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A058055 a(n) is the smallest positive number m such that m^2 + n is the next prime > m^2.

%I A058055 #48 Nov 02 2023 05:45:38
%S A058055 1,3,8,5,12,11,18,51,82,49,234,23,42,75,86,231,174,107,288,63,80,69,
%T A058055 102,325,166,765,128,143,822,727,276,597,226,835,702,461,254,693,592,
%U A058055 797,1284,349,370,2337,596,645,3012,1033,590,4083,1490,757,882,833,1668
%N A058055 a(n) is the smallest positive number m such that m^2 + n is the next prime > m^2.
%C A058055 The primes are in A058056.
%H A058055 Zak Seidov, <a href="/A058055/b058055.txt">Table of n, a(n) for n = 1..500</a> (first 400 terms from T. D. Noe)
%F A058055 a(n) = Min{ m > 0 | m^2 + n is the next prime after m^2}.
%F A058055 A053000(a(n)) = n. - _Zak Seidov_, Apr 12 2013
%e A058055 n=6: a(6)=11 and 11^2+6 is 127, a prime; n=97: a(97) = 2144 and 2144^2+97 = 4596833, the least prime of the form m^2+97.
%p A058055 for m from 1 to 10^5 do
%p A058055    r:= nextprime(m^2)-m^2;
%p A058055    if not assigned(R[r]) then R[r]:= m end if;
%p A058055 end do:
%p A058055 J:= map(op,{indices(R)}):
%p A058055 N:= min({$1..J[-1]} minus J)-1:
%p A058055 [seq(R[j],j=1..N)]; # _Robert Israel_, Aug 10 2012
%t A058055 nn = 100; t = Table[0, {nn}]; found = 0; m = 0; While[found < nn, m++; k = NextPrime[m^2] - m^2; If[k <= nn && t[[k]] == 0, t[[k]] = m; found++]]; t (* _T. D. Noe_, Aug 10 2012 *)
%o A058055 (Sage)
%o A058055 R = {}   # After _Robert Israel_'s Maple script.
%o A058055 for m in (1..2^12) :
%o A058055     r = next_prime(m^2) - m^2
%o A058055     if r not in R : R[r] = m
%o A058055 L = sorted(R.keys())
%o A058055 for i in (1..len(L)-1) :
%o A058055     if L[i] != L[i-1]+1 : break
%o A058055 [R[k] for k in (1..i)]  # _Peter Luschny_, Aug 11 2012
%Y A058055 Cf. A053000, A070316, A070317.
%Y A058055 See A085099, A215249 for other versions.
%K A058055 nonn
%O A058055 1,2
%A A058055 _Labos Elemer_, Nov 20 2000
%E A058055 Definition corrected by _Zak Seidov_, Mar 03 2008, and again by _Franklin T. Adams-Watters_, Aug 10 2012