A259266 a(n) is the unique odd-valued residue modulo 5^n of a number m such that m^2+1 is divisible by 5^n.
3, 7, 57, 443, 2057, 14557, 45807, 110443, 1672943, 6139557, 25670807, 123327057, 123327057, 5006139557, 19407922943, 102662389557, 407838170807, 3459595983307, 3459595983307, 79753541295807, 110981321985443, 110981321985443, 9647724486047943, 9647724486047943
Offset: 1
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Examples
If m^2+1 is divisible by 5, then m mod 5 is either 2 or 3; the odd value is 3, so a(1)=3. If m^2+1 is divisible by 5^2, then m mod 5^2 is either 7 or 18; the odd value is 7, so a(2)=7. If m^2+1 is divisible by 5^3, then m mod 5^3 is either 57 or 68; the odd value is 57, so a(3)=57.
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