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A194627 a(1)=1, a(n+1) = p(n)^2 + q(n)^2 + 1, where p(n) and q(n) are the number of prime and nonprime numbers respectively in the sequence so far.

%I A194627 #32 Nov 11 2024 07:27:01
%S A194627 1,2,3,6,9,14,21,30,41,46,59,66,81,98,117,138,161,186,213,242,273,306,
%T A194627 341,378,417,458,501,546,593,602,651,702,755,810,867,926,987,1050,
%U A194627 1115,1182,1251,1322,1395,1470,1547,1626,1707,1790,1875,1962,2051,2142
%N A194627 a(1)=1, a(n+1) = p(n)^2 + q(n)^2 + 1, where p(n) and q(n) are the number of prime and nonprime numbers respectively in the sequence so far.
%H A194627 Robert Israel, <a href="/A194627/b194627.txt">Table of n, a(n) for n = 1..10000</a>
%e A194627 For n=1, we have no primes and one nonprime (a(1)=1), so a(2)=0^2+1^2+1=2. Now we have one prime (a(2)=2) and one nonprime, so a(3)=1^2+1^2+1=3.
%p A194627 R:= 1: v:= 1: p:= 0: q:= 0:
%p A194627 for i from 2 to 100 do
%p A194627   if isprime(v) then p:= p+1 else q:= q+1 fi;
%p A194627   v:= p^2 + q^2 + 1;
%p A194627   R:= R,v
%p A194627 od:
%p A194627 R; # _Robert Israel_, Nov 10 2024
%t A194627 t = {1}; Do[ps = Count[t, _?(PrimeQ[#] &)]; AppendTo[t, ps^2 + (n - ps - 1)^2 + 1], {n, 2, 100}]; t (* _T. D. Noe_, Sep 15 2011 *)
%o A194627 (PARI) p=q=0;for(n=1,50,print1(k=p^2+q^2+1", ");if(isprime(k),p++,q++)) \\ _Charles R Greathouse IV_, Sep 16 2011
%Y A194627 Cf. A079545, A377791, A377882.
%K A194627 nonn,easy
%O A194627 1,2
%A A194627 _Greg Knowles_, Sep 15 2011