A075557 - OEIS

%I #14 Dec 08 2024 04:00:55

%S 3,5,7,13,37,31,23,17,53,11,251,29,79,43,73,61,97,67,107,173,59,103,

%T 199,163,71,149,47,101,509,89,151,283,229,271,211,109,257,113,269,157,

%U 241,331,83,389,41,313,1543,19,307,463,373,277,811,457,137,191,419,311,197

%N a(n) is the smallest odd prime such that (1) a(n) doesn't already appear in the sequence; (2) the n-th partial sum is divisible by n; and (3) the n-th partial sum is relatively prime to n+1.

%C Condition (3) is needed to ensure that a(n+1) exists.

%H Robert Israel, <a href="/A075557/b075557.txt">Table of n, a(n) for n = 1..10000</a>

%e a(5)=37: The 4th partial sum is 28. 7 is the smallest odd prime that satisfies (2) (28+7=35), but 7 has already been used. 17 satisfies (1) and (2) (28+17=45), but 45+a(6) must be a multiple of 6 and the only odd prime satisfying that requirement is 3, which has already been used. 37 works (28+37=65).

%p N:= 10000: # for terms until the first term > N.

%p Cands:= select(isprime, [seq(i,i=3..N,2)]):

%p nC:= nops(Cands):

%p R:= NULL: s:= 0:

%p for n from 1 do

%p   found:= false;

%p   for i from 1 to nC do

%p     if s + Cands[i] mod n = 0 and igcd(s + Cands[i],n+1) = 1 then

%p        R:= R, Cands[i];

%p        s:= s + Cands[i];

%p        Cands:= subsop(i=NULL,Cands);

%p        nC:= nC-1;

%p        found:= true;

%p        break

%p      fi

%p   od;

%p   if not found then break fi

%p od:

%p R; # _Robert Israel_, Dec 07 2024

%Y Cf. A065091 (odd primes).

%K nonn,easy,look

%O 1,1

%A _Amarnath Murthy_, Sep 24 2002

%E Edited by _David Wasserman_, Jun 27 2003

%E Corrected by _Robert Israel_, Dec 07 2024