The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A118229 Triangle, read by rows, equal to the matrix inverse of triangle A054431; the inverse transformation that obtains {a(n)} from b(n) = Sum_{1<=k<=n, gcd(k,n)=1} a(k). 2
 1, -1, 1, -1, 0, 1, 1, -1, -1, 1, -1, 0, 0, 0, 1, 1, 0, 0, -1, -1, 1, 1, 0, -1, 0, -1, 0, 1, -1, 0, 2, -1, 0, 0, -1, 1, -1, 0, 0, 0, 1, 0, -1, 0, 1, 1, 0, -1, 1, 0, -1, 1, -1, -1, 1, -1, 0, 1, 0, 0, 0, -1, 0, 0, 0, 1, 1, 0, -1, 0, 0, 0, 1, 0, 0, -1, -1, 1, 3, 0, -2, 0, -2, 0, 2, 0, -1, 0, -1, 0, 1, -3, 0, 1, 0, 3, 0, -1, -1, 1, 0, 0, 0, -1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,31 COMMENTS Column 1 is A096433. Column 2 = [0,1,0,-1,0,0,0,...(zero for n>4)]. Column 3 is A118230. LINKS Hamed Mousavi, Maxie D. Schmidt, Factorization Theorems for Relatively Prime Divisor Sums, GCD Sums and Generalized Ramanujan Sums, arXiv:1810.08373 [math.NT], 2018. See Figure 2.1, p. 6. FORMULA For column k > 1: Sum_{i = 2..n, gcd(n,i) = 1} T(i,k) = 1 when n = k+1, 0 elsewhere; for column k = 1: Sum_{i = 2..n, gcd(n,i) = 1} T(i,1) = 1 when n = 1 or 2, 0 elsewhere. EXAMPLE Describes a sequence transformation as follows. Say we have the arbitrary sequence {a(k)}. We define {b(k)}, based on {a(k)}, by: b(n) = Sum_{1<=k<=n, gcd(k,n)=1} a(k). So given {b(k)} (which must have b(1) = b(2)), how do we get the sequence {a(k)}? If a(n) = Sum_{k >= 2} b(k) * T(n,k), then there is a triangular array {T(n,k)} which begins as follows:    1;   -1,  1;   -1,  0,  1;    1, -1, -1,  1;   -1,  0,  0,  0,  1;    1,  0,  0, -1, -1,  1;    1,  0, -1,  0, -1,  0,  1;   -1,  0,  2, -1,  0,  0, -1,  1;   -1,  0,  0,  0,  1,  0, -1,  0,  1;    1,  0, -1,  1,  0, -1,  1, -1, -1,  1;   -1,  0,  1,  0,  0,  0, -1,  0,  0,  0,  1;    1,  0, -1,  0,  0,  0,  1,  0,  0, -1, -1, 1;    3,  0, -2,  0, -2,  0,  2,  0, -1,  0, -1, 0,  1;   -3,  0,  1,  0,  3,  0, -1, -1,  1,  0,  0, 0, -1, 1; ... MATHEMATICA M[n_] := M[n] = Inverse[Table[If[r >= c, If[GCD[r-c+1, c] == 1, 1, 0], 0], {r, 1, n}, {c, 1, n}]]; T[n_, k_] := If[n=c, if(gcd(r-c+1, c)==1, 1, 0)))^-1)[n, k]) CROSSREFS Cf. A054431 (matrix inverse), A096433 (column 1), A118230 (column 3). Sequence in context: A321928 A321917 A115201 * A172250 A309047 A255317 Adjacent sequences:  A118226 A118227 A118228 * A118230 A118231 A118232 KEYWORD sign,tabl AUTHOR Leroy Quet, Paul D. Hanna, Apr 16 2006 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified July 26 06:53 EDT 2021. Contains 346294 sequences. (Running on oeis4.)