A128708 -id:A128708 - cp's OEIS frontend

A128707 Least number having the maximal distance between consecutive integers coprime to n.

1, 1, 2, 1, 4, 1, 6, 1, 2, 3, 10, 1, 12, 5, 4, 1, 16, 1, 18, 3, 5, 9, 22, 1, 4, 11, 2, 5, 28, 1, 30, 1, 10, 15, 13, 1, 36, 17, 11, 3, 40, 5, 42, 9, 4, 21, 46, 1, 6, 3, 16, 11, 52, 1, 9, 5, 17, 27, 58, 1, 60, 29, 5, 1, 24, 7, 66, 15, 22, 3, 70, 1, 72, 35, 4, 17, 20, 11, 78, 3, 2, 39, 82, 5, 33

Offset: 1

T. D. Noe, Mar 24 2007

nonn

Let j(n) be the Jacobsthal function (A048669): maximal distance between consecutive integers coprime to n. Then a(n) is the least k>0 such that k+1,k+2,...k+j(n)-1 are not coprime to n. If n is prime and e>0, then j(n^e)=2 and a(n^e)=n-1. If n>2 is prime, then a(2n)=n-2. If m is the squarefree kernel of n (A007947), then j(n)=j(m) and a(n)=a(m). For composite n, a(n)A055932. When n is the product of the first r primes (A002110), then a(n)+1 begins (or is inside) a prime gap of size at least A048670(r).

			The numbers coprime to 10 are 1,3,7,9,11,13,17,19,... Observe that the differences are periodic: 2,4,2,2,2,4,2,... The largest distance between the coprime numbers is 4, which first occurs between 3 and 7. Hence j(10)=4 and a(10)=3.

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

Cf. A128708 (number of times the maximal value occurs).

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

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A128707 Least number having the maximal distance between consecutive integers coprime to n.

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