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 1-7 of 7 results.

A062373 Ratio of totient to Carmichael's lambda function is 2.

Original entry on oeis.org

8, 12, 15, 16, 20, 21, 28, 30, 32, 33, 35, 36, 39, 42, 44, 45, 51, 52, 55, 57, 64, 66, 68, 69, 70, 75, 76, 77, 78, 87, 90, 92, 93, 95, 99, 100, 102, 108, 110, 111, 114, 115, 116, 119, 123, 124, 128, 129, 135, 138, 141, 143, 147, 148, 150, 153, 154, 155, 159, 161
Offset: 1

Views

Author

Vladeta Jovovic, Jun 17 2001

Keywords

Comments

Numbers k such that the highest order of elements in (Z/kZ)* is phi(n)/2, (Z/kZ)* = the multiplicative group of integers modulo k. Also numbers k such that (Z/kZ)* = C_2 X C_(2r). - Jianing Song, Jul 28 2018
Contains the powers of 2 greater than 4, 4 times primes, and semiprimes pq where (p-1)/2 and (q-1)/2 are coprime. If n is odd and in this sequence then so is 2n. - Charlie Neder, May 27 2019

Examples

			From _Jianing Song_, Jul 28 2018: (Start)
(Z/8Z)* = C_2 X C_2, so 8 is a term.
(Z/21Z)* = C_2 X C_6, so 21 is a term.
(Z/35Z)* = C_2 X C_12, so 35 is a term. (End)

Links

Crossrefs

Cf. A000010, A002322, A034380, A033948, A062374, A062375, A062376, A062377.

Programs

  • Haskell
    a062373 n = a062373_list !! (n-1)
a062373_list = filter ((== 2) . a034380) [1..]
-- Reinhard Zumkeller, Sep 02 2014
  • Mathematica
    Reap[ For[ n = 1, n <= 161, n++, If[ EulerPhi[n] / CarmichaelLambda[n] == 2, Sow[n]]]][[2, 1]] (* Jean-François Alcover, Mar 26 2013 *)
Select[Range[200],EulerPhi[#]/CarmichaelLambda[#]==2&] (* Harvey P. Dale, Jun 27 2018 *)
  • PARI
    isok(n) = eulerphi(n)/lcm(znstar(n)[2]) == 2; \\ Michel Marcus, Jul 28 2018

Formula

Solutions to phi(k)/lambda(k) = 2.

Extensions

More terms from Reiner Martin, Dec 22 2001

A062377 Euler phi(n) / Carmichael lambda(n) = 10.

Original entry on oeis.org

275, 341, 451, 550, 671, 682, 775, 781, 902, 1111, 1271, 1342, 1375, 1441, 1550, 1562, 1661, 1775, 1991, 2101, 2201, 2222, 2321, 2542, 2651, 2750, 2761, 2882, 2911, 2981, 3025, 3091, 3131, 3275, 3322, 3421, 3550, 3641, 3751, 3775, 3875, 3982, 4061
Offset: 1

Views

Author

Vladeta Jovovic, Jun 17 2001

Keywords

Comments

Solutions to A000010(n)/A002322(n)=10.

Links

Crossrefs

Cf. A000010, A002322, A034380, A033948, A062373, A062374, A062375, A062376.

Programs

  • Mathematica
    Reap[ For[ n = 1, n <= 4061, n++, If[ EulerPhi[n] / CarmichaelLambda[n] == 10, Sow[n]]]][[2, 1]] (* Jean-François Alcover, Mar 26 2013 *)
Select[Range[4100],EulerPhi[#]/CarmichaelLambda[#]==10&] (* Harvey P. Dale, Dec 22 2022 *)
  • 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 } {A062377(n)=eulerphi(n)/cl(n)} for(x=1,10001, if(A062377(x)==10,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

A066695 Numbers k such that Euler phi(k) / Carmichael lambda(k) = 12.

Original entry on oeis.org

252, 273, 315, 364, 399, 468, 481, 532, 546, 630, 651, 665, 684, 693, 741, 756, 777, 793, 798, 855, 868, 903, 945, 949, 962, 988, 1001, 1036, 1071, 1085, 1116, 1204, 1209, 1261, 1281, 1287, 1302, 1330, 1332, 1386, 1395, 1404, 1407, 1417, 1449, 1463
Offset: 1

Views

Author

Randall L Rathbun, Jan 12 2002

Keywords

Links

Crossrefs

Cf. A000010, A002322.
Continuation of A062373, A062374, A062375, A062376, A062377.

Programs

  • Mathematica
    Select[Range[2000], EulerPhi[#]/CarmichaelLambda[#] == 12 &] (* Alonso del Arte, Apr 17 2017 *)
  • 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 }
{A0(n)=eulerphi(n)/cl(n)}
for(x=1,10001, if(A0(x)==12,print1(x, ",")))
  • PARI
    isok(k) = eulerphi(k)/lcm(znstar(k)[2]) == 12; \\ Michel Marcus, May 25 2022

A066696 Numbers k such that Euler phi(k) / Carmichael lambda(k) = 14.

Original entry on oeis.org

1247, 1421, 2059, 2494, 2842, 3053, 3479, 3683, 4118, 4859, 5537, 6106, 6119, 6931, 6958, 7366, 8023, 8471, 9017, 9653, 9718, 9947, 10277, 10991, 11074, 11711, 12083, 12238, 13427, 13769, 13862, 13987, 14239, 14351, 15863, 16046, 16942
Offset: 1

Views

Author

Randall L Rathbun, Jan 12 2002

Keywords

Links

Crossrefs

Cf. A000010, A002322.
Cf. A062373, A062374, A062375, 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)= my(k=factor(f), l=1); for(x=1,omega(f),l=lcm(l,cmf([k[x,1], k[x,2]]))); l }
{A062377(n)=eulerphi(n)/cl(n)}
for(x=1,30001, if(A062377(x)==14,print1(x,",")))
  • PARI
    isok(k) = eulerphi(k)/lcm(znstar(k)[2]) == 14; \\ Michel Marcus, May 25 2022

Extensions

Terms joined (twice) by Georg Fischer, Jul 08 2022

A066698 Numbers k such that Euler phi(k) / Carmichael lambda(k) = 34.

Original entry on oeis.org

14111, 24617, 28222, 29767, 32743, 42059, 45629, 49234, 59534, 60691, 65486, 66641, 69071, 73373, 84118, 88639, 88723, 91258, 97751, 98159, 105877, 121382, 125903, 128027, 129677, 133282, 136001, 138142, 140183, 146507, 146746, 153851
Offset: 1

Views

Author

Randall L Rathbun, Jan 12 2002

Keywords

Links

Crossrefs

Cf. A000010, A002322.
Continuation of A062373, A062374, A062375, A062376, A062377.

Programs

  • Mathematica
    Select[ Range[2 10^5], EulerPhi[ # ] == 34CarmichaelLambda[ # ] &]
  • 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 }
{A0(n)=eulerphi(n)/cl(n)}
for(x=1,30001, if(A0(x)==34,print1(x,",")))
  • PARI
    isok(k) = eulerphi(k)/lcm(znstar(k)[2]) == 34; \\ Michel Marcus, May 25 2022

Extensions

More terms from Robert G. Wilson v, Jan 13 2002
Showing 1-7 of 7 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.