A051217 -id:A051217 - cp's OEIS frontend

A200506 Least m>0 such that n = 6^x-y^2 (mod m) has no solution, or 0 if no such m exists.

0, 0, 0, 5, 5, 0, 0, 9, 5, 5, 7, 0, 63, 5, 5, 36, 9, 7, 5, 5, 0, 44, 9, 5, 5, 9, 16, 0, 5, 5, 16, 7, 0, 5, 5, 0, 0, 21, 5, 5, 9, 16, 16, 5, 5, 7, 12, 0, 5, 5, 28, 36, 7, 5, 5, 12, 192, 16, 5, 5, 37, 9, 16, 5, 5, 24, 7, 9, 5, 5, 9, 0, 0, 5, 5, 36, 9, 52, 5, 5

Offset: 0

M. F. Hasler, Nov 18 2011

nonn

If such an m>0 exists, this proves that n is not in A051217, i.e., not of the form 6^x-y^2. On the other hand, if there are integers x, y such that n=6^x-y^2, then we know that a(n)=0.

			See A200507.

M. F. Hasler, Table of n, a(n) for n = 0..1000

Cf. A051204-A051221, A200522, A200523, A200524, A200505-A200520.

A200506(n,b=6,p=3)={ my( x=0, qr, bx, seen ); for( m=2,9e9, while( x^p < m, issquare(b^x-n) & return(0); x++); qr=vecsort(vector(m,i,i^2+n)%m,,8); seen=0; bx=1; until( bittest(seen+=1<bx & break; next(3))); return(m))}

a(3+5k)=a(4+5k)=5, a(10+35k)=a(17+35k)=a(31+35k)=7 for all k>=0.

a(n)=9 for n=7, 16, 22, 70, 76 and 85 (mod 90).

A201122 Differences between odd powers of 6 and the next smaller square.

Original entry on oeis.org

2, 20, 32, 95, 3420, 8847, 89927, 494576, 1347932, 48525552, 265807127, 682379927, 3237653360, 52571448911, 356954431580, 1333226567615, 6534477744687, 69394484050880, 10500704463815, 378025360697340, 13608912985104240, 60046182811381232, 227226500274052935, 442409686123219952, 15926748700435918272

Offset: 1

Hugo Pfoertner, Nov 27 2011

nonn

			a(1)=6^1-2^2=2, a(2)=6^3-14^2=216-196=20, a(3)=6^5-88^2=7776-7744=32

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

Cf. A051217, A200440, A200506

#-Floor[Sqrt[#]]^2&/@(6^Range[1,51,2]) (* Harvey P. Dale, Aug 12 2012 *)

a(n)=6^(2*n-1) - floor(sqrt(6^(2*n-1)))^2

A134989 Numbers expressible in more than one way as 6^x-y^2.

Original entry on oeis.org

0, 20, 32, 95, 207, 720, 1152, 1215, 1287, 3420, 3807, 6255, 6407, 7452, 7767, 18095, 23247, 25920, 41472, 43740, 46332, 46647, 69255, 123120, 137052, 174087, 211815, 217935, 225180, 230652, 268272, 279612, 279927, 651420, 836892, 933120, 1416447

Offset: 1

Zak Seidov, Feb 05 2008

nonn

Numbers n such that equation 6^x-y^2=n has more than one solution.

			0=6^(2k)-(6^k)^2, k=1,2,..
20=6^2-4^2=6^3-14^2,
32=6^2-2^2=6^5-88^2,
95=6^3-11^2=6^7-529^2,
207=6^3-3^2=6^4-33^2=6^5-87^2,
720=6^4-24^2=6^5-84^2,
1152=6^4-12^2=6^7-528^2,
1215=6^4-9^2=6^5-81^2,
1287=6^4-3^2=6^6-213^2.

Cf. A051217.

lst = {}; Do[ t = 6^x - y^2; If[t < 10^7/7, AppendTo[lst, t]], {x, 185}, {y, (a = Floor@Sqrt[6^x - 10^7]; If[Element[a, Reals], a, 0]), Floor@Sqrt[6^x]}]; lst = Sort@lst; lsu = {}; Do[ If[lst[[n]] == lst[[n + 1]], AppendTo[lsu, lst[[n]]]], {n, -1 + Length@lst}]; Union@lsu (* Robert G. Wilson v, Feb 09 2008 *)

More terms from Robert G. Wilson v, Feb 09 2008

A135476 Positive numbers of the form 6^x-y^2 with repetition.

Original entry on oeis.org

1, 2, 5, 6, 11, 20, 20, 27, 32, 32, 35, 36, 47, 71, 72, 95, 116, 135, 140, 152, 167, 180, 191, 200, 207, 207, 207, 212, 215, 216, 272, 335, 380, 396, 455, 512, 551, 567, 620, 671, 720, 720, 767, 812, 855, 887, 896, 935, 972, 1007, 1040, 1052, 1071, 1100, 1127

Offset: 1

Zak Seidov, Feb 06 2008

nonn

Cf. A051217, A134989.

cp's OEIS Frontend

A200506 Least m>0 such that n = 6^x-y^2 (mod m) has no solution, or 0 if no such m exists.

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A201122 Differences between odd powers of 6 and the next smaller square.

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A134989 Numbers expressible in more than one way as 6^x-y^2.

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A135476 Positive numbers of the form 6^x-y^2 with repetition.

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