A224892 Dirichlet g.f.: Product_{k>=2} (1 - 1/k^(s-1)).

1, -2, -3, -4, -5, 0, -7, 0, -9, 0, -11, 12, -13, 0, 0, 0, -17, 18, -19, 20, 0, 0, -23, 24, -25, 0, 0, 28, -29, 30, -31, 32, 0, 0, 0, 36, -37, 0, 0, 40, -41, 42, -43, 44, 45, 0, -47, 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

Offset: 1

Author

Mats Granvik, Jul 24 2013

Keywords

• sign

Comments

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."

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).

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..512

Programs

Mathematica

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]]

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

Formula

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

Extensions

Definition edited by N. J. A. Sloane, Apr 24 2017

New name from Jon E. Schoenfield, Jan 06 2020