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.

Previous Showing 11-13 of 13 results.

A162290 Let A087788(n) = p*q*r, where p

Original entry on oeis.org

7, 23, 48, 22, 47, 45, 45, 21, 44, 163, 162, 43, 161, 280, 1684, 1363, 159, 351, 950, 1675, 1358, 949, 158, 345, 1829, 947, 1353, 510, 938, 1660, 2796, 1820, 820, 10208, 2779, 935, 1650, 817, 937, 1822
Offset: 1

Views

Author

A.K. Devaraj, Jul 01 2009

Keywords

Comments

A.K. Devaraj conjectured that a(n) is always an integer, and this was proved by Carl Pomerance.
a(n) may be called the Pomerance index of the n-th 3-Carmichael number.
An application of Pomerance index: The index for the Carmichael number 561 is 7. This can be used to prove that 561 is the only 3-factor Carmichael number with 3 as one of its factors. Proof: Let N be a 3-factor composite number. Keep 3 fixed and increase the other two prime factors indefinitely. The relevant Pomerance index is a number less than 7 but greater than 6. As the other two prime factors are increased indefinitely the Pomerance index becomes asymptotic to 6. Hence 561 is the only 3-factor Carmichael number with 3 as a factor. - A.K. Devaraj, Jul 27 2010
Let p be a prime number. Then, along the lines indicated above, it can be proved that there are only a finite number of 3-Carmichael numbers divisible by p. - A.K. Devaraj, Aug 06 2010

Links

Crossrefs

Cf. A002997, A087788, A162990.

Programs

  • PARI
    do(lim)=my(v=List()); forprime(p=3, sqrtnint(lim\=1,3), forprime(q=p+1, sqrtint(lim\p), forprime(r=q+1, lim\(p*q), if((q*r-1)%(p-1)||(p*r-1)%(q-1)||(p*q-1)%(r-1), , listput(v, [p*q*r,(p*q*r-1)*(p-1)/(q-1)/(r-1)]))))); v=vecsort(v,1); vector(#v,i,v[i][2]) \\ Charles R Greathouse IV, Sep 07 2016

Extensions

Edited by N. J. A. Sloane, Sep 14 2009, based on email messages from David Broadhurst and M. F. Hasler, Jul 10 2009
Spelling corrected by Jason G. Wurtzel, Aug 23 2010

A027451 First diagonal of A027447.

Original entry on oeis.org

1, 1, 4, 9, 144, 100, 3600, 11025, 78400, 63504, 6350400, 5336100, 768398400, 662547600, 577152576, 2029052025, 519437318400, 463325262400, 150117385017600, 135480939978384, 122885206329600, 111967718990400, 54192375991353600, 49770428644836900
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

Equals the denominators of MN(z;n)/(n!)^2 for n =>1, see A162990. - Johannes W. Meijer, Jul 21 2009
It appears that a(n) = denominator of n^2*sum(1/k^2,k=1..n). - Gary Detlefs, May 29 2010

Crossrefs

From Johannes W. Meijer, Jul 21 2009: (Start)
Equals A002944(n)^2.
Equals A001044(n-1)/A025527(n)^2.
(End)

Programs

Formula

Numerators of sequence a[ n, n ] in (a[ i, j ])^3 where a[ i, j ] = 1/i if j<=i, 0 if j>i.
a(n) = (lcm($1..n)/n)^2. - Johannes W. Meijer, Jul 21 2009

Extensions

More terms from Sean A. Irvine, Nov 04 2019

A163956 Multiplicative order of 2 in Z/mZ with m = A002997(n).

Original entry on oeis.org

40, 24, 36, 56, 60, 660, 198, 252, 45, 180, 60, 144, 153, 1012, 36, 120, 300, 72, 36, 1160, 60, 36, 300, 56, 36, 660, 4284, 264, 420, 3060, 2268, 180, 540, 1680, 120, 4900, 1080, 396, 72, 72, 60, 60, 144, 2970, 612, 396, 324, 210, 180, 540, 504, 792, 198, 180
Offset: 1

Views

Author

A.K. Devaraj, Aug 28 2009

Keywords

Comments

Related sequence: A162990. - A.K. Devaraj, Aug 31 2009

References

  • A. K. Devaraj, "Minimum universal exponent generalisation of Fermat's theorem" (ISSN 1550-3747)

Links

Crossrefs

Cf. A002326, A002997, A007733, A104016, A162990, A165139.

Programs

  • Mathematica
    MultiplicativeOrder[2, #] & /@ Select[Range[1, 10^6, 2], CompositeQ[#] && Divisible[# - 1, CarmichaelLambda[#]] &] (* Amiram Eldar, Jul 30 2020 *)

Formula

a(n) = A002326((A002997(n)-1)/2) = A007733(A002997(n)). - Amiram Eldar, Jul 30 2020

Extensions

Corrected and extended by M. F. Hasler, Sep 23 2009
Edited by N. J. A. Sloane, Sep 23 2009, following suggestions from M. F. Hasler
More terms from Amiram Eldar, Jul 30 2020
Previous Showing 11-13 of 13 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.