A055553 -id:A055553 - cp's OEIS frontend

A367320 Carmichael numbers k such that (k-1)/lambda(k) > (m-1)/lambda(m) for all Carmichael numbers m < k, where lambda is the Carmichael lambda function (A002322).

561, 1105, 1729, 29341, 41041, 63973, 172081, 825265, 852841, 1773289, 5310721, 9890881, 12945745, 18162001, 31146661, 93869665, 133205761, 266003101, 417241045, 496050841, 509033161, 1836304561, 1932608161, 2414829781, 4579461601, 9799928965, 11624584621, 12452890681

Offset: 1

Amiram Eldar, Nov 14 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 1..169 (terms below 10^22 calculated using data from Claude Goutier)
Amiram Eldar, Table of n, a(n), (a(n)-1)/A002322(a(n)) for n = 1..169.
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Index entries for sequences related to Carmichael numbers.

Subsequence of A002997.

Cf. A002322, A174590, A226216, A306414, A367319.

seq[kmax_] := Module[{s = {}, r, rm = 0, lam}, Do[If[CompositeQ[k], lam = CarmichaelLambda[k]; If[Mod[k, lam] == 1, r = (k - 1)/lam; If[r > rm, rm = r; AppendTo[s, k]]]], {k, 9, kmax, 2}]; s]; seq[10^6]

lista(kmax) = {my(r, rm = 0, lam); forcomposite(k = 4, kmax, if(k % 2, lam = lcm(znstar(k)[2]); if(k % lam == 1, r = (k-1)/lam; if(r > rm, rm = r; print1(k, ", ")))));}

A258839 Carmichael numbers whose prime factors all have the form p=1+x^2+y^2 for some x,y in Z.

Original entry on oeis.org

561, 162401, 410041, 488881, 656601, 2433601, 36765901, 109393201, 171454321, 176659201, 178837201, 189941761, 221884001, 288120421, 600892993, 618068881, 721244161, 931694401, 985052881, 1183104001, 1828377001, 1848112761, 1943951041, 2361232477, 2438403661

Offset: 1

Michel Marcus, Jun 12 2015

nonn

Banks & Freiberg show that this sequence is infinite.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Claude Goutier; terms 1..733 from Giovanni Resta)
William D. Banks and Tristan Freiberg, Carmichael numbers and the sieve, Journal of Number Theory, Vol. 165 (2016), pp. 15-29; arXiv preprint, arXiv:1506.03497 [math.NT], 2015.
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 (Carmichael numbers), A079545 (primes of the form x^2 + y^2 + 1).

has(n)=for(x=sqrtint(n\2),sqrtint(n-1), if(issquare(n-x^2-1), return(1)));0
Korselt(n,f=factor(n))=for(i=1,#f~,if(f[i, 2]>1||(n-1)%(f[i, 1]-1),return(0))); 1
is(n)=my(f); if(n%2==0||isprime(n)||!Korselt(n,f=factor(n))||n<9, return(0)); for(i=1,#f~,if(!has(f[i,1]), return(0))); 1 \\ Charles R Greathouse IV, Jun 12 2015

A267462 Carmichael numbers that are not of the form x^2 + y^2 + z^2 where x, y and z are integers.

Original entry on oeis.org

8911, 1152271, 10267951, 14913991, 64377991, 67902031, 139952671, 178482151, 612816751, 652969351, 743404663, 2000436751, 2560600351, 3102234751, 3215031751, 5615659951, 5883081751, 7773873751, 8863329511, 9462932431, 10501586767, 11335174831, 12191597551, 13946829751, 16157879263, 21046047751

Offset: 1

Altug Alkan, Jan 15 2016

nonn

Intersection of A002997 and A004215.
Carmichael numbers that are the sum of 4 but no fewer nonzero squares.
Carmichael numbers of the form 8*k + 7.
Subsequence of A185321.
Carmichael numbers of the form x^2 + y^2 + z^2 where x, y and z are integers are 561, 1105, 1729, 2465, 2821, 6601, 10585, 15841, 29341, 41041, 46657, 52633, 62745, 63973, 75361, 101101, 115921, 126217, 162401, 172081, 188461, 252601, 278545, 294409, 314821, 334153, 340561, 399001, 410041, 449065, 488881, 512461, 530881, 552721, ...

			Carmichael number 561 is not a term of this sequence because 561 = 2^2 + 14^2 + 19^2.
Carmichael number 8911 is a term because there is no integer values of x, y and z for the equation 8911 = x^2 + y^2 + z^2.
Carmichael number 10585 is not a term because 10585 = 0^2 + 37^2 + 96^2.

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.
G. Tarry, I. Franel, A. Korselt, and G. Vacca. Problème chinois. L'intermédiaire des mathématiciens 6 (1899), pp. 142-144.
Eric Weisstein's World of Mathematics, Carmichael Number
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A004215, A185321, A265237.

filter:= proc(n)
  local q;
  if isprime(n) then return false fi;
  if 2 &^ (n-1) mod n <> 1 then return false fi;
  for q in ifactors(n)[2] do
    if q[2] > 1 or (n-1) mod (q[1]-1) <> 0 then return false fi
    od;
    true
end proc:
select(filter, [seq(8*k+7, k=0..10^7)]); # Robert Israel, Jan 18 2016

Select[8*Range[1,8000000]+7, CompositeQ[#] && Divisible[#-1, CarmichaelLambda[#]] &] (* Amiram Eldar, Jun 26 2019 *)

isA004215(n) = { my(fouri, j) ; fouri=1 ; while( n >=7*fouri, if( n % fouri ==0, j= n/fouri -7 ; if( j % 8 ==0, return(1) ) ; ) ; fouri *= 4 ; ) ; return(0) ; } { for(n=1, 400, if(isA004215(n), print1(n, ", ") ; ) ; ) ; }
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 }
for(n=0, 1e10, if(isA002997(n) && isA004215(n), print1(n, ", ")));

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 }
for(n=0, 1e10, if(isA002997(k=8*n+7), print1(k, ", ")));

A277366 Composite numbers k such that phi(k)*lambda(k) divides (k-1)^2, where phi(k) = A000010(k) and lambda(k) = A002322(k).

Original entry on oeis.org

1729, 670033, 6840001, 83099521, 193708801, 321197185, 367804801, 484662529, 1752710401, 2320690177, 5064928705, 12820178449, 32220147601, 257124585601, 270177600001, 301036080385, 7043394657601, 13237329899521, 14276860416001, 85661522006401, 119377939968001

Offset: 1

Thomas Ordowski, Oct 11 2016

nonn

Are there infinitely many such numbers?

Such k must be a Carmichael number since phi(k)*lambda(k) = m*lambda(k)^2 for some integer m. - Nathan McNew, Oct 11 2016

Amiram Eldar, Table of n, a(n) for n = 1..115 (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.

Subsequence of A002997 and of A173703.

Cf. A000010, A002322, A002997, A173703.

Select[Range[10^8], CompositeQ[#] && Divisible[(# - 1)^2, EulerPhi[#] * CarmichaelLambda[#]] &] (* Amiram Eldar, Feb 02 2019 *)

lista(nn) = forcomposite(n=4, nn, if (((n-1)^2 % (eulerphi(n)*lcm(znstar(n)[2]))) == 0, print1(n, ", "));); \\ Michel Marcus, Oct 11 2016

is(n,f=factor(n))=(n-1)^2%(eulerphi(f)*lcm(znstar(f)[2])) == 0 && !isprime(n) && n>1 \\ Charles R Greathouse IV, Oct 11 2016

a(2)-a(3) from Michel Marcus, Oct 11 2016
a(4)-a(8) from Charles R Greathouse IV, Oct 11 2016
a(9)-a(13) from David A. Corneth, Oct 11 2016
More terms from Amiram Eldar, Feb 02 2019

A277720 Numbers k > 2 such that lambda(k)^2 divides k-1, where lambda(k) = A002322(k).

Original entry on oeis.org

2320690177, 17069520863233, 42182344790209, 65465530560001, 3432376805760001, 13322002122777601, 20388795375960001, 129009714848870401, 580007888606160001, 1096591987029196801, 3029756968906340401, 5806765663003468801, 6213994663149504001, 6367205158826803201, 7802569551798000001, 10319507991273499201

Offset: 1

Thomas Ordowski and Charles R Greathouse IV, Oct 28 2016

nonn

Squarefree numbers k > 2 such that (p-1)^2 | k-1 for every prime p|k.

For the first five terms, lambda(k)^2 | phi(k). - Thomas Ordowski, Apr 11 2017

Amiram Eldar, Table of n, a(n) for n = 1..47 (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.

Subsequence of A002997 and of A277389.

Cf. A002322.

isok(n) = (n % lcm(znstar(n)[2])^2)  == 1; \\ Michel Marcus, Apr 22 2017

a(7)-a(16) from Max Alekseyev, Apr 23 2017

A290281 Numbers k such that (k-1) mod phi(k) = lambda(k), where phi = A000010 and lambda = A002322.

Original entry on oeis.org

6601, 11972017, 34657141, 67902031, 139952671, 258634741, 2000436751, 8801128801, 9116583841, 9462932431, 38069223721, 326170416001, 359316634951, 1860929324101, 2022188518351, 2283475947391, 2648686458601, 2697891108151, 4513362899761, 5020030521001, 5472940991761, 6163867710001, 7507903975951, 19288340548471

Offset: 1

Robert Israel and Thomas Ordowski, Jul 25 2017

nonn

Numbers k such that A215486(k) = A002322(k).
Subsequence of the Carmichael numbers (A002997).
Composite numbers k such that (k-1) == lambda(k) (mod phi(k)).
Composite numbers k such that A277127(k) == 1 (mod A000010(k)).
Problem: are there infinitely many such numbers?
Conjecture: these are numbers k such that phi(k) + lambda(k) = k - 1. Checked up to 2^64. - Amiram Eldar and Thomas Ordowski, Dec 06 2019

Amiram Eldar, Table of n, a(n) for n = 1..239 (terms below 10^22 calculated using data from Claude Goutier; terms 1..79 from Robert Israel)
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. A000010, A002322, A002997, A215486.

Subsequence of A264012.

# Using data files for A002997
count:= 0:
for cfile in ["carmichael-16","carmichael17","carmichael18"] do
do
    S:= readline(cfile);
    if S = 0 then break fi;
    L:= map(parse, StringTools:-Split(S));
    n:= L[1]; pm:= map(`-`,L[2..-1],1);
    phin:= convert(pm,`*`);
    lambdan:= ilcm(op(pm));
    if n-1 - lambdan mod phin = 0 then
      count:= count+1; A[count]:= n;
    fi
od:
   fclose(cfile);
od:
seq(A[i],i=1..count); # Robert Israel, Jul 26 2017

Select[Range[10^8], Divisible[# - 1, (lam = CarmichaelLambda[#])] && Mod[# - 1, EulerPhi[#]] == lam &] (* Amiram Eldar, Dec 06 2019 *)

A290497 Carmichael numbers with a record number of aliquot divisors that are also Carmichael numbers.

Original entry on oeis.org

561, 63973, 31146661, 509033161, 84127131361, 11985185775745, 712484043821641, 24349841028259201, 53545320695780641, 141125066711098561, 16223841675726285601, 562477984940049379201

Offset: 1

Amiram Eldar, Aug 04 2017

The number of aliquot divisors is 0, 1, 2, 5, 7, 8, 10, 11, 17, 20, 26, 27, ...

The terms were calculated using Pinch's tables of Carmichael numbers (see link below).

			509033161 is in the sequence since it is a Carmichael number, and 5 of its divisors are also Carmichael numbers (1729, 63973, 126217, 188461 and 294409), more than for any smaller Carmichael number.

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.
Index entries for sequences related to Carmichael numbers.

Cf. A002997.

A002997 =  Cases[Range[1, 100000, 2], n_ /; Mod[n, CarmichaelLambda[n]] == 1 && ! PrimeQ[n]]; carmichaelQ[n_] := Not[PrimeQ[n]] && Divisible[n - 1, CarmichaelLambda[n]]; numSol[n_] := Module[{m = 0}, ds = Divisors[n];   Do[d = ds[[k]]; If[! carmichaelQ[d], Continue[]]; m++, {k, 2, Length[ds] - 1}]; m]; numSolmax = -1; seq = {}; Do[n = A002997[[j]]; m = numSol[n]; If[m > numSolmax, AppendTo[seq, n]; numSolmax = m], {j, 1, Length[A002997]}];seq

a(11) from Amiram Eldar, Jun 29 2019

a(12) calculated using data from Claude Goutier and added by Amiram Eldar, Apr 24 2024

A291616 Carmichael numbers k such that 2^d == 2^(k/d) (mod k) for all d|k.

Original entry on oeis.org

1105, 294409, 852841, 3828001, 17098369, 118901521, 150846961, 172947529, 186393481, 200753281, 686059921, 771043201, 1001152801, 1207252621, 1269295201, 1299963601, 1632785701, 1772267281, 2301745249, 4215885697, 4562359201, 4765950001, 4897161361

Offset: 1

Max Alekseyev, Thomas Ordowski, Altug Alkan, Aug 28 2017

nonn

Intersection of A002997 and A291601.

			Carmichael number 294409 = 37*73*109 is a term because 2^37 == 2^(73*109) (mod 294409), 2^73 == 2^(37*109) (mod 294409), 2^109 == 2^(37*73) (mod 294409).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Claude Goutier; terms 1..3648 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.

Cf. A002997, A291601, A291612.

A300949 Carmichael numbers whose prime factors form an arithmetic progression.

Original entry on oeis.org

1729, 2465, 29341, 294409, 1152271, 1857241, 6189121, 19384289, 56052361, 64377991, 118901521, 172947529, 216821881, 228842209, 625482001, 775368901, 1213619761, 1299963601, 2301745249, 4562359201, 8346731851, 9293756581, 9624742921, 9701285761, 11346205609, 13079177569, 13946829751, 14386156093

Offset: 1

Robert Israel and Thomas Ordowski, Mar 16 2018

nonn

All terms < 10^18 have three prime factors.  There are terms with more, e.g., 97888020200929464481 = 34471 * 91921 * 149371 * 206821, 147681255946700149193521 = 214831 * 572881 * 930931 * 1288981, and 2393527068197020059464161 = 431047 * 1149457 * 1867867 * 2586277.
A term with 3 prime factors is of the form (p-d)p(p+d), where p-d, p and p+d are prime, and p-d-1 | d(2d+3), p-1 | d^2, and p+d-1 | d(2d-3). Thus for each d there are only finitely many possible p that make this work. Note that 6|d, see A262723.
Conjecture: if n is a Carmichael number and lpf(n)gpf(n)(lpf(n)+gpf(n))/2 = n, then (lpf(n)+gpf(n))/2 is prime; and thus n has exactly three prime factors. Such numbers n form a proper subsequence of this sequence, also subsequence of A262723. - Charles R Greathouse IV and Thomas Ordowski, Mar 17 2018. Edited by Max Alekseyev, Mar 17 2018
Proof of the above conjecture: Say n = paq with 2 < p < q being primes and a = (p+q)/2, with (a,p!)=1. If n is a Carmichael number, then pa == 1 (mod q-1), so p^2 + pq == 2 (mod q-1), so p^2 + p == 2 (mod q-1). In particular, p^2 + p - 2 >= q-1, which implies that (p+1)^2 > q. Say a has k prime factors, so that a >= (p+2)^k. But a < q, so q > (p+2)^k. Thus, (p+1)^2 > q > (p+2)^k. This implies k=1. - Carl Pomerance (in a letter to the second author), Mar 18 2018
Note: this does not exclude the existence of the Carmichael numbers m = pq(p+q)/2 with more than three prime factors, where p and q are prime. - Thomas Ordowski, Mar 19 2018

			29341 = 13*37*61 is a Carmichael number, and [13, 37, 61] is an arithmetic progression of length 3 and with common difference of 37 - 13 = 61 - 37 = 24. We have 37 = (13 + 61)/2.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Claude Goutier; terms 1..3841 from Max Alekseyev)
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. A002997, A087788, A262723.

# using Richard Pinch data file carmichael-16
cfile:= "carmichael-16":
Res:= NULL:
do
  S:= readline(cfile);
  if S = 0 then break fi;
  L:= map(parse, StringTools:-Split(S));
  R:= sort(L[2..-1]);
  d:= (R[-1]-R[1])/(nops(R)-1);
  if R = [seq(i,i=R[1]..R[-1],d)] then
    Res:= Res, L[1];
  fi
od:
fclose(cfile):
Res;

Select[Cases[Range[1, 10^7, 2], ?(And[Mod[#, CarmichaelLambda@ #] == 1, ! PrimeQ[#]] &)], Length@ Union@ Differences@ FactorInteger[#][[All, 1]] == 1 &] (* _Michael De Vlieger, Mar 17 2018, after Artur Jasinski at A002997 *)

A303791 Carmichael numbers (A002997) that are central polygonal numbers (A002061).

Original entry on oeis.org

5310721, 2278677961, 9593125081, 29859667201, 467593730289953281, 98538479002618905601, 146842414757227736821

Offset: 1

Max Alekseyev, Apr 30 2018

Also, Carmichael numbers of the form k^2 + k + 1.
Also, Carmichael numbers of the form k^2 - k + 1.
There are no other terms below 10^22.
Carmichael numbers m such that 4m - 3 is square. - Thomas Ordowski, Apr 30 2018

Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.

Intersection of A002997 and A002061.

Cf. A265285, A265628.

a(6)-a(7) calculated using data from Claude Goutier and added by Amiram Eldar, Apr 20 2024

cp's OEIS Frontend

A367320 Carmichael numbers k such that (k-1)/lambda(k) > (m-1)/lambda(m) for all Carmichael numbers m < k, where lambda is the Carmichael lambda function (A002322).

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A258839 Carmichael numbers whose prime factors all have the form p=1+x^2+y^2 for some x,y in Z.

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A267462 Carmichael numbers that are not of the form x^2 + y^2 + z^2 where x, y and z are integers.

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A277366 Composite numbers k such that phi(k)*lambda(k) divides (k-1)^2, where phi(k) = A000010(k) and lambda(k) = A002322(k).

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A277720 Numbers k > 2 such that lambda(k)^2 divides k-1, where lambda(k) = A002322(k).

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A290281 Numbers k such that (k-1) mod phi(k) = lambda(k), where phi = A000010 and lambda = A002322.

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A290497 Carmichael numbers with a record number of aliquot divisors that are also Carmichael numbers.

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A291616 Carmichael numbers k such that 2^d == 2^(k/d) (mod k) for all d|k.

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