A234299 a(n) = |A| is the smallest order of a set A of consecutive integers which has Euler-phi(3*5*7*11*...*Pn) members coprime to 3*5*7*..*Pn, where Pn is the n-th odd prime.

2, 13, 101, 1149, 15005, 255243, 4849829, 111546416, 3234846593, 100280245037, 3710369067373, 152125131763569, 6541380665834971, 307444891294245656, 16294579238595022313, 961380175077106319477, 58644190679703485491570, 3929160775540133527939470

Offset

1,1

Comments

The sequence is strongly associated with A072752.

In A072752, you are looking for a maximum sized set of consecutive numbers where none are counted by Euler's phi(3*5*7*...*Pn); this sequence looks for a minimum sized set of consecutive numbers where all the numbers counted by Euler are included.

One candidate for A (not necessarily of minimum size) is the set {1, 2, 3,..., 3*5*..*Pn}, which has the requested number of coprime elements. This yields the simple upper bound a(n) <= A070826(n+1). - R. J. Mathar, May 03 2017

Links

Table of n, a(n) for n=1..18.

Formula

Let A = { set of any k consecutive integers}, |A| its size.

Let B = {x IN A | gcd(x, 3*5*7...Prime(n))=1}.

Condition: |B| = phi(3*5*7...Prime(n))= A005867(n+1).

a(n) = minimum(|A|) which meets the above condition.

a(n) = A070826(n+1) - A072752(n+1).

Example

a(1)=2,  phi(3) = 2, A={1,2 }, B={1,2}, |B|=2  gcd(1,3) = 1; gcd(2,3) = 1; minimum(|A|) = 2.
a(2)=13, phi(3*5) = 8, A={7,8,9,10,...,19}, B={7, 8, 11, 13, 14, 16, 17, 19}, |B|=8, A was chosen so |A| is a minimum.

Crossrefs

Cf. A005867, A070826, A072752.

Sequence in context: A030519 A141116 A330349 * A301355 A077246 A365148

Adjacent sequences: A234296 A234297 A234298 * A234300 A234301 A234302

Keyword

hard,nonn

Author

John F. Morack, Dec 22 2013

Extensions

Edited by R. J. Mathar, May 03 2017

Status

approved