A271047 A prime number sieve defined by the recurrence: T(n, k) = If n = k then 1 else if k divides n then -Sum_{i=k+1..n} T(n, i) else T(n,k) = 0.

1, -1, 1, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, 0, -1, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1

Offset

1

Comments

Same negative sum as in the recurrence for the Möbius function except that it is applied at all the divisors and not only in the first column. The table therefore acts as a prime number sieve giving the characteristic sequence of prime numbers in the first column. Row sums are 1,0,0,0,0,0,0,0,0,...

Links

Table of n, a(n) for n=1..78.

Formula

T(n, k) = If n = k then 1 else if k divides n then -Sum_{i=k+1..n} T(n, i) else T(n,k) = 0.

Example

{
{1},
{-1, 1},
{-1, 0, 1},
{0, -1, 0, 1},
{-1, 0, 0, 0, 1},
{0, 0, -1, 0, 0, 1},
{-1, 0, 0, 0, 0, 0, 1},
{0, 0, 0, -1, 0, 0, 0, 1},
{0, 0, -1, 0, 0, 0, 0, 0, 1},
{0, 0, 0, 0, -1, 0, 0, 0, 0, 1},
{-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1},
{0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1}
}

Mathematica

(* recurrence *) Clear[t, n, k, nn]; nn = 12; t[n_, k_] := t[n, k] = If[n == k, 1, If[Mod[n, k] == 0, -Sum[t[n, i], {i, k + 1, n}], 0]]; Flatten[Table[Table[t[n, k], {k, 1, n}], {n, 1, nn}]]

Crossrefs

Sequence in context: A115524 A370122 A117198 * A054525 A174852 A341517

Adjacent sequences: A271044 A271045 A271046 * A271048 A271049 A271050

Keyword

sign

Author

Mats Granvik, Mar 29 2016

Status

approved