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A224892 Dirichlet g.f.: Product_{k>=2} (1 - 1/k^(s-1)). 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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
FORMULA
Dirichlet g.f.: Product_{k>=2} (1 - 1/k^(s-1)).
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]]
PROG
(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
CROSSREFS
Sequence in context: A279385 A267000 A365430 * A351233 A265517 A063972
KEYWORD
sign
AUTHOR
Mats Granvik, Jul 24 2013
EXTENSIONS
Definition edited by N. J. A. Sloane, Apr 24 2017
New name from Jon E. Schoenfield, Jan 06 2020
STATUS
approved

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Last modified February 21 13:52 EST 2024. Contains 370235 sequences. (Running on oeis4.)