A124934 -id:A124934 - cp's OEIS frontend

A094178 Numbers n such that 4n+1 is divisible only by primes of form 4m+1 (i.e., by the Pythagorean primes A002144).

1, 3, 4, 6, 7, 9, 10, 13, 15, 16, 18, 21, 22, 24, 25, 27, 28, 31, 34, 36, 37, 39, 42, 43, 45, 46, 48, 49, 51, 55, 57, 58, 60, 64, 66, 67, 69, 70, 72, 73, 76, 78, 79, 81, 84, 87, 88, 91, 93, 94, 97, 99, 100, 102, 105, 106, 108, 111, 112, 114, 115, 120, 121, 123, 126, 127

Offset: 1

Lekraj Beedassy, May 06 2004

nonn

For the actual numbers 4n+1, see A008846(n).

Complement of A124934; A125203(a(n)) = 0; A000290 and A000217 are subsequences. - Reinhard Zumkeller, Nov 24 2006

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

import Data.List (elemIndices)
a094178 n = a094178_list !! (n-1)
a094178_list = map (+ 1) $ elemIndices 0 a125203_list
-- Reinhard Zumkeller, Jan 02 2013

More terms from Ray Chandler, Jun 20 2004

A125199 Triangle read by rows: T(n,k) = 4nk - n - k, 1<=k<=n.

Original entry on oeis.org

2, 5, 12, 8, 19, 30, 11, 26, 41, 56, 14, 33, 52, 71, 90, 17, 40, 63, 86, 109, 132, 20, 47, 74, 101, 128, 155, 182, 23, 54, 85, 116, 147, 178, 209, 240, 26, 61, 96, 131, 166, 201, 236, 271, 306, 29, 68, 107, 146, 185, 224, 263, 302, 341, 380, 32, 75, 118, 161, 204, 247

Offset: 1

Reinhard Zumkeller, Nov 24 2006

A124934 gives the range: for n,k with 1<=k<=n exists at least one m such that A124934(m)=T(n,k);
row sums give A125200; central terms give A125201;
T(n,1) = A016789(n-1);
T(n,2) = A017041(n-1) for n>1;
T(n,3) = A017485(n-1) for n>2;
T(n,n-1) = A125202(n) for n>1;
T(n,n) = A002939(n).

Flatten[Table[4*n*k-n-k,{n,15},{k,n}]] (* Harvey P. Dale, Nov 15 2014 *)

A125203 Number of ways to write n as 4xy - x - y with 1<=x<=y.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Nov 24 2006

nonn

a(A094178(n))=0; a(A124934(n))>0; a(A125217(n))=1; a(A125218(n))>1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A094178, A124934, A125217, A125218.

Cf. A193773.

a125203 n = length [() | x <- [1 .. (n + 1) `div` 3],
                         let (y,m) = divMod (x + n) (4 * x - 1),
                         x <= y, m == 0]
-- Reinhard Zumkeller, Jan 02 2013

a[n_] := Solve[1<=x<=y && n == 4 x y - x - y, {x, y}, Integers] // Length;
Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Oct 12 2021 *)

A125217 Numbers that can be written uniquely as 4xy-x-y with 1<=x<=y.

Original entry on oeis.org

2, 5, 8, 11, 12, 14, 17, 19, 20, 23, 29, 30, 32, 33, 35, 38, 40, 44, 50, 52, 53, 54, 59, 61, 62, 63, 65, 75, 77, 80, 82, 83, 85, 90, 92, 95, 98, 103, 104, 109, 113, 117, 118, 119, 122, 124, 125, 129, 132, 134, 137, 138, 143, 145, 147, 149, 151, 158, 159, 162, 164, 167

Offset: 1

Reinhard Zumkeller, Nov 24 2006

nonn

A125203(a(n)) = 1.

A124934 is the union of this sequence and A125218.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

import Data.List (elemIndices)
a125217 n = a125217_list !! (n-1)
a125217_list = map (+ 1) $ elemIndices 1 a125203_list
-- Reinhard Zumkeller, Jan 02 2013

A125218 Numbers having at least two representations as 4xy-x-y with 1<=x<=y.

Original entry on oeis.org

26, 41, 47, 56, 68, 71, 74, 86, 89, 96, 101, 107, 110, 116, 128, 131, 140, 146, 152, 155, 161, 166, 173, 176, 182, 185, 191, 194, 201, 206, 209, 215, 221, 224, 236, 239, 242, 250

Offset: 1

Reinhard Zumkeller, Nov 24 2006

nonn

A125203(a(n)) > 1.

A124934 is the union of this sequence and A125217.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

import Data.List (findIndices)
a125218 n = a125218_list !! (n-1)
a125218_list = map (+ 1) $ findIndices (> 1) a125203_list
-- Reinhard Zumkeller, Jan 02 2013

A380644 Numbers that can be expressed as 4jk+j+k, j,k >= 1, as well as 4jk-j-k, j,k >= 2.

Original entry on oeis.org

26, 41, 47, 56, 61, 68, 71, 74, 86, 89, 96, 101, 107, 110, 116, 128, 131, 140, 146, 151, 152, 155, 159, 161, 166, 173, 176, 182, 185, 191, 194, 201, 206, 208, 209, 215, 221, 224, 236, 239, 242, 250, 251, 257, 261, 263, 266, 271, 272, 278, 281, 290, 293, 296, 299

Offset: 1

Hugo Pfoertner, Jan 29 2025

nonn

			a(1) = 26, because 26 = 4*1*5 + 1 + 5 = 4*2*4 - 2 - 4.
a(2) = 41: 41 = 4*1*8 + 1 + 8 = 4*3*4 - 3 - 4.

Cf. A054520, A124934, A125217, A125218, A380140, A380509, A380572.

cp's OEIS Frontend

A094178 Numbers n such that 4n+1 is divisible only by primes of form 4m+1 (i.e., by the Pythagorean primes A002144).

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A125199 Triangle read by rows: T(n,k) = 4*n*k - n - k, 1<=k<=n.

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Mathematica

A125203 Number of ways to write n as 4*x*y - x - y with 1<=x<=y.

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Mathematica

A125217 Numbers that can be written uniquely as 4*x*y-x-y with 1<=x<=y.

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Haskell

A125218 Numbers having at least two representations as 4*x*y-x-y with 1<=x<=y.

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Haskell

A380644 Numbers that can be expressed as 4*j*k+j+k, j,k >= 1, as well as 4*j*k-j-k, j,k >= 2.

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Examples

Crossrefs

A125199 Triangle read by rows: T(n,k) = 4nk - n - k, 1<=k<=n.

A125203 Number of ways to write n as 4xy - x - y with 1<=x<=y.

A125217 Numbers that can be written uniquely as 4xy-x-y with 1<=x<=y.

A125218 Numbers having at least two representations as 4xy-x-y with 1<=x<=y.

A380644 Numbers that can be expressed as 4jk+j+k, j,k >= 1, as well as 4jk-j-k, j,k >= 2.