A234299 - OEIS

%I #35 Nov 25 2024 12:40:20

%S 2,13,101,1149,15005,255243,4849829,111546416,3234846593,100280245037,

%T 3710369067373,152125131763569,6541380665834971,307444891294245656,

%U 16294579238595022313,961380175077106319477,58644190679703485491570,3929160775540133527939470

%N 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.

%C The sequence is strongly associated with A072752.

%C 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.

%C 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

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

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

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

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

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

%e 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.

%e 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.

%Y Cf. A005867, A070826, A072752.

%K hard,nonn

%O 1,1

%A _John F. Morack_, Dec 22 2013

%E Edited by _R. J. Mathar_, May 03 2017