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 1-5 of 5 results.

A115905 Numbers k such that k + phi(k) is a triangular number.

Original entry on oeis.org

2, 4, 9, 11, 20, 23, 25, 27, 49, 53, 92, 93, 105, 121, 127, 147, 150, 163, 169, 220, 228, 231, 233, 281, 286, 289, 294, 332, 361, 431, 474, 529, 541, 543, 548, 613, 660, 675, 710, 726, 743, 825, 827, 841, 888, 891, 961, 977, 993, 1014, 1028, 1265, 1310, 1369
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Examples

			121 + phi(121) = 231 = T(21).

Links

Crossrefs

Cf. A000010, A000217, A069066, A115908, A115910.

Programs

  • Mathematica
    Select[Range[1500],OddQ[Sqrt[8(#+EulerPhi[#])+1]]&] (* Harvey P. Dale, May 11 2015 *)
  • PARI
    isok(n) = ispolygonal(n + eulerphi(n), 3); \\ Michel Marcus, Jan 25 2014

A287472 Triangular numbers k such that phi(k) is also a triangular number, where phi(k) is the Euler totient function (A000010).

Original entry on oeis.org

1, 231, 1035, 6786, 190036, 193131, 766941, 1237951, 1348903, 3069003, 3396921, 8034036, 9152781, 11875501, 15694003, 28001386, 29587278, 35149920, 61643856, 63196903, 130758706, 178161126, 198214005, 227751153, 268111746, 339210081, 402102261, 654224878
Offset: 1

Views

Author

Amiram Eldar, May 25 2017

Keywords

Comments

The indices of these triangular numbers are: 1, 21, 45, 116, 616, 621, 1238, 1573, 1642, 2477, 2606, 4008, 4278, 4873, 5602, 7483, 7692, 8384, 11103, 11242, 16171, 18876, 19910, 21342, 23156, 26046, 28358, 36172, 46196, 46621, 67572, 72816, ...
The indices of the triangular phi values are: 1, 15, 32, 63, 384, 495, 927, 1440, 1599, 1856, 2015, 2240, 3200, 4640, 5375, 4895, 4095, 4095, 6400, 9855, 10880, 9855, 13824, 16128, 12095, 19520, 21504, 25344, 25983, 45584, 37184, 40959, ...

Examples

			231 = 21*22/2 is triangular, phi(231)=120=15*16/2 is also triangular, thus 231 is in the sequence.

Links

Crossrefs

Cf. A000010, A000217, A083674, A083675, A115910, A116541.

Programs

  • Mathematica
    triQ[n_] := IntegerQ@Sqrt[8n+1]; Select[Accumulate[Range[1000]], triQ[EulerPhi[#]]&]
  • PARI
    isok(n) = ispolygonal(n, 3) && ispolygonal(eulerphi(n), 3); \\ Michel Marcus, May 25 2017

A115909 Numbers k such that sigma(k)*k is a triangular number.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 57, 64, 65, 85, 87, 128, 256, 512, 1024, 1245, 1421, 2048, 3146, 3330, 3790, 4096, 6695, 7257, 8192, 10137, 16384, 25884, 32768, 34420, 34551, 34947, 65536, 131072, 208495, 262144, 348161, 440495, 524288, 530270, 534430
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Comments

Infinite, since all the powers of 2 belong to the sequence.

Examples

			sigma(1421)*1421 = 2429910 = T(2204).

Links

Crossrefs

Cf. A064987, A000217, A115904, A115910.

Programs

  • Mathematica
    Select[Range[540000],IntegerQ[(Sqrt[1+8(#*DivisorSigma[1,#])]-1)/2]&] (* Harvey P. Dale, Dec 16 2011 *)
  • PARI
    isok(n) = ispolygonal(n*sigma(n), 3); \\ Amiram Eldar, Apr 06 2023

A287473 Triangular numbers k such that phi(k) is a square number, where phi(k) is the Euler totient function (A000010).

Original entry on oeis.org

1, 10, 136, 630, 2016, 7875, 9180, 18915, 32896, 37128, 46056, 58311, 66430, 103740, 131841, 198135, 225456, 301476, 323610, 332520, 408156, 499500, 738720, 786885, 839160, 862641, 922761, 924120, 1065070, 1079715, 1183491, 1385280, 1851850, 1906128, 1925703
Offset: 1

Views

Author

Amiram Eldar, May 25 2017

Keywords

Comments

The indices of these triangular numbers are: 1, 4, 16, 35, 63, 125, 135, 194, 256, 272, 303, 341, 364, 455, 513, 629, 671, 776, 804, 815, 903, 999, 1215, 1254, 1295, 1313, 1358, 1359, 1459, 1469, 1538, 1664, 1924, 1952, 1962, ... and their phi values are the squares of: 1, 2, 8, 12, 24, 60, 48, 96, 128, 96, 120, 180, 144, 144, 288, 288, 240, 288, 264, 288, 336, 360, 432, 600, 432, 720, 720, 480, 648, 672, 864, 576, 720, 720, 1080, ...
Similar to A115910, since A115910(n)^2 are squares whose phi is a triangular number.

Examples

			136=16*17/2 is triangular, phi(136)=64=8^2 is a square, thus 136 is in the sequence.

Links

Crossrefs

Cf. A000010, A000217, A000290, A039770, A115910, A256151.
Intersection of A000217 and A039770.

Programs

  • Mathematica
    Select[Accumulate[Range[1000]],IntegerQ[Sqrt[EulerPhi[#]]]&]
  • PARI
    isok(n) = ispolygonal(n, 3) && issquare(eulerphi(n)); \\ Michel Marcus, May 25 2017

A292063 Triangular numbers n such that psi(n) is also a triangular number, where psi is the Dedekind psi function (A001615).

Original entry on oeis.org

1, 780, 2775, 5050, 474825, 681528, 1727011, 5286126, 5911641, 6604795, 17325441, 21612025, 27799696, 45025305, 386767578, 1538599128, 2086160121, 3679490220, 5718242211, 7092226351, 8019794628, 16505718895, 36604197735, 55541611986, 56693041356, 89369984476
Offset: 1

Views

Author

Amiram Eldar, Sep 08 2017

Keywords

Comments

The indices of these triangular numbers are 1, 39, 74, 100, 974, 1167, 1858, 3251, 3438, 3634, 5886, 6574, 7456, 9489, ...
The indices of the triangular psi values are 1, 63, 95, 135, 1280, 1664, 2015, 4607, 4095, 4095, 7424, 7424, 9152, 12543, ...

Examples

			780 is in the sequence since 780 = 39*40/2 is triangular and psi(780) = 2016 = 63*64/2 is also triangular.

Links

Crossrefs

Cf. A000217, A001615, A083674, A083675, A115910, A116541, A287472.

Programs

  • Mathematica
    psi[n_] := If[n<1, 0, n*Sum[MoebiusMu[d]^2/d, {d, Divisors @ n}]]; triQ[n_] := IntegerQ@ Sqrt[8n+1]; Select[Accumulate[Range[1000]], triQ[psi[#]]&]

Extensions

a(18)-a(26) from Giovanni Resta, Sep 11 2017
Showing 1-5 of 5 results.

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

Content is available under The OEIS End-User License Agreement.