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A233998 Values of n such that numbers of the form x^2+n*y^2 for some integers x, y cannot have prime factor of 5 raised to an odd power.

%I A233998 #44 Apr 22 2025 04:33:26
%S A233998 2,3,7,8,12,13,17,18,22,23,27,28,32,33,37,38,42,43,47,48,50,52,53,57,
%T A233998 58,62,63,67,68,72,73,75,77,78,82,83,87,88,92,93,97,98,102,103,107,
%U A233998 108,112,113,117,118,122,123,127,128,132,133,137,138,142,143,147,148,152,153,157,158
%N A233998 Values of n such that numbers of the form x^2+n*y^2 for some integers x, y cannot have prime factor of 5 raised to an odd power.
%C A233998 Equivalently, this sequence is the union of numbers of the form 25^n*(5*n+2) and numbers of the form 25^n*(5*n+3).
%F A233998 a(n) = 2.4 n + O(log n). - _Charles R Greathouse IV_, Dec 19 2013
%o A233998 (PARI) is(n)=n/=25^valuation(n, 25); n%5==2||n%5==3 \\ _Charles R Greathouse IV_ and _V. Raman_, Dec 19 2013
%o A233998 (Python)
%o A233998 from sympy import integer_log
%o A233998 def A233998(n):
%o A233998     def bisection(f,kmin=0,kmax=1):
%o A233998         while f(kmax) > kmax: kmax <<= 1
%o A233998         kmin = kmax >> 1
%o A233998         while kmax-kmin > 1:
%o A233998             kmid = kmax+kmin>>1
%o A233998             if f(kmid) <= kmid:
%o A233998                 kmax = kmid
%o A233998             else:
%o A233998                 kmin = kmid
%o A233998         return kmax
%o A233998     def f(x): return n+x-sum(((m:=x//25**i)-2)//5+(m-3)//5+2 for i in range(integer_log(x,25)[0]+1))
%o A233998     return bisection(f,n,n) # _Chai Wah Wu_, Mar 19 2025
%Y A233998 Cf. A055046, A055047, A233999.
%K A233998 nonn,easy
%O A233998 1,1
%A A233998 _V. Raman_, Dec 18 2013