cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A260647 Numbers that are the sum of two distinct nonzero triangular numbers.

Original entry on oeis.org

4, 7, 9, 11, 13, 16, 18, 21, 22, 24, 25, 27, 29, 31, 34, 36, 37, 38, 39, 42, 43, 46, 48, 49, 51, 55, 56, 57, 58, 60, 61, 64, 65, 66, 67, 69, 70, 72, 73, 76, 79, 81, 83, 84, 87, 88, 91, 92, 93, 94, 97, 99, 100, 101, 102, 106, 108, 111, 112, 114, 115, 119, 120
Offset: 1

Views

Author

Arkadiusz Wesolowski, Dec 02 2015

Keywords

Comments

The sequence contains every square greater than 1.
Conjecture: the sequence contains infinitely many primes.

Examples

			24 = 3 + 21, so 24 is in the sequence.

Crossrefs

Cf. A000217, A265140 (exactly one way), A262749 (more than one way), A265134 (exactly two ways), A265135 (more than two ways), A265136 (exactly three ways), A265137 (more than three ways), A265138 (exactly four ways).
Subsequence of A051533.

Programs

  • Mathematica
    r = 120; lst = Table[0, {r}]; lim = Floor[Sqrt[8*r - 7]]; Do[num = (i^2 + i)/2 + (j^2 + j)/2; If[num <= r, lst[[num]]++], {i, lim}, {j, i - 1}]; Flatten@Position[lst, n_ /; n > 0]
With[{nn=20},Select[Union[Total/@Subsets[Accumulate[Range[nn]],{2}]],#<= (nn(nn+1))/2+1&]] (* Harvey P. Dale, Jul 26 2020 *)

Formula

{k: A307597(k) > 0 }. - R. J. Mathar, Apr 28 2020

A265134 Numbers that are the sum of two distinct nonzero triangular numbers in exactly two ways.

Original entry on oeis.org

16, 31, 46, 51, 76, 94, 111, 121, 123, 126, 133, 141, 146, 156, 157, 172, 174, 186, 191, 196, 198, 216, 225, 226, 231, 237, 241, 246, 259, 268, 281, 286, 289, 291, 297, 301, 310, 315, 321, 326, 328, 336, 346, 354, 366, 367, 379, 384, 391, 396, 412, 416
Offset: 1

Views

Author

Arkadiusz Wesolowski, Dec 02 2015

Keywords

Crossrefs

Cf. A000217, A051533, A260647, A265140 (exactly one way), A262749 (more than one way), A265135 (more than two ways), A265136 (exactly three ways), A265137 (more than three ways), A265138 (exactly four ways).

Programs

  • Mathematica
    r = 416; lst = Table[0, {r}]; lim = Floor[Sqrt[8*r - 7]]; Do[num = (i^2 + i)/2 + (j^2 + j)/2; If[num <= r, lst[[num]]++], {i, lim}, {j, i - 1}]; Flatten@Position[lst, 2]
Module[{nn=40,trnos},trnos=Accumulate[Range[nn]];Select[PositionIndex[ Sort[ Counts[Total/@Subsets[trnos,{2}]]]][2],#<=Last[trnos]&]] (* The program uses the PositionIndex and Counts functions from Mathematica version 10 *) (* Harvey P. Dale, Dec 25 2015 *)

A262749 Numbers that are the sum of two distinct nonzero triangular numbers in more than one way.

Original entry on oeis.org

16, 31, 46, 51, 76, 81, 94, 106, 111, 121, 123, 126, 133, 141, 146, 156, 157, 172, 174, 181, 186, 191, 196, 198, 211, 216, 225, 226, 231, 237, 241, 246, 256, 259, 268, 276, 281, 286, 289, 291, 297, 301, 310, 315, 321, 326, 328, 331, 336, 346, 354, 361, 366
Offset: 1

Views

Author

Arkadiusz Wesolowski, Dec 02 2015

Keywords

Comments

The magic sum of any 3 X 3 semimagic square composed of triangular numbers is a(n) + A000217(m) for some m and n.

Examples

			16 = 1 + 15 = 6 + 10.

Crossrefs

Cf. A000217, A051533, A260647, A265140 (exactly one way), A265134 (exactly two ways), A265135 (more than two ways), A265136 (exactly three ways), A265137 (more than three ways), A265138 (exactly four ways).

Programs

  • Mathematica
    r = 366; lst = Table[0, {r}]; lim = Floor[Sqrt[8*r - 7]]; Do[num = (i^2 + i)/2 + (j^2 + j)/2; If[num <= r, lst[[num]]++], {i, lim}, {j, i - 1}]; Flatten@Position[lst, n_ /; n > 1]
Module[{nn=30,trnos},trnos=Accumulate[Range[nn]];Select[Sort[Flatten[ Table[ PositionIndex[Counts[Total/@Subsets[trnos,{2}]]][i],{i,2,nn}]]], #<= Last[trnos]&]] (* The program uses the PositionIndex and Counts functions from Mathematica version 10 *)  (* Harvey P. Dale, Dec 26 2015 *)

A265136 Numbers that are the sum of two distinct nonzero triangular numbers in exactly three ways.

Original entry on oeis.org

81, 106, 181, 211, 256, 276, 331, 361, 381, 406, 456, 556, 606, 631, 666, 681, 706, 718, 731, 781, 856, 861, 931, 939, 956, 981, 1051, 1131, 1206, 1231, 1456, 1506, 1563, 1606, 1631, 1681, 1731, 1732, 1756, 1831, 1891, 1901, 1956, 1981, 2081, 2095, 2101
Offset: 1

Views

Author

Arkadiusz Wesolowski, Dec 02 2015

Keywords

Crossrefs

Cf. A000217, A051533, A260647, A265140 (exactly one way), A262749 (more than one way), A265134 (exactly two ways), A265135 (more than two ways), A265137 (more than three ways), A265138 (exactly four ways).

Programs

  • Mathematica
    r = 2101; lst = Table[0, {r}]; lim = Floor[Sqrt[8*r - 7]]; Do[num = (i^2 + i)/2 + (j^2 + j)/2; If[num <= r, lst[[num]]++], {i, lim}, {j, i - 1}]; Flatten@Position[lst, 3]
Module[{nn=70,trnos},trnos=Accumulate[Range[nn]];Select[ PositionIndex[ Sort[ Counts[Total/@Subsets[trnos,{2}]]]][3],#<=Last[trnos]&]] (* The program uses the PositionIndex and Counts functions from Mathematica version 10 *) (* Harvey P. Dale, Dec 25 2015 *)

A265138 Numbers that are the sum of two distinct nonzero triangular numbers in exactly four ways.

Original entry on oeis.org

471, 531, 601, 616, 786, 871, 906, 991, 1056, 1126, 1156, 1186, 1281, 1296, 1341, 1446, 1486, 1551, 1576, 1602, 1641, 1656, 1771, 1806, 1836, 1896, 1906, 1921, 2044, 2061, 2146, 2226, 2265, 2281, 2316, 2401, 2421, 2451, 2486, 2551, 2646, 2716, 2746, 2781
Offset: 1

Views

Author

Arkadiusz Wesolowski, Dec 02 2015

Keywords

Crossrefs

Cf. A000217, A051533, A260647, A265140 (exactly one way), A262749 (more than one way), A265134 (exactly two ways), A265135 (more than two ways), A265136 (exactly three ways), A265137 (more than three ways).

Programs

  • Mathematica
    r = 2781; lst = Table[0, {r}]; lim = Floor[Sqrt[8*r - 7]]; Do[num = (i^2 + i)/2 + (j^2 + j)/2; If[num <= r, lst[[num]]++], {i, lim}, {j, i - 1}]; Flatten@Position[lst, 4]
Module[{nn=80,trnos},trnos=Accumulate[Range[nn]];Select[PositionIndex[ Sort[ Counts[Total/@Subsets[trnos,{2}]]]][4],#<=Last[trnos]&]] (* The program uses the PositionIndex and Counts functions from Mathematica version 10 *) (* Harvey P. Dale, Dec 25 2015 *)

A265140 Numbers that are the sum of two distinct nonzero triangular numbers in exactly one way.

Original entry on oeis.org

4, 7, 9, 11, 13, 18, 21, 22, 24, 25, 27, 29, 34, 36, 37, 38, 39, 42, 43, 48, 49, 55, 56, 57, 58, 60, 61, 64, 65, 66, 67, 69, 70, 72, 73, 79, 83, 84, 87, 88, 91, 92, 93, 97, 99, 100, 101, 102, 108, 112, 114, 115, 119, 120, 127, 130, 135, 136, 137, 139, 142, 144
Offset: 1

Views

Author

Arkadiusz Wesolowski, Dec 02 2015

Keywords

Crossrefs

Cf. A000217, A051533, A260647, A262749 (more than one way), A265134 (exactly two ways), A265135 (more than two ways), A265136 (exactly three ways), A265137 (more than three ways), A265138 (exactly four ways).

Programs

  • Mathematica
    r = 144; lst = Table[0, {r}]; lim = Floor[Sqrt[8*r - 7]]; Do[num = (i^2 + i)/2 + (j^2 + j)/2; If[num <= r, lst[[num]]++], {i, lim}, {j, i - 1}]; Flatten@Position[lst, 1]

A265137 Numbers that are the sum of two distinct nonzero triangular numbers in more than three ways.

Original entry on oeis.org

471, 531, 601, 616, 786, 871, 906, 991, 1056, 1126, 1156, 1186, 1281, 1296, 1341, 1381, 1446, 1486, 1551, 1576, 1602, 1641, 1656, 1771, 1806, 1836, 1896, 1906, 1921, 2031, 2044, 2061, 2146, 2226, 2265, 2281, 2316, 2356, 2401, 2421, 2451, 2486, 2551, 2646
Offset: 1

Views

Author

Arkadiusz Wesolowski, Dec 02 2015

Keywords

Comments

If there exists any 3 X 3 magic square composed of triangular numbers, then its magic sum is a(n) + A000217(m) for some m and n.

Crossrefs

Cf. A000217, A051533, A260647, A265140 (exactly one way), A262749 (more than one way), A265134 (exactly two ways), A265135 (more than two ways), A265136 (exactly three ways), A265138 (exactly four ways).

Programs

  • Mathematica
    r = 2646; lst = Table[0, {r}]; lim = Floor[Sqrt[8*r - 7]]; Do[num = (i^2 + i)/2 + (j^2 + j)/2; If[num <= r, lst[[num]]++], {i, lim}, {j, i - 1}]; Flatten@Position[lst, n_ /; n > 3]
Module[{nn=80,trnos},trnos=Accumulate[Range[nn]];Select[Sort[ Flatten[ Table[ PositionIndex[Counts[Total/@Subsets[trnos,{2}]]][i],{i,4,nn}]]], #<= Last[trnos]&]] (* The program uses the PositionIndex and Counts functions from Mathematica version 10 *) (* Harvey P. Dale, Dec 26 2015 *)
Showing 1-7 of 7 results.

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