A055553 -id:A055553 - cp's OEIS frontend

A257643 Carmichael numbers k such that k-1 is squarefree.

139952671, 74689102411, 121254376891, 187054437571, 231440115271, 236359158267, 303008129971, 306252926071, 380574791611, 426951670531, 556303918171, 639109148371, 660950414671, 1101375141511, 1483826843731, 1487491483171, 1861175569891, 2794268624071

Offset: 1

Thomas Ordowski, Nov 05 2015

nonn

If k is a Carmichael number with k-1 squarefree, then gcd(phi(k),k-1) = lambda(k), i.e., Carmichael lambda function A002322.

Amiram Eldar, Table of n, a(n) for n = 1..5287 (terms below 10^22, calculated using data from Claude Goutier; terms 1..164 from Robert Israel, terms 165..1037 from Charles R Greathouse IV)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.

Subsequence of A185321.

Cf. A002997, A264012, A007674.

t(n) = my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]>1||(n-1)%(f[i, 1]-1), return(0))); 1;
is(n) = n%2 && !isprime(n) && t(n) && n>1;
isok(n) = is(n) && issquarefree(n-1); \\ Altug Alkan, Nov 06 2015

is(n) = my(f=factor(n)); for(i=1, #f~, if(f[i,1]%4<3 || f[i, 2]>1 || (n-1)%(f[i, 1]-1), return(0))); !isprime(n) && issquarefree(n-1)
is(n) = n%2 && !isprime(n) && t(n) && n>1 \\ Charles R Greathouse IV, Nov 09 2015

A277389 Numbers k such that lambda(k)^3 divides (k-1)^2, where lambda(k) = A002322(k).

Original entry on oeis.org

1, 2, 1729, 19683001, 367804801, 631071001, 2064236401, 2320690177, 24899816449, 40017045601, 110592000001, 137299665601, 432081216001, 479534887801, 760355883001, 1111195454401, 3176523000001, 3495866888449, 3837165696001, 8571867768001, 14373832968001

Offset: 1

Thomas Ordowski, Oct 12 2016

nonn

Carmichael numbers are composite numbers n such that k = 1 (mod lambda(k)); equivalently, lambda(k)^2 divides (k-1)^2. As a result, all composite terms of the sequence are Carmichael numbers A002997. But there are no primes in this sequence except for 2 (since lambda(p) = p-1 and (p-1)^3 > (p-1)^2 for p > 2) and so all terms in this sequence other than 1 and 2 are Carmichael numbers. - Charles R Greathouse IV, Oct 15 2016

Is this sequence infinite?

Amiram Eldar, Table of n, a(n) for n = 1..1469 (terms below 10^22, calculated using data from Claude Goutier; terms 1..58 from Robert Israel, terms 59..101 from Charles R Greathouse IV)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.

Cf. A002322, A002997, A265628.

isok(n) = ((n-1)^2 % (lcm(znstar(n)[2])^3)) == 0; \\ Michel Marcus, Oct 12 2016

a(4) from Michel Marcus, Oct 12 2016
a(5)-a(6) from Michel Marcus, Oct 13 2016
More terms from Robert Israel, Oct 13 2016

A290692 Carmichael numbers of the form p - 2 where p is a prime number.

Original entry on oeis.org

561, 2465, 656601, 1909001, 174352641, 230996949, 275283401, 939947009, 1534274841, 3264820001, 5860426881, 6025532241, 25536531021, 36709177121, 53388707681, 54519328481, 56222911361, 101536702401, 105528976961, 180481509681, 196866607601, 239862350001, 329245587161, 347469383801, 347511324161

Offset: 1

Altug Alkan, Aug 09 2017

nonn

Rotkiewicz mentioned the first six terms of this sequence at the end of page 59 of his article (Links section). But his list includes 2821 and 46657 (2823 = 3 * 941 and 46659 = 3 * 103 * 151), which should not be there.

Carmichael numbers of the form p + 2 where p is a prime number are 1105, 2821, 6601, 29341, 41041, 52633, ... (see also A272754 for corresponding prime numbers).

Amiram Eldar, Table of n, a(n) for n = 1..5901 (terms below 10^22 calculated using data from Claude Goutier; terms 1..591 from Robert Israel)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
R. G. E. Pinch, Carmichael numbers up to 10^16, 10^16 to 10^17, 10^17 to 10^18
Andrzej Rotkiewicz, On pseudoprimes having special forms and a solution of K. Szymiczek's problem, Acta Mathematica Universitatis Ostraviensis, Vol. 13, No. 1 (2005), pp. 57-71.
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A272754, A287591.

# Using data file from Richard Pinch
infile:= "carmichael-16": Res:= NULL;
do
  S:= readline(infile);
  if S = 0 then break fi;
  L:= sscanf(S,"%d");
  if nops(L) <> 1 then break fi;
  if isprime(L[1]+2) then Res:= Res, L[1]; fi
od:
Res; # Robert Israel, Jun 03 2019

Cases[Range[1, 10^7, 2], n_ /; And[Mod[n, CarmichaelLambda@ n] == 1, ! PrimeQ@ n, PrimeQ[n + 2]]] (* Michael De Vlieger, Aug 09 2017, after Artur Jasinski at A002997 *)

isA002997(n) = {my(f); bittest(n, 0) && !for(i=1, #f=factor(n)~, (f[2, i]==1 && n%(f[1, i]-1)==1)||return) && #f>1}
isok(n) = isprime(n+2) && isA002997(n)

More terms from Robert Israel, Jun 03 2019

A290805 Least Carmichael number whose Euler totient function value is an n-th power.

Original entry on oeis.org

561, 1729, 63973, 1729, 367939585, 63973, 294409, 232289960085001, 11570858964626401, 79939760257, 509033161, 611559276803883001, 13079177569, 27685385948423487745, 26979791457662785, 287290964059686145, 13046319747121261903830001, 7847507962539316696504321, 993942550111105, 6280552422566791778305, 24283361157780097, 759608966313690599499265, 6657107145346817668085761, 283219223388059484626764342346640001

Offset: 1

Amiram Eldar, Aug 11 2017

nonn

Banks proved that for each positive integer N there are an infinite number of Carmichael numbers whose Euler totient function value is an N-th power. Therefore this sequence is infinite.

For any n > 26, a(n) > 10^22. - Amiram Eldar, Apr 20 2024

			phi(1729) = 36^2 = 6^4 while phi(561) and phi(1105) are not perfect powers, therefore a(2) = a(4) = 1729.

Max Alekseyev, Table of n, a(n) for n = 1..60
William D. Banks, Carmichael Numbers with a Square Totient, Canadian Mathematical Bulletin, Vol. 52, No. 1 (2009), pp. 3-8.
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
R. G. E. Pinch, Tables relating to Carmichael numbers.

Cf. A000010, A002997, A272798, A292573.

Terms up to a(13) were calculated using Pinch's tables of Carmichael numbers.
a(1) prepended by David A. Corneth, Aug 11 2017
a(14)-a(16), a(19)-a(21), a(25)-a(26) calculated using data from Claude Goutier and added by Amiram Eldar, Apr 20 2024
a(17)-a(18), a(22)-a(24) from Max Alekseyev, Apr 25 2024
Edited by Max Alekseyev, Dec 04 2024

A291637 Carmichael numbers (A002997) that are super-Poulet numbers (A050217).

Original entry on oeis.org

294409, 1299963601, 4215885697, 4562359201, 7629221377, 13079177569, 19742849041, 45983665729, 65700513721, 147523256371, 168003672409, 227959335001, 459814831561, 582561482161, 1042789205881, 1297472175451, 1544001719761, 2718557844481, 3253891093249, 4116931056001, 4226818060921, 4406163138721, 4764162536641, 4790779641001, 5419967134849, 7298963852041, 8470346587201

Offset: 1

Max Alekseyev and Thomas Ordowski, Aug 28 2017

nonn

Problem: are there infinitely many such numbers?
From Daniel Suteu, Sep 17 2020: (Start)
If we consider f(n) to be the smallest number in the sequence with n prime factors, then we have:
f(3) = 294409,
f(4) = 3018694485093841,
f(5) <= 521635331852681575100906881,
f(6) <= 2835402730651853232634509813787097410561,
f(7) <= 165784025660216242122027716057592895796242004385542265601. (End)

Amiram Eldar, Table of n, a(n) for n = 1..5328 (calculated using data from Claude Goutier; terms 1..983 from Max Alekseyev)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Index entries for sequences related to Carmichael numbers.

Intersection of A178997 and A002997.

A329538 Odd composite numbers k such that A111076(k)^((k-1)/2) == -1 (mod k).

Original entry on oeis.org

29341, 1152271, 5481451, 14913991, 15247621, 36765901, 133800661, 178482151, 299736181, 579606301, 652969351, 702683101, 739444021, 743404663, 775368901, 3215031751, 4340265931, 5871134179, 8657319259, 9293756581, 12191597551, 13734086221, 14386156093, 19331388805

Offset: 1

Amiram Eldar and Thomas Ordowski, Nov 16 2019

nonn

Carmichael numbers k such that A111076(k)^((k-1)/2) == -1 (mod k).
Note that if p is an odd prime, then A111076(p)^((p-1)/2) == -1 (mod p).
Max Alekseyev proved (in a letter to the second author) that all these numbers have an odd number of prime factors, showing that if k is a term, then k is a Carmichael number m such that p-1 does not divide (m-1)/2 for every prime p|m (the numbers m form the supersequence A329799).
There are 6469 terms k of this sequence below 2^64:
4240 with 3 prime factors, least is 29341 = 13*37*61,
1790 with 5 prime factors, least is 4340265931 = 19*43*107*131*379,
437 with 7 prime factors, least is 37038179683765 = 5*13*29*37*317*757*2213,
2 with 9 prime factors, least is 1025735495681200591 = 7*19*31*67*79*163*199*271*5347.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Claude Goutier)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A111076, A329799.

Subsequence of A262043.

f[1, lam_] = 1; f[n_, lam_] := If[n < 5, n - 1, Module[{k = 1}, While[GCD[k, n] > 1 || MultiplicativeOrder[k, n] < lam, k++]; k]]; aQ[n_] := CompositeQ[n] && Divisible[n - 1 , (lam = CarmichaelLambda[n])] && PowerMod[f[n, lam], (n - 1)/2, n] == n - 1; Select[Range[1, 6*10^6, 2], aQ] (* after the Charles R Greathouse IV at A111076 *)

A329948 Carmichael numbers m that have at least 3 prime factors p such that p+1 | m+1.

Original entry on oeis.org

9857524690572481, 33439671284716801, 96653613831890401, 270136961300544031, 528096456788419441, 650643395658753601, 710238404427321601, 1822922951416158241, 4011563714063821201, 4525693104167627041, 4631812281009523441, 7049793086137296001, 8605736094003523201, 10449416165574628801, 11175581620177915681, 12746447178170148001, 12769123623410580481, 17705945296667070001

Offset: 1

Daniel Suteu, Nov 25 2019

nonn

It is not known whether any Carmichael number (A002997) is also Lucas-Carmichael number (A006972). If such a number exists, then it would be a term of this sequence.

			m = 9857524690572481 is a term because it is a Carmichael number and it has at least 3 prime factors p, {13, 61, 433}, such that p+1 | m+1.

Amiram Eldar, Table of n, a(n) for n = 1..179 (terms below 10^22, calculated using data from Claude Goutier)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Wikipedia, Carmichael number.
Wikipedia, Lucas-Carmichael number.

Cf. A002997, A006972.

use bigint; use ntheory ':all'; sub isok { my $m = $[0]; is_carmichael($m) && (grep { ($m+1) % ($+1) == 0 } factor($m)) >= 3 };

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

Tim Johannes Ohrtmann, Oct 28 2020

nonn

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Claude Goutier; terms 1..1142 from Daniel Suteu)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Richard Pinch, Tables relating to Carmichael numbers.

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).

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

Equals A002997 intersect A069272.

A346569 Carmichael numbers (A002997) k such that A003961(k) is also a Carmichael number.

Original entry on oeis.org

938531360353681, 6178246534322281, 518705522457928921, 7019247908645553241, 16242056799655920481, 94812683932464811561, 94986212971063089241, 408133613144935002601, 418441276466266605481, 453648717063017803081, 556606627235843071681, 1140359076998537247001

Offset: 1

Amiram Eldar, Jul 23 2021

nonn

Each of the first 17 terms has 3 distinct prime divisors. [updated Apr 22 2024]

a(6) <= 94812683932464811561. A term with 4 prime factors is 9584146525723596902470058833132261. - Daniel Suteu, Jul 24 2021

			938531360353681 = 53881 * 107761 * 161641 is a term since it is a Carmichael number, and A003961(938531360353681) = 53887 * 107773 * 161659 = 938844932257009 is also a Carmichael number.

Amiram Eldar, Table of n, a(n) for n = 1..17 (terms below 10^22 calculated using data from Claude Goutier)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A003961.

Subsequence of A346568.

a(6) verified and a(7)-a(13) calculated using using data from Claude Goutier by Amiram Eldar, Apr 22 2024

A367231 Carmichael numbers k such that the multiplicative order of 2 modulo k is odd.

Original entry on oeis.org

15841, 52633, 5049001, 68154001, 104852881, 238244041, 382536001, 3215031751, 3863326897, 7211236033, 8214723001, 15462960481, 22008493921, 23000028481, 29392867201, 31708772257, 41217865921, 53125756201, 60518537641, 74190097801, 77874636001, 83828294551, 103387371361

Offset: 1

Amiram Eldar, Nov 11 2023

nonn

These Carmichael numbers seem to be relatively rare: among the 4279356 Carmichael numbers below 2^64 only 3097 are terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Claude Goutier)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Index entries for sequences related to Carmichael numbers.

Intersection of A002997 and A036259.
Subsequence of A367230.
Cf. A002326, A163956.

Select[2*Range[3*10^6] + 1, Mod[#, CarmichaelLambda[#]] == 1 && CompositeQ[#] && OddQ[MultiplicativeOrder[2, #]] &]

is(n) = n > 1 && n % 2 && !isprime(n) && n % lcm(znstar(n)[2]) == 1 && znorder(Mod(2, n)) % 2;

cp's OEIS Frontend

A257643 Carmichael numbers k such that k-1 is squarefree.

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A277389 Numbers k such that lambda(k)^3 divides (k-1)^2, where lambda(k) = A002322(k).

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A290692 Carmichael numbers of the form p - 2 where p is a prime number.

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A290805 Least Carmichael number whose Euler totient function value is an n-th power.

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A291637 Carmichael numbers (A002997) that are super-Poulet numbers (A050217).

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A329538 Odd composite numbers k such that A111076(k)^((k-1)/2) == -1 (mod k).

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Mathematica

A329948 Carmichael numbers m that have at least 3 prime factors p such that p+1 | m+1.

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Perl

A338443 Carmichael numbers with 11 prime factors.

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A346569 Carmichael numbers (A002997) k such that A003961(k) is also a Carmichael number.

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