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 1-8 of 8 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 *)

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]

A342326 a(n) is the smallest nonnegative integer that can be written as a sum of two distinct nonzero triangular numbers in exactly n ways or -1 if no such integer exists.

Original entry on oeis.org

0, 4, 16, 81, 471, 2031, 1381, 11781, 6906, 17956, 34531, 123256, 40056, 305256, 863281, 448906, 200281, 1957231, 520731, 10563906, 1001406, 11222656, 7631406, 3454506, 1482081, 75865156, 7172606106, 8852431, 25035156, 334020781, 13018281, 38531031, 7410406, 7014160156
Offset: 0

Views

Author

Robert G. Root, Mar 08 2021

Keywords

Comments

Conjecture: This sequence has a positive a(n) for every positive integer n, and each sequence in the infinite indexed family, of which this sequence offers the initial terms, is infinite, as well.
From David A. Corneth, Mar 08 2021: (Start)
a(40) = 37052031, a(45) = 221310781, a(48) = 60765331, a(39) <= 2782318906, a(42) <= 325457031, a(47) <= 927577056, a(50) <= 2200089531, a(54) <= 327539956, a(56) <= 926300781, a(60) <= 481676406, a(63) <= 4598740656, a(64) <= 303826656, a(71) <= 4579579956, a(72) <= 789949306, a(80) <= 1519133281, a(96) <= 3220562556. Terms for n <= 96 not listed here and terms for which only upper bounds are known are >= 3*10^8.
Is a(n) == 6 (mod 25) for n >= 5? It holds for all terms known to date.
The triangular numbers mod 25 are periodic with period 25. Constructing all 25*25 = 625 sums of two distinct triangular numbers mod 25 gives 65 cases for 6 (mod 25). The second largest occurs 40 times. (End)
a(47) = 550240551, a(59) = 7629645156, a(67) = 6418012656, a(81) = 9498658731, a(90) = 8188498906. All upper bounds listed in the above comments for n other than 47 are the exact values of a(n). For all n for which no value is listed here or above, a(n) > 10^10 (or a(n) = -1). - Jon E. Schoenfield, Mar 09 2021
From Martin Ehrenstein, Mar 09 2021: (Start)
a(44) = 15646972656. For n<=51, all terms not mentioned here or above, a(n) >= 6.5*10^10 (or a(n) = -1).
a(47) == 1 (mod 25) and a(95) = 47652012541 == 16 (mod 25). Thus the answer to Corneth's question is 'No'. (End)

Examples

			a(1) =  4 = 1 +  3;
a(2) = 16 = 1 + 15 =  6 + 10;
a(3) = 81 = 3 + 78 = 15 + 66 = 36 + 45.

Links

Crossrefs

Cf. A000217, A260647, A051533, A262749, A265140, A265134, A265136, A265138, A307597.

Programs

  • Mathematica
    r = 125000; (* generates the first 12 terms of the sequence *)
lst = Table[0, {r}];
lim = Floor[Sqrt[2r]];
Do[ num = (i^2 + i)/2 + (j^2 + j)/2;
If[num <= r, lst[[num]]++], {i, lim}, {j,  i - 1}];
First /@ (Flatten@Position[lst, #] & /@ Range[Max[lst]])
  • PARI
    upto(n) = {my(v = vector(n)); res = vector(10); for(i = 1, (sqrtint(8*n + 1)-1)\2, bi = binomial(i + 1, 2); for(j = i+1, (sqrtint(8*(n - bi))-1)\2, v[bi + binomial(j+1, 2)]++ ) ); for(i = 1, #v, if(v[i] > 0, if(v[i] > #res, res = concat(res, vector(v[i] - #res)); ); if(res[v[i]] == 0, res[v[i]] = i ) ) ); concat(0, res) } \\ David A. Corneth, Mar 08 2021

Formula

a(n) = min { m >= 0 : A307597(m) = n }. - Alois P. Heinz, Mar 08 2021

Extensions

a(13)-a(18) from Alois P. Heinz, Mar 08 2021
a(19)-a(25) from David A. Corneth, Mar 08 2021
a(26)-a(33) from Jon E. Schoenfield, Mar 09 2021 (some terms first found by David A. Corneth)

A265135 Numbers that are the sum of two distinct nonzero triangular numbers in more than two ways.

Original entry on oeis.org

81, 106, 181, 211, 256, 276, 331, 361, 381, 406, 456, 471, 531, 556, 601, 606, 616, 631, 666, 681, 706, 718, 731, 781, 786, 856, 861, 871, 906, 931, 939, 956, 981, 991, 1051, 1056, 1126, 1131, 1156, 1186, 1206, 1231, 1281, 1296, 1341, 1381, 1446, 1456
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), A265136 (exactly three ways), A265137 (more than three ways), A265138 (exactly four ways).

Programs

  • Mathematica
    r = 1456; 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 > 2]
Module[{nn=60,trnos},trnos=Accumulate[Range[nn]];Select[Sort[Flatten[ Table[ PositionIndex[Counts[Total/@Subsets[trnos,{2}]]][i],{i,3,nn}]]], #<= Last[trnos]&]] (* The program uses the PositionIndex and Counts functions from Mathematica version 10 *) (* Harvey P. Dale, Dec 26 2015 *)

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 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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