A204994 -id:A204994 - cp's OEIS frontend

A204999 a(n) = (1/n)*A204998(n).

3, 4, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

For a guide to related sequences, see A204892.

Positions of 3's seem to be given by a subsequence of A104777. - Antti Karttunen, Sep 29 2018

Antti Karttunen, Table of n, a(n) for n = 1..23005

Cf. A204996, A204997, A204998, A204892.

Mathematica
```
(See the program at A204994.)
```

A204999(n) = { my(d); for(k=sqrtint(1+n), oo, for(j=1,k-1,if(!((d=(k^2)-(j^2))%n),return(d/n),if(dAntti Karttunen, Sep 28 2018

a(n) = (A204996(n)-A204997(n))/n.

More terms from Antti Karttunen, Sep 28 2018

A204996 Least k^2 such that n divides k^2-j^2 for some j

Original entry on oeis.org

4, 9, 4, 9, 9, 16, 16, 9, 25, 36, 36, 16, 49, 64, 16, 25, 81, 81, 100, 36, 25, 144, 144, 25, 100, 196, 36, 64, 225, 64, 256, 36, 49, 324, 36, 81, 361, 400, 64, 49, 441, 100, 484, 144, 49, 576, 576, 49, 196, 225, 100, 196, 729, 144, 64, 81, 121, 900, 900, 64

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

For a guide to related sequences, see A204892.

Cf. A204892, A204994.

Mathematica
```
(See the program at A204994.)
```

A204997 The square j^2 such that n divides k^2-j^2>0, where k is the least positive integer for which such a j exists.

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 9, 1, 16, 16, 25, 4, 36, 36, 1, 9, 64, 9, 81, 16, 4, 100, 121, 1, 25, 144, 9, 36, 196, 4, 225, 4, 16, 256, 1, 9, 324, 324, 25, 9, 400, 16, 441, 100, 4, 484, 529, 1, 49, 25, 49, 144, 676, 36, 9, 25, 64, 784, 841, 4

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

For a guide to related sequences, see A204892.

Cf. A204892, A204994.

Mathematica
```
(See the program at A204994.)
```

A204998 a(n) = k^2 - j^2, where (k^2,j^2) is the least pair of distinct squares for which n divides their difference.

Original entry on oeis.org

3, 8, 3, 8, 5, 12, 7, 8, 9, 20, 11, 12, 13, 28, 15, 16, 17, 72, 19, 20, 21, 44, 23, 24, 75, 52, 27, 28, 29, 60, 31, 32, 33, 68, 35, 72, 37, 76, 39, 40, 41, 84, 43, 44, 45, 92, 47, 48, 147, 200, 51, 52, 53, 108, 55, 56, 57, 116, 59, 60, 61, 124, 63, 64, 65, 132, 67, 68, 69, 140, 71, 72, 73, 148, 75, 76, 77, 156, 79, 80, 81, 164

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

For a guide to related sequences, see A204892.

Antti Karttunen, Table of n, a(n) for n = 1..23005

Cf. A204994, A204892, A204996, A204997, A204999.

Mathematica
```
(See the program at A204994.)
```

A204998(n) = { my(d); for(k=sqrtint(1+n), oo, for(j=1,k-1,if(!((d=(k^2)-(j^2))%n),return(d),if(dAntti Karttunen, Sep 28 2018

a(n) = A204996(n) - A204997(n).

More terms from Antti Karttunen, Sep 28 2018

A204905 Least k such that n divides k^2-j^2 for some j satisfying 1<=j

Original entry on oeis.org

2, 3, 2, 3, 3, 4, 4, 3, 5, 6, 6, 4, 7, 8, 4, 5, 9, 9, 10, 6, 5, 12, 12, 5, 10, 14, 6, 8, 15, 8, 16, 6, 7, 18, 6, 9, 19, 20, 8, 7, 21, 10, 22, 12, 7, 24, 24, 7, 14, 15, 10, 14, 27, 12, 8, 9, 11, 30, 30, 8

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

See A204892 for a discussion and guide to related sequences.

			1 divides 2^2-1^2, so a(1)=2
2 divides 3^2-1^2, so a(2)=3
3 divides 2^2-a^2, so a(3)=2
4 divides 3^2-a^2, so a(4)=3

Cf. A000290, A204892.

s[n_] := s[n] = n^2; z1 = 600; z2 = 60;
Table[s[n], {n, 1, 30}]     (* A000290 *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]     (* A120070 *)
v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
d[n_] := d[n] = First[Delete[w[n], Position[w[n], 0]]]
Table[d[n], {n, 1, z2}]   (* A204994 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]
m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]
j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2
Table[k[n], {n, 1, z2}]       (* A204905 *)
Table[j[n], {n, 1, z2}]       (* A204995 *)
Table[s[k[n]], {n, 1, z2}]    (* A204996 *)
Table[s[j[n]], {n, 1, z2}]    (* A204997 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A204998 *)
Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A204999 *)

A204995 The index jA204905) for which such j exists.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 1, 4, 4, 5, 2, 6, 6, 1, 3, 8, 3, 9, 4, 2, 10, 11, 1, 5, 12, 3, 6, 14, 2, 15, 2, 4, 16, 1, 3, 18, 18, 5, 3, 20, 4, 21, 10, 2, 22, 23, 1, 7, 5, 7, 12, 26, 6, 3, 5, 8, 28, 29, 2

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

For a guide to related sequences, see A204892.

Cf. A204892, A204994.

Mathematica
```
(See the program at A204994.)
```

cp's OEIS Frontend

A204999 a(n) = (1/n)*A204998(n).

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A204996 Least k^2 such that n divides k^2-j^2 for some j

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Mathematica

A204997 The square j^2 such that n divides k^2-j^2>0, where k is the least positive integer for which such a j exists.

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Mathematica

A204998 a(n) = k^2 - j^2, where (k^2,j^2) is the least pair of distinct squares for which n divides their difference.

Views

Author

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Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A204905 Least k such that n divides k^2-j^2 for some j satisfying 1<=j

Views

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Examples

Crossrefs

Programs

Mathematica

A204995 The index jA204905) for which such j exists.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica