A115039 Exponent n for the smallest x such that x^2 + p = y^n over the set of primes p.

3, 2, 2, 3, 3, 2, 4, 2, 3, 2, 5, 2, 2, 2, 7, 6, 2, 3, 2, 9, 2, 7, 2, 3, 4, 2, 7, 2, 3, 2, 7, 2, 2, 2, 2, 9, 2, 2, 3, 2, 5, 2, 3, 2, 2, 3, 2, 9, 5, 2, 2, 5, 4, 7, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 4, 2, 2, 2, 3, 11, 2, 2, 2, 2, 2, 2, 2, 2, 5, 9, 3, 3, 2, 4, 2, 2, 9, 2, 2, 9, 2, 2, 9, 2, 2, 2, 2, 3, 3, 2, 2, 2

Offset

1,1

Comments

Conjecture: There will always be an x,y,n such that x^2 + p = y^n for all primes p.

Links

Table of n, a(n) for n=1..105.

Example

5 is the smallest number that when we add its square to prime 2, we get a perfect power, 3^3. So 3 is the first entry.

Prog

(PARI) sqplusp(n) = { local(p, x, y, c=0); forprime(p=2, n, for(x=1, n, y=x^2+p; if(ispower(y), print1(exponent(y)", "); c++; break ) ) ); print(); print(c) } exponent(n) = \ Return the exponent if n is a perfect power { local(x, ln, j, e=0); ln=omega(n); x=factor(n); e=x[1, 2]; for(j=2, ln, if(x[j, 2] < e, e=x[j, 2]) ); return(e) }

Crossrefs

Sequence in context: A347381 A075392 A069901 * A032536 A374167 A115061

Adjacent sequences: A115036 A115037 A115038 * A115040 A115041 A115042

Keyword

easy,nonn

Author

Cino Hilliard, Feb 26 2006

Status

approved