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 1-4 of 4 results.

A069674 Triangular numbers whose digit permutations yield at least two further triangular numbers.

Original entry on oeis.org

120, 210, 1035, 2080, 4095, 4560, 4950, 5460, 9045, 13530, 15400, 17020, 17205, 17578, 19306, 20100, 20503, 20706, 23005, 30135, 30628, 36046, 37401, 40186, 43071, 43660, 46360, 46971, 50721, 51040, 52003, 53301, 60031, 63190, 66430
Offset: 1

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Author

Amarnath Murthy, Apr 06 2002

Keywords

Examples

			120 yields 21 and 210; 210 yields 21 and 120.

Crossrefs

Cf. A000217, A077671, A095869, A095870.

Extensions

Edited, corrected and extended by Ray Chandler, Jun 29 2004
Offset changed to 1 by Jinyuan Wang, Aug 06 2021

A095869 Triangular numbers whose digit permutations yield no further triangular numbers.

Original entry on oeis.org

1, 3, 6, 15, 21, 28, 36, 45, 55, 66, 78, 91, 136, 171, 231, 276, 378, 406, 435, 465, 528, 561, 595, 666, 703, 741, 861, 903, 990, 1081, 1128, 1176, 1225, 1275, 1326, 1431, 1540, 1596, 1653, 1711, 1770, 1953, 2016, 2211, 2278, 2346, 2485, 2556, 2628, 2775
Offset: 1

Views

Author

Ray Chandler, Jun 29 2004

Keywords

Crossrefs

Cf. A000217, A069674, A077671, A095870.

Extensions

Offset changed to 1 by Jinyuan Wang, Aug 06 2021

A095870 Least triangular number whose digit permutations yield exactly n further triangular numbers.

Original entry on oeis.org

1, 10, 120, 17020, 20503, 156520, 1087075, 1021735, 1026028, 1053426, 104653, 10623745, 10280845, 1004653, 10568503, 10725396, 10762480, 12890503, 10348975, 10394520, 101495628, 100543290, 102495403, 100359028, 104972805, 124780503
Offset: 0

Views

Author

Ray Chandler, Jun 29 2004

Keywords

Examples

			20503 yields 253, 325, 23005 and 52003.

Crossrefs

Cf. A069674, A077671, A095869.

A343811 Triangular numbers k such that every permutation of the digits of k is a triangular number.

Original entry on oeis.org

0, 1, 3, 6, 10, 55, 66, 666
Offset: 1

Views

Author

Ctibor O. Zizka, Apr 30 2021

Keywords

Comments

I assume this sequence is finite.
From Jon E. Schoenfield, Mar 05 2022: (Start)
If k is a term of this sequence and is not a repdigit, then at least one of its permutations (a triangular number, call it k1) will have a digit d0 in its ones place and a digit d1 != d0 in its tens place, and the number that results from reversing those last two digits will be a triangular number k2 = k1 + 9*(d0 - d1), so the two distinct triangular numbers k2 and k1 will differ by no more than 9*(9-0) = 81. But the j-th triangular number T(j) differs from the nearest other triangular number by T(j) - T(j-1) = j, so if two distinct triangular numbers T(k1) > T(k2) differ by no more than 81, then k1 <= 81. No triangular numbers <= T(81) = 3321 are terms that exceed 666, so if there exists any term > 666, it is a repdigit triangular number.
However, Ballew and Weger proved (see A045914) that the only repdigit triangular numbers are 0, 1, 3, 6, 55, 66, and 666. Thus, 666 is the last term of this sequence. (End)

Examples

			10 = 4*5/2, 01 = 1*2/2.

Crossrefs

Cf. A000217, A003459, A045914, A077671.

Programs

  • Mathematica
    triQ[n_] := IntegerQ @ Sqrt[8*n + 1]; Select[Range[0, 1000], AllTrue[Permutations[ IntegerDigits[#] ], triQ[FromDigits[#1]] &] &] (* Amiram Eldar, Apr 30 2021 *)
pdtQ[n_]:=AllTrue[FromDigits/@Permutations[IntegerDigits[n]],OddQ[ Sqrt[ 8#+1]]&]; Select[Accumulate[Range[0,5000]],pdtQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 04 2021 *)
Showing 1-4 of 4 results.

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

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