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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A272594 Numbers n such that the multiplicative group modulo n is the direct product of 4 cyclic groups.

Original entry on oeis.org

120, 168, 240, 264, 280, 312, 336, 360, 408, 420, 440, 456, 480, 504, 520, 528, 552, 560, 600, 616, 624, 660, 672, 680, 696, 720, 728, 744, 760, 780, 792, 816, 880, 888, 912, 920, 924, 936, 952, 960, 984, 1008, 1020, 1032, 1040, 1056, 1064, 1080, 1092, 1104, 1120, 1128, 1140, 1144, 1155, 1160, 1176, 1200
Offset: 1

Views

Author

Joerg Arndt, May 05 2016

Keywords

Comments

Numbers n such that A046072(n) = 4.

Crossrefs

Direct product of k groups: A033948 (k=1), A272592 (k=2), A272593 (k=3), A272595 (k=5), A272596 (k=6), A272597 (k=7), A272598 (k=8), A272599 (k=9).

Programs

  • Mathematica
    A046072[n_] := Which[n == 1 || n == 2, 1,
     OddQ[n], PrimeNu[n],
     EvenQ[n] && !Divisible[n, 4], PrimeNu[n] - 1,
     Divisible[n, 4] && !Divisible[n, 8], PrimeNu[n],
     Divisible[n, 8], PrimeNu[n] + 1];
Select[Range[1200], A046072[#] == 4&] (* Jean-François Alcover, Dec 22 2021, after Geoffrey Critzer in A046072 *)
  • PARI
    for(n=1, 3*10^3, my(t=#(znstar(n)[2])); if(t==4, print1(n, ", ")));

A272595 Numbers n such that the multiplicative group modulo n is the direct product of 5 cyclic groups.

Original entry on oeis.org

840, 1320, 1560, 1680, 1848, 2040, 2184, 2280, 2520, 2640, 2760, 2856, 3080, 3120, 3192, 3360, 3432, 3480, 3640, 3696, 3720, 3864, 3960, 4080, 4200, 4368, 4440, 4488, 4560, 4620, 4680, 4760, 4872, 4920, 5016, 5040, 5160, 5208, 5280, 5304, 5320, 5460, 5520, 5544, 5640, 5712, 5720, 5880, 5928, 6072, 6120
Offset: 1

Views

Author

Joerg Arndt, May 05 2016

Keywords

Comments

Numbers n such that A046072(n) = 5.

Crossrefs

Direct product of k groups: A033948 (k=1), A272592 (k=2), A272593 (k=3), A272594 (k=4), A272596 (k=6), A272597 (k=7), A272598 (k=8), A272599 (k=9).

Programs

  • Mathematica
    A046072[n_] := Which[n == 1 || n == 2, 1,
     OddQ[n], PrimeNu[n],
     EvenQ[n] && !Divisible[n, 4], PrimeNu[n] - 1,
     Divisible[n, 4] && !Divisible[n, 8], PrimeNu[n],
     Divisible[n, 8], PrimeNu[n] + 1];
Select[Range[10^4], A046072[#] == 5&] (* Jean-François Alcover, Dec 22 2021, after Geoffrey Critzer in A046072 *)
  • PARI
    for(n=1, 10^4, my(t=#(znstar(n)[2])); if(t==5, print1(n, ", ")));

A272596 Numbers n such that the multiplicative group modulo n is the direct product of 6 cyclic groups.

Original entry on oeis.org

9240, 10920, 14280, 15960, 17160, 18480, 19320, 21840, 22440, 24024, 24360, 25080, 26040, 26520, 27720, 28560, 29640, 30360, 31080, 31416, 31920, 32760, 34320, 34440, 35112, 35880, 36120, 36960, 37128, 38280, 38640, 38760, 39480, 40040, 40920, 41496, 42504, 42840, 43680, 44520, 44880, 45240, 46200
Offset: 1

Views

Author

Joerg Arndt, May 05 2016

Keywords

Comments

Numbers n such that A046072(n) = 6.

Crossrefs

Direct product of k groups: A033948 (k=1), A272592 (k=2), A272593 (k=3), A272594 (k=4), A272595 (k=5), A272597 (k=7), A272598 (k=8), A272599 (k=9).

Programs

  • Mathematica
    A046072[n_] := Which[n == 1 || n == 2, 1,
     OddQ[n], PrimeNu[n],
     EvenQ[n] && !Divisible[n, 4], PrimeNu[n] - 1,
     Divisible[n, 4] && !Divisible[n, 8], PrimeNu[n],
     Divisible[n, 8], PrimeNu[n] + 1];
Select[Range[5*10^4], A046072[#] == 6&] (* Jean-François Alcover, Dec 22 2021, after Geoffrey Critzer in A046072 *)
  • PARI
    for(n=1, 10^5, my(t=#(znstar(n)[2])); if(t==6, print1(n, ", ")));

A272597 Numbers n such that the multiplicative group modulo n is the direct product of 7 cyclic groups.

Original entry on oeis.org

120120, 157080, 175560, 185640, 207480, 212520, 240240, 251160, 267960, 271320, 286440, 291720, 314160, 316680, 326040, 328440, 338520, 341880, 351120, 360360, 367080, 371280, 378840, 394680, 397320, 404040, 408408, 414120, 414960, 425040, 426360, 434280, 442680, 447720, 456456, 462840, 469560, 471240
Offset: 1

Views

Author

Joerg Arndt, May 05 2016

Keywords

Comments

Numbers n such that A046072(n) = 7.

Crossrefs

Direct product of k groups: A033948 (k=1), A272592 (k=2), A272593 (k=3), A272594 (k=4), A272595 (k=5), A272596 (k=6), A272598 (k=8), A272599 (k=9).

Programs

  • Mathematica
    A046072[n_] := Which[n == 1 || n == 2, 1,
     OddQ[n], PrimeNu[n],
     EvenQ[n] && !Divisible[n, 4], PrimeNu[n] - 1,
     Divisible[n, 4] && !Divisible[n, 8], PrimeNu[n],
     Divisible[n, 8], PrimeNu[n] + 1];
Select[Range[5*10^5], A046072[#] == 7&] (* Jean-François Alcover, Dec 22 2021, after Geoffrey Critzer in A046072 *)
  • PARI
    for(n=1, 10^6, my(t=#(znstar(n)[2])); if(t==7, print1(n, ", ")));

A272598 Numbers n such that the multiplicative group modulo n is the direct product of 8 cyclic groups.

Original entry on oeis.org

2042040, 2282280, 2762760, 2984520, 3483480, 3527160, 3612840, 3723720, 4037880, 4084080, 4269720, 4444440, 4555320, 4564560, 4772040, 4869480, 4924920, 5091240, 5165160, 5383560, 5442360, 5525520, 5542680, 5645640, 5754840, 5811960, 5969040, 6016920, 6126120, 6163080, 6240360, 6366360, 6431880, 6440280
Offset: 1

Views

Author

Joerg Arndt, May 05 2016

Keywords

Comments

Numbers n such that A046072(n) = 8.

Crossrefs

Direct product of k groups: A033948 (k=1), A272592 (k=2), A272593 (k=3), A272594 (k=4), A272595 (k=5), A272596 (k=6), A272597 (k=7), A272599 (k=9).

Programs

  • Mathematica
    A046072[n_] := Which[n == 1 || n == 2, 1,
     OddQ[n], PrimeNu[n],
     EvenQ[n] && !Divisible[n, 4], PrimeNu[n] - 1,
     Divisible[n, 4] && !Divisible[n, 8], PrimeNu[n],
     Divisible[n, 8], PrimeNu[n] + 1];
Select[Range[120, 120*10^5, 120], A046072[#] == 8&] (* Jean-François Alcover, Dec 22 2021, after Geoffrey Critzer in A046072 *)
  • PARI
    for(n=1, 10^7, my(t=#(znstar(n)[2])); if(t==8, print1(n, ", ")));

A272599 Numbers n such that the multiplicative group modulo n is the direct product of 9 cyclic groups.

Original entry on oeis.org

38798760, 46966920, 52492440, 59219160, 63303240, 66186120, 68643960, 70750680, 75555480, 77597520, 80120040, 81124680, 83723640, 84444360, 85645560, 86551080, 87807720, 92520120, 93573480, 93933840, 95975880, 98138040, 102222120, 102287640, 104772360, 104984880, 107267160, 107987880, 108228120, 109341960, 110427240
Offset: 1

Views

Author

Joerg Arndt, May 05 2016

Keywords

Comments

Numbers n such that A046072(n) = 9.

Crossrefs

Direct product of k groups: A033948 (k=1), A272592 (k=2), A272593 (k=3), A272594 (k=4), A272595 (k=5), A272596 (k=6), A272597 (k=7), A272598 (k=8).

Programs

  • Mathematica
    A046072[n_] := Which[n == 1 || n == 2, 1,
     OddQ[n], PrimeNu[n],
     EvenQ[n] && !Divisible[n, 4], PrimeNu[n] - 1,
     Divisible[n, 4] && ! Divisible[n, 8], PrimeNu[n],
     Divisible[n, 8], PrimeNu[n] + 1];
Select[Range[840, 840*140000, 840], A046072[#] == 9&] (* Jean-François Alcover, Dec 22 2021, after Geoffrey Critzer in A046072 *)
  • PARI
    for(n=1, 10^9, my(t=#(znstar(n)[2])); if(t==9, print1(n, ", ")));

A062374 Euler phi(n) / Carmichael lambda(n) = 4.

Original entry on oeis.org

24, 40, 48, 56, 60, 65, 72, 84, 85, 88, 96, 104, 105, 112, 130, 132, 136, 140, 144, 145, 152, 156, 165, 170, 176, 180, 184, 185, 192, 200, 204, 205, 210, 216, 220, 221, 224, 228, 231, 232, 248, 265, 276, 285, 288, 290, 296, 300, 304, 305, 308, 325, 328, 330
Offset: 1

Views

Author

Vladeta Jovovic, Jun 17 2001

Keywords

Comments

Solutions to A000010(n)/A002322(n)=4.

Links

Crossrefs

Cf. A000010, A002322, A034380, A033948, A062373, A062375, A062376, A062377.

Programs

  • Mathematica
    Select[Range[400],EulerPhi[#]/CarmichaelLambda[#]==4&] (* Harvey P. Dale, May 23 2011 *)
  • PARI
    {cmf(f)=if( ((f[1]==2)&&(f[2]>2)),eulerphi(f[1]^f[2])/2, eulerphi(f[1]^f[2])) } {cl(f)= k=factor(f); l=1; for(x=1,omega(f),l=lcm(l,cmf([k[x,1], k[x,2]]))); l } {A062374(n)=eulerphi(n)/cl(n)} for(x=1,10001, if(A062374(x)==4,print1(x,",")))

Extensions

More terms from Randall L Rathbun, Jan 12 2002

A062375 Euler phi(n) / Carmichael lambda(n) = 6.

Original entry on oeis.org

63, 91, 117, 126, 133, 171, 182, 189, 217, 234, 247, 259, 266, 279, 301, 333, 342, 351, 378, 387, 403, 427, 434, 441, 469, 494, 511, 518, 549, 553, 558, 559, 567, 589, 602, 603, 637, 657, 666, 679, 702, 711, 721, 763, 774, 806, 817, 837, 854, 871, 873, 882
Offset: 1

Views

Author

Vladeta Jovovic, Jun 17 2001

Keywords

Comments

Solutions to A000010(n)/A002322(n)=6.

Links

Crossrefs

Cf. A000010, A002322, A034380, A033948, A062373, A062374, A062376, A062377.

Programs

  • PARI
    {cmf(f)=if( ((f[1]==2)&&(f[2]>2)),eulerphi(f[1]^f[2])/2, eulerphi(f[1]^f[2])) } {cl(f)= k=factor(f); l=1; for(x=1,omega(f),l=lcm(l,cmf([k[x,1], k[x,2]]))); l } {A062375(n)=eulerphi(n)/cl(n)} for(x=1,10001, if(A062375(x)==6,print1(x,",")))

Extensions

More terms from Randall L Rathbun, Jan 12 2002

A062376 Numbers k such that Euler phi(k) / Carmichael lambda(k) = 8.

Original entry on oeis.org

80, 120, 160, 168, 195, 208, 255, 260, 264, 272, 280, 312, 320, 336, 340, 360, 390, 400, 408, 416, 420, 435, 440, 456, 464, 510, 528, 552, 555, 580, 592, 595, 600, 615, 616, 640, 656, 660, 663, 672, 696, 697, 715, 740, 744, 760, 765, 792, 795, 800, 820
Offset: 1

Views

Author

Vladeta Jovovic, Jun 17 2001

Keywords

Comments

Solutions to A000010(n)/A002322(n)=8.

Links

Crossrefs

Cf. A000010, A002322, A034380, A033948, A062373, A062374, A062375, A062377.

Programs

  • PARI
    {cmf(f)=if( ((f[1]==2)&&(f[2]>2)),eulerphi(f[1]^f[2])/2, eulerphi(f[1]^f[2])) }
{cl(f)= k=factor(f); l=1; for(x=1,omega(f),l=lcm(l,cmf([k[x,1], k[x,2]]))); l }
{A062376(n)=eulerphi(n)/cl(n)}
for(x=1,1000, if(A062376(x)==8, print1(x,", ")))

Extensions

More terms from Randall L Rathbun, Jan 12 2002

A076942 Smallest k > 0 such that n*k+1 is a square.

Original entry on oeis.org

3, 4, 1, 2, 3, 4, 5, 1, 7, 8, 9, 2, 11, 12, 1, 3, 15, 16, 17, 4, 3, 20, 21, 1, 23, 24, 25, 6, 27, 4, 29, 7, 3, 32, 1, 8, 35, 36, 5, 2, 39, 4, 41, 10, 8, 44, 45, 1, 47, 48, 5, 12, 51, 52, 8, 3, 7, 56, 57, 2, 59, 60, 1, 15, 3, 8, 65, 16, 7, 12, 69, 4, 71, 72, 9, 18, 15, 8, 77, 1, 79, 80, 81
Offset: 1

Views

Author

Amarnath Murthy, Oct 19 2002

Keywords

Comments

a(n) <= n-2 for n > 2; a(p) = p-2 if p is a prime > 2. [Comment corrected by Floris P. van Doorn, Jan 31 2009]
a(n) = n - 2 precisely when n > 2 has a primitive root; that is, for 4, and p^k and 2*p^k for p an odd prime and k > 0. - Franklin T. Adams-Watters, Apr 13 2009

Links

Crossrefs

Cf. A033948, A033949, A215653. [From Franklin T. Adams-Watters, Apr 13 2009]

Programs

  • Mathematica
    Do[k = 1; While[ !IntegerQ[Sqrt[n*k + 1]], k++ ]; Print[k], {n, 1, 85}]
  • PARI
    a(n) = {my(m = n + 1, k = 1); while(!issquare(m), m += n; k++); k;} \\ Amiram Eldar, Mar 16 2025

Formula

a(n) = ((A215653(n))^2-1)/n.

Extensions

Edited and extended by Robert G. Wilson v, Oct 21 2002
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