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.

2, 3, 7, 8, 12, 13, 17, 18, 22, 23, 27, 28, 32, 33, 37, 38, 42, 43, 47, 48, 50, 52, 53, 57, 58, 62, 63, 67, 68, 72, 73, 75, 77, 78, 82, 83, 87, 88, 92, 93, 97, 98, 102, 103, 107, 108, 112, 113, 117, 118, 122, 123, 127, 128, 132, 133, 137, 138, 142, 143, 147, 148, 152, 153, 157, 158

Offset: 1

Author

V. Raman, Dec 18 2013

Keywords

• nonn
• easy

Comments

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).

Crossrefs

Cf. A055046, A055047, A233999.

Programs

PARI

is(n)=n/=25^valuation(n, 25); n%5==2||n%5==3 \\ Charles R Greathouse IV and V. Raman, Dec 19 2013

Python

from sympy import integer_log
def A233998(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    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))
    return bisection(f,n,n) # Chai Wah Wu, Mar 19 2025

Formula

a(n) = 2.4 n + O(log n). - Charles R Greathouse IV, Dec 19 2013