A051205 -id:A051205 - cp's OEIS frontend

A247763 The number of representations of n as n=x^2-2^y, x>0, y>=0.

1, 1, 1, 1, 1, 0, 1, 2, 1, 0, 0, 1, 0, 1, 1, 1, 4, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 2, 2, 1, 1, 1, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 2, 0, 0, 4, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 2, 0, 0, 0, 1

Offset: 1

R. J. Mathar, Oct 19 2014

nonn

The n for which the count is nonzero are listed in A051204.

R. J. Mathar, Table of n, a(n) for n = 1..136
Maohua Le, On the number of solutions of the generalized Ramanujan-Nagell equation x^2-d=2^(n+2), Acta Arithm. 60 (1991) 149.
R. J. Mathar, The number of representations of n of the form n=x^2-2^y [PDF]
Nicholas Tzanakis, On the diophantine equation y^2-d=2^k, J. Number Theory 17 (1983) 144.

Cf. A051205.

A200513 Least m>0 such that n = y^2 - 3^x (mod m) has no solution, or 0 if no such m exists.

Original entry on oeis.org

0, 0, 8, 0, 8, 9, 0, 0, 0, 0, 8, 9, 8, 0, 9, 0, 0, 12, 8, 0, 8, 28, 0, 9, 0, 20, 8, 0, 8, 9, 20, 80, 9, 0, 8, 0, 8, 0, 9, 63, 0, 9, 8, 80, 8, 20, 0, 9, 0, 28, 8, 63, 8, 12, 0, 0, 9, 36, 8, 9, 8, 0, 12, 0, 532, 9, 8, 80, 8, 108, 20, 15, 0, 0, 8, 63, 8, 9, 0

Offset: 0

M. F. Hasler, Nov 18 2011

nonn

To prove that an integer n is in A051205, it is sufficient to find integers x,y such that y^2 - 3^x = n. In that case, a(n)=0. To prove that n is *not* in A051205, it is sufficient to find a modulus m for which the (finite) set of all possible values of 3^x and y^2 (mod m) allows us to deduce that y^2 - 3^x can never equal n. The present sequence lists the smallest such m>0, if it exists.

			See A200512.

Cf. A051204-A051221, A200505-A200520.

A200513(n,b=3,p=3)={ my( x=0, qr, bx, seen ); for( m=3,9e9, while( x^p < m, issquare(b^x+n) & return(0); x++); qr=vecsort(vector(m,y,y^2-n)%m,,8); seen=0; bx=1; until( bittest(seen+=1<bx & break; next(3))); return(m))}

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A247763 The number of representations of n as n=x^2-2^y, x>0, y>=0.

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A200513 Least m>0 such that n = y^2 - 3^x (mod m) has no solution, or 0 if no such m exists.

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