A128708 Let j(n) be the Jacobsthal function, A048669. Then a(n) is the number of times that the gap j(n) appears between consecutive numbers <= n+1 and coprime to n.

1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 2, 1, 1, 2, 8, 1, 3, 1, 2, 2, 1, 1, 4, 5, 1, 9, 2, 1, 2, 1, 16, 2, 1, 2, 6, 1, 1, 2, 4, 1, 2, 1, 2, 6, 1, 1, 8, 7, 5, 2, 2, 1, 9, 2, 4, 2, 1, 1, 4, 1, 1, 6, 32, 2, 2, 1, 2, 2, 2, 1, 12, 1, 1, 10, 2, 2, 2, 1, 8, 27, 1, 1, 4, 2, 1, 2, 4, 1, 6, 2, 2, 2, 1, 2, 16, 1, 7, 6, 10

Offset

1,4

Comments

Differs from A087653 starting at n=35. For prime n and e>0, a(n^e)=n^(e-1). The closely-related sequence A128707 satisfies the inequality a(n)*A128707(n) <= n-1, with equality for prime n. If m is the squarefree kernel of n (A007947), then a(n)/a(m) = n/m.

Links

T. D. Noe, Table of n, a(n) for n=1..10000

Example

The numbers coprime to 15 are 1,2,4,7,8,11,13,14,16,17,19,22,... Observe that the differences are periodic: 1,2,3,1,3,2,1,2,1,2,3,... The maximum value is 3, which occurs twice in the first period. Hence a(15)=2.

Mathematica

JacobsthalCount[n_] := Module[{g, d, mx}, g=Select[Range[n+1], GCD[n, # ]==1&]; d=Rest[g]-Most[g]; mx=Max@@d; Count[d, mx]]; Table[JacobsthalCount[n], {n, 100}]

Crossrefs

Sequence in context: A003557 A073752 A346487 * A087653 A295666 A355003

Adjacent sequences: A128705 A128706 A128707 * A128709 A128710 A128711

Keyword

nonn

Author

T. D. Noe, Mar 24 2007

Status

approved