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User: John F. Morack

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John F. Morack has authored 1 sequences.

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.

Original entry on oeis.org

2, 13, 101, 1149, 15005, 255243, 4849829, 111546416, 3234846593, 100280245037, 3710369067373, 152125131763569, 6541380665834971, 307444891294245656, 16294579238595022313, 961380175077106319477, 58644190679703485491570, 3929160775540133527939470
Offset: 1

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Author

John F. Morack, Dec 22 2013

Keywords

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

Examples

			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.

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).

Extensions

Edited by R. J. Mathar, May 03 2017

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