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a(n) is the least p such that p + 5*k*(k+1) is prime for 0 <= k <= n-1 but not for k=n.
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%I #23 Mar 12 2024 09:42:06

%S 2,3,31,43,37,7,709,8941,1723,163,1801,13,32077430821,313296437089,

%T 106776242048569,3345710409941689

%N a(n) is the least p such that p + 5*k*(k+1) is prime for 0 <= k <= n-1 but not for k=n.

%C a(n) is the least p such that p, p+10, p+10+20, ..., p+10+20+...+10*(n-1) are prime but p+10+20+...+10*n is composite.

%e a(4) = 43 because 43, 43+10=53, 53+20=73, 73+30=103 are prime but 103+40=143 is composite, and no number smaller than 43 works.

%p f:= proc(p) local k;

%p for k from 1 while isprime(p+k*(k+1)*5) do od:

%p k

%p end proc:

%p A:= Vector(12): count:= 0:

%p for i from 1 while count < 12 do

%p v:= f(ithprime(i));

%p if A[v] = 0 then count:= count+1; A[v]:= ithprime(i) fi

%p od:

%p convert(A,list);

%t Table[p=1;m=5;Monitor[Parallelize[While[True,If[And[MemberQ[PrimeQ[Table[p+m*k*(k+1),{k,0,n-1}]],False]==False,PrimeQ[p+m*n*(n+1)]==False],Break[]];p++];p],p],{n,1,10}] (* _J.W.L. (Jan) Eerland_, Mar 08 2024 *)

%Y Cf. A164926.

%K nonn,more

%O 1,1

%A _J. M. Bergot_ and _Robert Israel_, May 27 2021

%E a(13)-a(14) from _Martin Ehrenstein_, May 28 2021

%E a(15)-a(16) from _Bert Dobbelaere_, Jun 07 2021