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 1-9 of 9 results.

A006931 Least Carmichael number with n prime factors, or 0 if no such number exists.

Original entry on oeis.org

561, 41041, 825265, 321197185, 5394826801, 232250619601, 9746347772161, 1436697831295441, 60977817398996785, 7156857700403137441, 1791562810662585767521, 87674969936234821377601, 6553130926752006031481761, 1590231231043178376951698401
Offset: 3

Views

Author

N. J. A. Sloane and Richard Pinch

Keywords

Comments

Alford, Grantham, Hayman, & Shallue construct large Carmichael numbers, finding upper bounds for a(3)-a(19565220) and a(10333229505). - Charles R Greathouse IV, May 30 2013

References

  • J.-P. Delahaye, Merveilleux nombres premiers ("Amazing primes"), p. 269, Pour la Science, Paris 2000.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002997, A135717, A135719, A135720, A135721.
Cf. A087788, A141711, A074379, A112428, A112429, A112430, A112431, A112432.

Programs

  • Mathematica
    (* Program not suitable to compute more than a few terms *)
A2997 = Select[Range[1, 10^6, 2], CompositeQ[#] && Mod[#, CarmichaelLambda[#] ] == 1&];
(First /@ Split[Sort[{PrimeOmega[#], #}& /@ A2997], #1[[1]] == #2[[1]]&])[[All, 2]] (* Jean-François Alcover, Sep 11 2018 *)
  • PARI
    Korselt(n,f=factor(n))=for(i=1,#f[,1],if(f[i,2]>1||(n-1)%(f[i,1]-1),return(0)));1
a(n)=my(p=2,f);forprime(q=3,default(primelimit),forstep(k=p+2,q-2,2,f=factor(k);if(vecmax(f[,2])==1 && #f[,2]==n && Korselt(k,f), return(k)));p=q)
\\ Charles R Greathouse IV, Apr 25 2012
  • PARI
    carmichael(A, B, k) = A=max(A, vecprod(primes(k+1))\2); (f(m, l, lo, k) = my(list=List()); my(hi=sqrtnint(B\m, k)); if(lo > hi, return(list)); if(k==1, lo=max(lo, ceil(A/m)); my(t=lift(1/Mod(m,l))); while(t < lo, t += l); forstep(p=t, hi, l, if(isprime(p), my(n=m*p); if((n-1)%(p-1) == 0, listput(list, n)))), forprime(p=lo, hi, if(gcd(m, p-1) == 1, list=concat(list, f(m*p, lcm(l, p-1), p+1, k-1))))); list); vecsort(Vec(f(1, 1, 3, k)));
a(n) = if(n < 3, return()); my(x=vecprod(primes(n+1))\2,y=2*x); while(1, my(v=carmichael(x,y,n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Feb 24 2023

Extensions

Corrected by Lekraj Beedassy, Dec 31 2002
More terms from Ralf Stephan, from the Pinch paper, Apr 16 2005
Edited by N. J. A. Sloane, May 16 2008 at the suggestion of R. J. Mathar.
Escape clause added by Jianing Song, Dec 12 2021

A112428 Carmichael numbers equal to the product of 5 primes.

Original entry on oeis.org

825265, 1050985, 9890881, 10877581, 12945745, 13992265, 16778881, 18162001, 27336673, 28787185, 31146661, 36121345, 37167361, 40280065, 41298985, 41341321, 41471521, 47006785, 67371265, 67994641, 69331969, 74165065
Offset: 1

Views

Author

Shyam Sunder Gupta, Dec 11 2005

Keywords

Comments

A subsequence is given by (6n+1)*(12n+1)*(18n+1)*(36n+1)*(72n+1) with n in A206349. - M. F. Hasler, Apr 14 2015

Examples

			a(1)=825265=5*7*17*19*73

Links

Crossrefs

Cf. A002997, A014614, A206349.
Cf. A087788, A141711, A074379, A112429 - A112432, A006931.

Programs

Formula

A112428 = A002997 intersect A014614. - M. F. Hasler, Apr 14 2015

Extensions

Crossrefs added by M. F. Hasler, Apr 14 2015

A112429 Carmichael numbers equal to the product of 6 primes.

Original entry on oeis.org

321197185, 413631505, 417241045, 496050841, 509033161, 611397865, 612347905, 638959321, 672389641, 832060801, 834720601, 868234081, 945959365, 986088961, 1074363265, 1177800481, 1210178305, 1256855041, 1410833281, 1481619601
Offset: 1

Views

Author

Shyam Sunder Gupta, Dec 11 2005

Keywords

Examples

			a(1)=321197185=5*19*23*29*37*137

Links

Crossrefs

Cf. A002997, A087788, A141711, A074379, A112428, A112430, A112431, A112432, A006931.

Programs

A112430 Carmichael numbers equal to the product of 7 primes.

Original entry on oeis.org

5394826801, 6295936465, 12452890681, 13577445505, 15182481601, 20064165121, 22541365441, 24673060945, 26242929505, 26602340401, 27405110161, 28553256865, 33203881585, 38059298641, 39696166081, 40460634865
Offset: 1

Views

Author

Shyam Sunder Gupta, Dec 11 2005

Keywords

Examples

			a(1) = 5394826801 = 7*13*17*23*31*67*73.

Links

Crossrefs

Cf. A002997, A087788, A141711, A074379, A112428, A112429, A112431, A112432, A006931.

Programs

A112432 Carmichael numbers equal to the product of 9 primes.

Original entry on oeis.org

9746347772161, 11537919313921, 11985185775745, 14292786468961, 23239986511105, 24465723528961, 26491881502801, 27607174936705, 30614445878401, 30912473358481, 34830684315505, 51620128928641
Offset: 1

Views

Author

Shyam Sunder Gupta, Dec 11 2005

Keywords

Examples

			a(1) = 9746347772161 = 7*11*13*17*19*31*37*41*641.

Links

Crossrefs

Cf. A002997, A087788, A141711, A074379, A112428, A112429, A112430, A112431, A006931.

Programs

A112431 Carmichael numbers equal to the product of 8 primes.

Original entry on oeis.org

232250619601, 306177962545, 432207073585, 576480525985, 658567396081, 689702851201, 747941832001, 1013666981041, 1110495895201, 1111586883121, 1286317859905, 1292652236161, 1341323384401, 1471186523521, 1567214060545
Offset: 1

Views

Author

Shyam Sunder Gupta, Dec 11 2005

Keywords

Examples

			a(1) = 232250619601 = 7*11*13*17*31*37*73*163.

Links

Crossrefs

Cf. A002997, A087788, A141711, A074379, A112428, A112429, A112430, A112432, A006931.

Programs

A141711 Carmichael numbers with more than 3 prime factors.

Original entry on oeis.org

41041, 62745, 63973, 75361, 101101, 126217, 172081, 188461, 278545, 340561, 449065, 552721, 656601, 658801, 670033, 748657, 825265, 838201, 852841, 997633, 1033669, 1050985, 1082809, 1569457, 1773289, 2100901, 2113921, 2433601
Offset: 1

Views

Author

M. F. Hasler, Jul 01 2008

Keywords

Comments

Sequence A087788 gives Carmichael numbers with exactly 3 prime factors; since they cannot have fewer (cf. references in A002997), this sequence is the complement of A087788 in A002997.
The terms preceding a(17) = 825265 = A006931(5) have exactly 4 prime factors. See A112428 - A112432 for Carmichael numbers with exactly 5, ..., 9 prime factors. - M. F. Hasler, Apr 14 2015

Examples

			a(17)=825265 is the least Carmichael number having more than 4 divisors, thus the sequence differs from A074379 only from that term on.

Links

Crossrefs

Cf. A002997, A087788, A074379, A112428 - A112432, A006931.

Programs

  • Mathematica
    ok[n_] :=  Divisible[n - 1, CarmichaelLambda[n]] && Length[FactorInteger[n]] > 3; Select[ Range[3*10^6], ok] (* Jean-François Alcover, Sep 23 2011 *)
  • PARI
    A2997=readvec("b002997.gp"); A002997(n)=A2997[n]; for( n=1,100, omega( A002997(n) ) > 3 & print1( A002997(n)", "))

Formula

A141711 = A002997 \ A087788 = A074379 U A112428 U A112429 U A112430 U A112431 U A112432 U ...

A338442 Carmichael numbers with 10 prime factors.

Original entry on oeis.org

1436697831295441, 1493812621027441, 2094319836529921, 2349991949342401, 2842648863161185, 2859959706040801, 3455134500424321, 3871703982953521, 4177950872896801, 4289150794129201, 4937378437571041, 5071419883911745, 5778659093725441, 6665161459969441, 6682056104892961
Offset: 1

Views

Author

Tim Johannes Ohrtmann, Oct 28 2020

Keywords

Examples

			1436697831295441 = 11*13*19*29*31*37*41*43*71*127 and 10, 12, 18, 28, 30, 36, 40, 42, 70, 126 all divide 1436697831295440.

Links

Crossrefs

Cf. A002997 (Carmichael numbers).
Cf. A006931 (Least Carmichael number with n prime factors).
Cf. A299710 (Number of terms less than 10^n).
Cf. A087788, A074379, A112428, A112429, A112430, A112431, A112432, A338443 (Carmichael numbers with 3-9 and 11 prime factors).

Programs

  • PARI
    is(n)={omega(n)==10&&is_A002997(n)}

Formula

Equals A002997 intersect A046314.

A338443 Carmichael numbers with 11 prime factors.

Original entry on oeis.org

60977817398996785, 105083995864811041, 107473646345582881, 132819104923908481, 145671955835893201, 161802381510126721, 165167398073764801, 206063729626916161, 263076030916096321, 292433912163313921, 292561243007134465, 337365329710615921, 388219799621120545
Offset: 1

Views

Author

Tim Johannes Ohrtmann, Oct 28 2020

Keywords

Examples

			60977817398996785 = 5*7*17*19*23*37*53*73*79*89*233 and 4, 6, 16, 18, 22, 36, 52, 72, 78, 88, 232 all divide 60977817398996784.

Links

Crossrefs

Cf. A002997 (Carmichael numbers).
Cf. A006931 (Least Carmichael number with n prime factors).
Cf. A299710 (Number of terms less than 10^n).
Cf. A087788, A074379, A112428, A112429, A112430, A112431, A112432, A338442 (Carmichael numbers with 3-10 prime factors).

Programs

  • PARI
    is(n)={omega(n)==11&&is_A002997(n)}

Formula

Equals A002997 intersect A069272.
Showing 1-9 of 9 results.

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