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.

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A322697 Eleven-column table read by rows: Primitive distinct 11-tuples that have the same value of phi, sigma, and tau.

Original entry on oeis.org

5701466232, 5806935288, 5824716000, 5863036500, 5940941280, 5978840580, 5980026420, 6376872150, 6520948434, 6566627430, 6726168150, 28249479864, 29028683320, 29045194360, 29849886456, 29850170376, 31778709534, 31888948266, 33524950602, 33528569646, 33543272334, 33644878554
Offset: 1

Views

Author

Jud McCranie, Dec 30 2018

Keywords

Comments

The terms are consecutive 11-tuples, ordered so that (A) a(11i-10) < a(11i-9) < ... < a(11i) for i > 0, and (B) a(11i+1) < a(11i+12) for i >= 0. This sequence has primitive solutions only. If k is relatively prime to all of the terms in a primitive 11-tuple, then multiplying the terms in that 11-tuple by k gives another solution - see A322687.

Examples

			5701466232, 5806935288, 5824716000, 5863036500, 5940941280, 5978840580, 5980026420, 6376872150, 6520948434, 6566627430, and 6726168150 have the same value of phi (1443225600), sigma (20473689600), and tau (384), so these eleven numbers are in the sequence.

Links

Crossrefs

Cf. A134922, A322687, A322688, A322689, A322690, A322691, A322692, A322693, A322694, A322695, A322696.

Extensions

More terms from Jud McCranie, Jan 13 2019

A322679 Three-column table read by rows: Triples that have the same value of phi, sigma, and tau.

Original entry on oeis.org

63048, 67122, 70818, 106120, 115938, 122322, 121352, 122120, 137170, 133920, 140448, 141372, 206112, 207468, 218892, 226632, 242730, 254562, 227304, 228000, 229500, 247968, 260832, 262548, 273824, 290812, 290852, 282720, 284580, 298452, 302560, 319396, 342550, 315240, 335610, 354090
Offset: 1

Views

Author

Jud McCranie, Jan 15 2019

Keywords

Comments

The terms are consecutive triples, ordered so that (A) a(3i-2) < a(3i-1) < a(3i), and (B) a(3i-2) < a(3j-2) for i < j. Primitive triples are in A322689.

Examples

			phi(63048) = phi(67122) = phi(70818) = 20160, sigma(63048) = sigma(67122) = sigma(70818) = 164160, tau(63048) = tau(67122) = tau(70818) = 32, so 63048, 67122, and 70818 are in the sequence.

Crossrefs

Cf. A134922, A322689, A322678, A322680, A322681, A322682, A322683, A322684, A322685, A322686, A322687.

A322680 Four-column table read by rows: Quadruples that have the same value of phi, sigma, and tau.

Original entry on oeis.org

1167320, 1197912, 1275318, 1345542, 3502548, 3536232, 3684252, 3695412, 4846520, 4980792, 5302638, 5594622, 5196000, 5299680, 5857830, 6000150, 5276448, 5568672, 5605308, 5764220, 6115656, 6510834, 6866106, 6869346, 6119520, 6159780, 6586650, 6928350, 6595560, 6917064, 6960840, 7818690
Offset: 1

Views

Author

Jud McCranie, Jan 16 2019

Keywords

Comments

The terms are consecutive quadruples, ordered so that (A) a(4i-3) < a(4i-2) < a(4i-1) < a(4i) for i > 0, and (B) a(4i+1) < a(4i+5) for i >= 0. Primitive solutions are in A322690.

Examples

			1167320, 1197912, 1275318, and 1345542 have the same value of phi (362880), sigma (3283200), and tau (64), so these four numbers are in the sequence.

Crossrefs

Cf. A134922, A322689, A322678, A322679, A322681, A322682, A322683, A322684, A322685, A322686, A322687.

A322681 Five-column table read by rows: Quintuples that have the same value of phi, sigma, and tau.

Original entry on oeis.org

15132960, 15870624, 15966240, 15975036, 16854684, 15175160, 15572856, 16579134, 16629354, 17492046, 17671392, 18346968, 18644448, 20598318, 20608038, 26382240, 27668256, 27843360, 27850284, 28026540, 28020384, 29474016, 29563296, 29667924, 31301556, 30743000, 31130008, 31356440, 34531750, 35220790
Offset: 1

Views

Author

Jud McCranie, Jan 16 2019

Keywords

Comments

The terms are consecutive quintuples, ordered so that (A) a(5i-4) < a(5i-3) < ... < a(5i) for i > 0, and (B) a(5i+1) < a(5i+6) for i >= 0. Primitive terms are in A322691.

Examples

			15132960, 15870624, 15966240, 15975036,and 16854684 have the same value of phi (3870720), sigma (55157760), and tau (192), so these five numbers are in the sequence.

Crossrefs

Cf. A134922, A322689, A322678, A322679, A322680, A322682, A322683, A322684, A322685, A322686, A322687.

A322682 Six-column table read by rows: Sextuples that have the same value of phi, sigma, and tau.

Original entry on oeis.org

98422632, 98724000, 99373500, 100693920, 101356380, 111298770, 98724000, 99373500, 100693920, 101356380, 111298770, 114002850, 152611020, 152891400, 161014380, 162770850, 171652650, 171733650, 173949984, 185635044, 187420296, 195265356, 195856836, 210450786
Offset: 1

Views

Author

Jud McCranie, Jan 16 2019

Keywords

Comments

The terms are consecutive sextuples, ordered so that (A) a(6i-5) < a(6i-4) < ... < a(6i) for i > 0, and (B) a(6i+1) < a(6i+7) for i >= 0. Primitive solutions are in A322692.

Examples

			98422632, 98724000, 99373500, 100693920, 101356380, and 111298770 have the same value of phi (24883200), sigma (341228160), and tau (192), so these six numbers are in the sequence.

Crossrefs

Cf. A134922, A322692, A322679, A322680, A322681, A322683, A322684, A322685, A322686, A322687.

A322683 Seven-column table read by rows: Septuples that have the same value of phi, sigma, and tau.

Original entry on oeis.org

98422632, 98724000, 99373500, 100693920, 101356380, 111298770, 114002850, 306564000, 312681120, 314675820, 335624850, 345611970, 353938650, 354008850, 378502560, 387919200, 390471300, 399564396, 402010740, 428530050, 428773950, 620451360, 650334048, 650695584, 654615840, 654976476, 661275384, 691042044
Offset: 1

Views

Author

Jud McCranie, Jan 16 2019

Keywords

Comments

The terms are consecutive septuples, ordered so that (A) a(7i-6) < a(7i-5) < ... < a(7i) for i > 0, and (B) a(7i+1) < a(7i+8) for i >= 0. Primitive solutions are in A322693.

Examples

			98422632, 98724000, 99373500, 100693920, 101356380, 111298770, and 114002850 have the same value of phi (24883200), sigma (341228160), and tau (192), so these seven numbers are in the sequence.

Crossrefs

Cf. A134922, A322692, A322679, A322680, A322681, A322682, A322684, A322685, A322686, A322687.

A322684 Eight-column table read by rows: Octuples that have the same value of phi, sigma, and tau.

Original entry on oeis.org

1068277140, 1070239800, 1078550760, 1123696860, 1127100660, 1139395950, 1201568550, 1202135550, 2145948000, 2188767840, 2202730740, 2317003260, 2349373950, 2419283790, 2477570550, 2478061950, 2232975640, 2234245720, 2296145112, 2296166952, 2444516118, 2578842354, 2579120742, 2580251718
Offset: 1

Views

Author

Jud McCranie, Jan 16 2019

Keywords

Comments

The terms are consecutive octuples, ordered so that (A) a(8i-7) < a(8i-6) < ... < a(8i) for i > 0, and (B) a(8i+1) < a(8i+9) for i >= 0. Primitive solutions are in A322694.

Examples

			1068277140, 1070239800, 1078550760, 1123696860, 1127100660, 1139395950, 1201568550, 1202135550 have the same value of phi (232243200), sigma (3989744640), and tau (384), so these eight numbers are in the sequence.

Crossrefs

Cf. A134922, A322692, A322679, A322680, A322681, A322682, A322683, A322685, A322686, A322687.

A322685 Nine-column table read by rows: 9-tuples that have the same value of phi, sigma, and tau.

Original entry on oeis.org

4337267040, 4548689376, 4577446560, 4578614964, 4606647660, 4607561340, 5024337318, 5056654590, 5059532610, 4343835816, 4344550776, 4467333640, 4467573880, 4583778744, 4584409224, 4879970766, 4880641986, 5149389234
Offset: 1

Views

Author

Jud McCranie, Jan 16 2019

Keywords

Comments

The terms are consecutive 9-tuples, ordered so that (A) a(9i-8) < a(9i-7) < ... < a(9i) for i > 0, and (B) a(9i+1) < a(9i+10) for i >= 0. The primitive solutions are in A322995.

Examples

			4337267040, 4548689376, 4577446560, 4578614964, 4606647660, 4607561340, 5024337318, 5056654590, and 5059532610 have the same value of phi (1106472960), sigma (15850598400), and tau (384), so these nine numbers are in the sequence.

Crossrefs

Cf. A134922, A322692, A322679, A322680, A322681, A322682, A322683, A322684, A322686, A322687.

A322686 Ten-column table read by rows: 10-tuples that have the same value of phi, sigma, and tau.

Original entry on oeis.org

5701466232, 5806935288, 5824716000, 5863036500, 5940941280, 5978840580, 5980026420, 6376872150, 6520948434, 6566627430, 5806935288, 5824716000, 5863036500, 5940941280, 5978840580, 5980026420, 6376872150, 6520948434, 6566627430, 6726168150
Offset: 1

Views

Author

Jud McCranie, Jan 16 2019

Keywords

Comments

The terms are consecutive 10-tuples, ordered so that (A) a(10i-9) < a(10i-8) < ... < a(10i) for i > 0, and (B) a(10i+1) < a(10i+11) for i >= 0. Primitive solutions are in A322696.

Examples

			5701466232, 5806935288, 5824716000, 5863036500, 5940941280, 5978840580, 5980026420, 6376872150, 6520948434, and 6566627430 have the same value of phi (1443225600), sigma (20473689600), and tau (384), so these ten numbers are in the sequence.

Crossrefs

Cf. A134922, A322692, A322679, A322680, A322681, A322682, A322683, A322684, A322685, A322687.

A143189 Numbers k such that at least one of the following equations holds: phi(2*k-1) = phi(2*k), sigma(2*k-1) = sigma(2*k), tau(2*k-1) = tau(2*k).

Original entry on oeis.org

1, 2, 8, 11, 17, 29, 38, 43, 47, 67, 68, 71, 73, 74, 86, 89, 95, 101, 103, 107, 109, 116, 122, 127, 128, 143, 151, 188, 191, 194, 197, 215, 223, 227, 241, 248, 251, 254, 269, 277, 283, 302, 305, 317, 323, 332, 349, 359, 368, 371, 373, 397, 403, 409, 410, 433, 452
Offset: 1

Views

Author

Roger L. Bagula, Oct 17 2008

Keywords

Comments

Inspired by A134922.

Links

Crossrefs

Cf. A000005 (tau), A000010 (phi), A000203 (sigma), A134922.

Programs

  • Mathematica
    a[n_] =
If[EulerPhi[2*n - 1] - EulerPhi[2*n] == 0, n,
If[DivisorSigma[1, 2*n - 1] - DivisorSigma[1, 2*n] == 0, n,
If[Length[Divisors[2*n - 1]] - Length[Divisors[2*n]] == 0, n, {}]]];
Flatten[Table[a[n], {n, 1, 400}]]
  • PARI
    isok(k) = {my(f1 = factor(2*k-1), f2 = factor(2*k)); eulerphi(f1) == eulerphi(f2) || sigma(f1) == sigma(f2) || numdiv(f1) == numdiv(f2);} \\ Amiram Eldar, Jun 07 2025

Extensions

Name edited by Amiram Eldar, Jun 07 2025
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