn}], {n, 1, nn}];LOGPRODUCT = LOGPRODUCT.LOGi;, {i, 2, nn}], i]; LOGPRODUCT[[All, 1]]

%o (PARI) seq(n)={my(v=vector(n, k, k==1)); for(k=2, n, my(w=vector(n)); w[1]=1; w[k]=-k; v=dirmul(v, w)); v} \\ _Andrew Howroyd_, Dec 31 2019

%K sign

%O 1,2

%A _Mats Granvik_, Jul 24 2013

%E Definition edited by _N. J. A. Sloane_, Apr 24 2017

%E New name from _Jon E. Schoenfield_, Jan 06 2020