A087788 -id:A087788 - cp's OEIS frontend

A112429 Carmichael numbers equal to the product of 6 primes.

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

Offset: 1

Shyam Sunder Gupta, Dec 11 2005

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

R. J. Mathar, Table of n, a(n) for n = 1..14401

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

is(n)={omega(n)==6&&is_A002997(n)} \\ M. F. Hasler, Apr 14 2015

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

Shyam Sunder Gupta, Dec 11 2005

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

is(n)={bigomega(n)==7&&is_A002997(n)} \\ M. F. Hasler, Apr 14 2015

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

Shyam Sunder Gupta, Dec 11 2005

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

is(n)={omega(n)==9&&is_A002997(n)} \\ M. F. Hasler, Apr 14 2015

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

Shyam Sunder Gupta, Dec 11 2005

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

is(n)={bigomega(n)==8&&is_A002997(n)} \\ M. F. Hasler, Apr 14 2015

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

M. F. Hasler, Jul 01 2008

nonn

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

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

ok[n_] :=  Divisible[n - 1, CarmichaelLambda[n]] && Length[FactorInteger[n]] > 3; Select[ Range[3*10^6], ok] (* Jean-François Alcover, Sep 23 2011 *)

A2997=readvec("b002997.gp"); A002997(n)=A2997[n]; for( n=1,100, omega( A002997(n) ) > 3 & print1( A002997(n)", "))

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

A324456 Numbers m > 1 such that there exists a divisor g > 1 of m which satisfies s_g(m) = g.

Original entry on oeis.org

6, 10, 12, 15, 18, 20, 21, 24, 28, 33, 34, 36, 39, 40, 45, 48, 52, 57, 63, 65, 66, 68, 72, 76, 80, 85, 87, 88, 91, 93, 96, 99, 100, 105, 111, 112, 117, 120, 126, 130, 132, 133, 135, 136, 144, 145, 148, 153, 156, 160, 165, 171, 175, 176, 185, 186, 189, 190

Offset: 1

Bernd C. Kellner, Feb 28 2019

The function s_g(m) gives the sum of the base-g digits of m.
The sequence is infinite, since it contains A324460.
The sequence also contains the 3-Carmichael numbers A087788 and the primary Carmichael numbers A324316.
A term m must have at least 2 prime factors, and the divisor g satisfies the inequalities 1 < g < m^(1/(ord_g(m)+1)) <= sqrt(m), where ord_g(m) gives the maximum exponent e such that g^e divides m.
Note that the sequence contains the 3-Carmichael numbers, but not all Carmichael numbers. This is a nontrivial fact.
The subsequence A324460 mainly gives examples in which g is composite.
See Kellner 2019.
It appears that g is usually prime: compare with A324857 (g prime) and the sparser sequence A324858 (g composite). However, g is usually composite for higher values of m. - Jonathan Sondow, Mar 17 2019

			6 is a member, since 2 divides 6 and s_2(6) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..806 from Bernd C. Kellner)
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), Article #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), Article #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019.

Subsequences are A033502, A087788, A324316, A324458, A324460.
Subsequence of A324455.
Cf. A324315, A324457, A324459.
Union of A324857 and A324858.

s[n_, g_] := If[n < 1 || g < 2, 0, Plus @@ IntegerDigits[n, g]];
f[n_] := AnyTrue[Divisors[n], s[n, #] == # &];
Select[Range[5000], f[#] &]

isok(n) = {fordiv(n, d, if ((d>1) && (sumdigits(n, d) == d), return (1)););} \\ Michel Marcus, Mar 19 2019

A141702 a(n) is the number of Carmichael numbers of the form prime(n)prime(n')prime(n") with n > n' > n".

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 2, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 3, 1, 2, 0, 0, 0, 0, 1

Offset: 1

M. F. Hasler, Jun 30 2008

The formula and PARI code uses Korselt's criterion. This sequence is a somewhat trivial variant of the more interesting sequence giving the number of Carmichael numbers of the form prime(n)*prime(n')*prime(n") with n < n' < n" (known to be finite for given n).

			a(7)=2 is the first nonzero term since 561 = 3*11*17 and 1105 = 5*13*17 are the two smallest Carmichael numbers and there's no other Carmichael number having prime(7)=17 as largest factor.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002997 and references therein ; A087788 ; A141703 ff.

A141702(n) = { local( p=prime(n), c=0 ); forprime( q=5,p-2, forprime( r=3,q-2, (p*q*r-1)%(p-1)==0 && (p*q*r-1)%(q-1)==0 && (p*q*r-1)%(r-1)==0 && c++ ));c }

a(n) = # { pqr | p=prime(n) > q=prime(n') > r=prime(n") ; p-1 | pqr-1 ; q-1 | pqr-1 ; r-1 | pqr-1 }

A300629 a(1) = 561; a(n+1) = smallest Fermat pseudoprime to all natural bases up to lpf(a(n)).

Original entry on oeis.org

561, 1105, 1729, 29341, 162401, 252601, 1152271, 2508013, 3828001, 6733693, 17098369, 17236801, 29111881, 82929001, 172947529, 216821881, 228842209, 366652201, 413138881, 2301745249, 2438403661, 5255104513, 5781222721, 8251854001, 12173703001, 13946829751, 15906120889, 23224518901, 31876135201, 51436355851, 57274147841, 58094662081

Offset: 1

Thomas Ordowski, Mar 10 2018

nonn

It is sufficient to consider only prime bases: a(n+1) is the least composite number k such that p^(k-1) == 1 (mod k) for every prime p <= lpf(a(n)), with a(1) = 561.
Conjecture: a(n+1) is the smallest Carmichael number k such that lpf(k) > lpf(a(n)), with a(1) = 561. It seems that such Carmichael numbers have exactly three prime factors.
The above conjecture is true if A083876(n) < A285549(n) for all n > 1, but has not been proven; there is no counterexample up to a(n) < 2^64. - Max Alekseyev and Thomas Ordowski, Mar 13 2018
Carl Pomerance (in a letter to the author) wrote, Mar 13 2018: (Start)
Assuming a strong form of the prime k-tuples conjecture, if there are no small counterexamples, there are likely to be none.
Here's why.
Assuming prime k-tuples, there are infinitely many Carmichael numbers of the form (6k+1)(12k+1)(18k+1), where each factor is prime. And from Bateman-Horn, these are fairly thickly distributed. There are other even better triples such as (60k+41)(90k+61)(150k+101), where "better" means the least prime factor is not so far below the cube root.
So, to get into the sequence, a number needs to be a Fermat pseudoprime for every base up to nearly the cube root.
However, it's a theorem that a sufficiently large number with this property must be a Carmichael number. (End)
Theorem: if lpf(a(n)) < m < a(n), then m is prime if and only if p^(m-1) == 1 (mod m) for every prime p <= lpf(a(n)). - Thomas Ordowski, Mar 13 2018
lpf(a(n)) are listed in A300748. - Max Alekseyev, Mar 13 2018
For m > 1, A135720(m) >= A083876(m-1), with equality iff lpf(a(n)) = prime(m); by this conjecture in the second comment. - Thomas Ordowski, Mar 13 2018

Max Alekseyev, Table of n, a(n) for n = 1..138

Cf. A002997, A020639, A083876, A271221, A285549, A300748.

Subsequence of A087788 and of A135720.

A328935 Imprimitive Carmichael numbers: Carmichael numbers m such that if m = p_1 * p_2 * ... *p_k is the prime factorization of m then g(m) = gcd(p_1 - 1, ..., p_k - 1) > sqrt(lambda(m)), where lambda is the Carmichael lambda function (A002322).

Original entry on oeis.org

294409, 399001, 488881, 512461, 1152271, 1461241, 3057601, 3828001, 4335241, 6189121, 6733693, 10267951, 14676481, 17098369, 19384289, 23382529, 50201089, 53711113, 56052361, 64377991, 68154001, 79624621, 82929001, 84350561, 96895441, 115039081, 118901521, 133800661

Offset: 1

Amiram Eldar, Oct 31 2019

nonn

Granville and Pomerance separated the Carmichael numbers into two classes, primitive and imprimitive, according to whether g(m) <= sqrt(lambda(n)) or not.
They conjectured that most Carmichael numbers are primitive and most 3-Carmichael numbers (A087788) are imprimitive.
Comment from Jeppe Stig Nielsen, Apr 21 2021: (Start)
In cases n = 1, 3, 5, 7, 8, 10, 14, 15, 19, 20, ..., there exists a primitive Carmichael number in the same "family" (Carmichael numbers that share the ratio (p_1-1):(p_2-1):...:(p_k-1) belong to the same family). However, in the remaining cases, the entire family consists of imprimitive Carmichael numbers.
There can be more than one primitive Carmichael number in a family. For example, both Carmichael numbers 5828853661 and 965507554621 are primitive, and are in the family 1:3:6:70. The first imprimitive Carmichael number in the family 1:3:6:70 is a(1639)=59610715093021. (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000
Andrew Granville and Carl Pomerance, Two contradictory conjectures concerning Carmichael numbers, Mathematics of Computation, Vol. 71, No. 238 (2002), pp. 883-908.

Cf. A002322, A002997, A087788, A258409, A335584.

aQ[n_] := Length[(f = FactorInteger[n])] > 2 && Max[f[[;; , 2]]] == 1 && Divisible[n-1, (lambda = LCM @@ (f[[;; , 1]] - 1))] && GCD @@ (f[[;; , 1]] - 1) > Sqrt[lambda]; Select[Range[4*10^6], aQ]

isA328935(m)=f=factor(m);!(issquarefree(f)&&omega(f)>2)&&return(0);p=f[,1]~;r=apply(x->x-1,p);foreach(r,x,(m-1)%x!=0&&return(0));g=gcd(r);a=r/g;g>lcm(a) \\ p, g, and a are like in Granville & Pomerance, Jeppe Stig Nielsen, Apr 21 2021

Terms m of A002997 such that A258409(m) > sqrt(A002322(m)).

A036060 Number of 3-component Carmichael numbers C = (6M + 1)(12M + 1)(18M + 1) < 10^n.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 7, 10, 16, 25, 50, 86, 150, 256, 436, 783, 1435, 2631, 4765, 8766, 16320, 30601, 57719, 109504, 208822, 400643, 771735, 1494772, 2903761, 5658670, 11059937, 21696205, 42670184, 84144873, 66369603, 329733896, 655014986, 1303918824, 2601139051

Offset: 3

N. J. A. Sloane

nonn

Note that this is different from the count of 3-Carmichael numbers, A132195. The numbers counted here are neither those listed in A087788 (3 arbitrary prime factors) nor those listed in A033502 (where 6m + 1, 12m + 1 and 18m + 1 are all prime). - M. F. Hasler, Apr 14 2015

Posting by Harvey Dubner (harvey(AT)dubner.com) to Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU), Nov 23 1998.

Amiram Eldar, Table of n, a(n) for n = 3..42
H. Dubner, 3-Component Carmichael Numbers-correction, Post to Number Theory List, Nov 23 1998.
Harvey Dubner, Carmichael Numbers of the form (6m+1)(12m+1)(18m+1), Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.1.
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A033502, A055553, A087788, A132195.

Terms updated (from Dubner's paper) by Amiram Eldar, Aug 11 2017

cp's OEIS Frontend

A112429 Carmichael numbers equal to the product of 6 primes.

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A112430 Carmichael numbers equal to the product of 7 primes.

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A112432 Carmichael numbers equal to the product of 9 primes.

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A112431 Carmichael numbers equal to the product of 8 primes.

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A141711 Carmichael numbers with more than 3 prime factors.

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A324456 Numbers m > 1 such that there exists a divisor g > 1 of m which satisfies s_g(m) = g.

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A141702 a(n) is the number of Carmichael numbers of the form prime(n)*prime(n')*prime(n") with n > n' > n".

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A300629 a(1) = 561; a(n+1) = smallest Fermat pseudoprime to all natural bases up to lpf(a(n)).

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A328935 Imprimitive Carmichael numbers: Carmichael numbers m such that if m = p_1 * p_2 * ... *p_k is the prime factorization of m then g(m) = gcd(p_1 - 1, ..., p_k - 1) > sqrt(lambda(m)), where lambda is the Carmichael lambda function (A002322).

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A141702 a(n) is the number of Carmichael numbers of the form prime(n)prime(n')prime(n") with n > n' > n".