A051214 -id:A051214 - cp's OEIS frontend

A200523 Least m>0 such that n = 3^x-y^2 (mod m) has no solution, or 0 if no such m exists.

0, 0, 0, 0, 8, 0, 8, 9, 0, 0, 12, 0, 8, 9, 8, 20, 9, 0, 0, 12, 8, 80, 8, 0, 45, 9, 0, 0, 8, 80, 8, 9, 0, 45, 9, 20, 8, 21, 8, 80, 9, 80, 28, 9, 8, 0, 8, 0, 91, 9, 20, 36, 8, 0, 8, 12, 0, 80, 9, 80, 8, 9, 8, 28, 15, 0, 91, 9, 8, 45, 8, 0, 0, 15, 0, 20, 8, 0

Offset: 0

M. F. Hasler, Nov 18 2011

nonn

If such an m>0 exists, this proves that n is not in A051214, i.e., not of the form 3^x-y^2. On the other hand, if there are integers x, y such that n=3^x-y^2, then we know that a(n)=0.

			See A200507 for developed examples.
Some of the larger values include a(107)=17732, a(146)=1924, a(347)=4400, a(416)=2044, a(458)>30000.

M. F. Hasler, Table of n, a(n) for n = 0..457

Cf. A051204-A051221, A200522, A200524, A200505-A200520.

A200523(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))}

A201124 Differences between odd powers of 3 and the next smaller square.

Original entry on oeis.org

2, 2, 18, 71, 83, 747, 1679, 7538, 22394, 65186, 177578, 370898, 1497119, 2428283, 5285858, 47572722, 129918074, 274553378, 1128916463, 2107864523, 6892205234, 25794120683, 14732727599, 132594548391, 214986322034, 1934876898306, 8608610563379, 24645805942151, 63317186094563, 94369472696663

Offset: 1

Hugo Pfoertner, Nov 27 2011

nonn

			a(1)=3^1-1^2=2, a(2)=3^3-5^2=27-25=2, a(3)=3^5-15^2=243-225=18

Hugo Pfoertner, Table of n, a(n) for n = 1..500

Cf. A051214

 #-Floor[Sqrt[#]]^2&/@(3^Range[1,61,2])  (* Harvey P. Dale, Nov 01 2022 *)

a(n) = 3^(2*n-1) - floor(sqrt(3^(2*n-1)))^2

cp's OEIS Frontend

A200523 Least m>0 such that n = 3^x-y^2 (mod m) has no solution, or 0 if no such m exists.

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