A055553 -id:A055553 - cp's OEIS frontend

A135719 a(n) is the index of the smallest Carmichael number (A002997) with n prime divisors, or 0 if no such number exists.

1, 11, 40, 403, 1224, 4886, 19096, 120137, 485941, 2974628, 25293838

Offset: 3

Artur Jasinski, Nov 25 2007

Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Richard Pinch, Tables relating to Carmichael numbers (All the Carmichael numbers up to 10^18).

A002997(a(n)) = A006931(n). - M. F. Hasler, Apr 14 2015

a(8)-a(11) from Donovan Johnson, Feb 23 2012
a(12) from Amiram Eldar, Jul 08 2019
Escape clause added by Jianing Song, Dec 12 2021
a(13) calculated using data from Claude Goutier and added by Amiram Eldar, Apr 20 2024

A175531 Carmichael numbers of order 2.

Original entry on oeis.org

443372888629441, 39671149333495681, 842526563598720001, 2380296518909971201, 3188618003602886401, 4208895375600667752001

Offset: 1

Max Alekseyev, Jun 08 2010

Odd composite integer k is in this sequence if k == 1 or p (mod p^2 - 1) for every prime p|k.

Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Everett W. Howe, Higher-order Carmichael numbers, Math. Comp. 69 (2000), 1711-1719. doi:10.1090/S0025-5718-00-01225-4

Subsequence of A002997, A175530, and A299799.

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

A290793 Carmichael numbers k such that Euler totient function of k (phi(k)) is a cube.

Original entry on oeis.org

63973, 18162001, 26921089, 133205761, 225745345, 490503601, 496050841, 698548201, 1031750401, 1100674561, 1384157161, 2178944461, 3805181281, 11351100241, 12648201841, 26498875681, 26542598401, 28553256865, 28645206001, 37590868801, 39866123377, 40527674881

Offset: 1

Amiram Eldar, Aug 10 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.

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

			phi(63973) = 36^3.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Claude Goutier)
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.
Index entries for sequences related to Carmichael numbers.

Intersection of A002997 (Carmichael numbers) and A039771.

Cf. A000010, A000578, A272798.

With[{s = Import["b002997.txt", "Data"][[All, -1]]}, Select[s, IntegerQ@ Power[EulerPhi@ #, 1/3] &]] (* Michael De Vlieger, Aug 14 2017, using b-file at A002997 *)

A290945 Triangular Carmichael numbers.

Original entry on oeis.org

561, 8911, 10585, 41041, 115921, 314821, 334153, 6313681, 8134561, 14913991, 32914441, 60957361, 67902031, 135556345, 289766701, 321197185, 329769721, 368113411, 471905281, 765245881, 842202361, 962442001, 1507746241, 2489462641, 2588653081, 3104207821

Offset: 1

Amiram Eldar, Aug 14 2017

nonn

Intersection of A000217 and A002997.
The least triangular Carmichael numbers with the number of prime factors = 3, 4, 5, 6, 7, ... are: 561, 41041, 765245881, 321197185, 1583892181303201, ...
The number of terms below 10^k for k = 3, 4, ... are: 1, 2, 4, 7, 9, 13, 22, 32, 53, 77, 137, 211, 358, 545, 879, 1423, ...
Jonathan Vos Post discovered in 2004 that a(21) = 842202361 = A000217(41041) = A002817(286) is also a doubly triangular Carmichael number. The next number with this property is a(1108) = 292800629576356021 = A000217(765245881) = A002817(39121) (41041 and 765245881 are triangular Carmichael numbers that are also indices of triangular numbers that are also Carmichael numbers).

			8911 = A000217(133) = A002997(7) therefore 8911 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..8920 (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.
Jonathan Vos Post, Table of Figurate Numbers, Sorted, Through 10,000. [Wayback Machine link]
Index entries for sequences related to Carmichael numbers.

Cf. A000217, A002817, A002997.

seqQ[n_]:=IntegerQ[Sqrt[8n+1]] && !PrimeQ[n] && (Mod[n, CarmichaelLambda[n]] == 1); Select[Range[10^6], seqQ]

A291612 Carmichael numbers k that satisfy 2^d == 2^(k/d) (mod k) for all d|k and are not Super-Poulet numbers (A050217).

Original entry on oeis.org

1105, 852841, 3828001, 17098369, 118901521, 150846961, 172947529, 186393481, 200753281, 686059921, 771043201, 1001152801, 1207252621, 1269295201, 1632785701, 1772267281, 2301745249, 4765950001, 4897161361, 5278692481, 6030849889, 8251854001, 12121569601, 12456671569

Offset: 1

Max Alekseyev, Thomas Ordowski, Altug Alkan, Aug 27 2017

nonn

Intersection of A002997 and A291602.

			Carmichael number 1105 = 5*13*17 is a term because 2^5 == 2^(13*17) (mod 1105), 2^13 == 2^(5*17) (mod 1105), 2^17 == 2^(5*13) (mod 1105) and it is not a Super-Poulet number.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Claude Goutier; terms 1..2665 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, A050217, A291602.

A329417 Carmichael numbers m that have at least 3 prime factors p such that (p-1)*p^2 divides m-p.

Original entry on oeis.org

12876480001, 102293818705, 162303632569, 639554081761, 783962120161, 3224063844001, 4553777859841, 10276904735461, 40867660260505, 51496980091921, 51641004415105, 52412615611201, 52933062609505, 73892907966241, 97388953462801, 107862864807061, 182236335107905, 210587050134721

Offset: 1

Amiram Eldar and Daniel Suteu, Nov 29 2019

nonn

In 1950, Giuga conjectured that there are no composite numbers n for which 1^(n-1) + 2^(n-1) + ... + (n-1)^(n-1) == -1 (mod n). If such a number exists, then it must be a Carmichael number n such that (p-1)*p^2 divides n-p for every prime p dividing n.

			m = 12876480001 is a term because it is a Carmichael number, and it has at least 3 prime factors p, {7, 11, 37}, such that (p-1)*p^2 divides m-p.

Giuseppe Giuga, Su una presumibile proprietà caratteristica dei numeri primi (in Italian), Istituto Lombardo Scienze e Lettere, Rendiconti di Classe di scienze matematiche e naturali, Vol. 83 (1950), pp. 511-528.

Amiram Eldar, Table of n, a(n) for n = 1..8950 (terms below 10^22 calculated using data from Claude Goutier; terms 1..854 from Daniel Suteu)
Takashi Agoh, On Giuga's conjecture, Manuscripta Mathematica, Vol. 87, No. 1 (1995), pp. 501-510.
William D. Banks, C. Wesley Nevans and Carl Pomerance, A remark on Giuga's conjecture and Lehmer's totient problem, Albanian Journal of Mathematics, Vol. 3, No. 2 (2009), pp. 81-85; alternative link.
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Eric Weisstein's World of Mathematics, Giuga's Conjecture.
Wikipedia, Agoh-Giuga conjecture.
Index entries for sequences related to Carmichael numbers.

Cf. A002997.

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

A202562 Carmichael numbers whose prime factors do not divide any smaller Carmichael number.

Original entry on oeis.org

561, 84350561, 851703301, 2436691321, 34138047673, 60246018673, 63280622521, 83946864769, 110296864801, 114919915021, 155999871721, 225593397919, 342267565249, 534919693681, 660950414671, 733547013841, 1079942171239, 1301203515361, 1333189866793

Offset: 1

Arkadiusz Wesolowski, Dec 21 2011

nonn

Note that all terms so far have only three prime factors.

			a(2) = 84350561 because 84350561 = 107*743*1061 and the smaller Carmichael numbers do not have the factors 107, 743 or 1061.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Claude Goutier; terms 1..1377, terms below 10^18, from Donovan Johnson)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Eric Weisstein's World of Mathematics, Carmichael Number.
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A087788.

A207080 The smallest Carmichael number k such that phi(k) does not divide (k-1)^n, where phi is the Euler totient function.

Original entry on oeis.org

561, 2821, 838201, 41471521, 45496270561, 776388344641, 344361421401361, 375097930710820681, 330019822807208371201, 4971170854788923506051

Offset: 1

José María Grau Ribas, Feb 15 2012

Conjecture: phi(a(n)) divides (a(n)-1)^(n+1).

a(10) <= 9645020063586019926451. - Daniel Suteu, Dec 25 2020

Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
José María Grau and Antonio M. Oller-Marcén, On k-Lehmer numbers, Integers, 12 (2012), #A37; alternative link; arXiv preprint, arXiv:1012.2337 [math.NT], 2010-2012.
Nathan McNew, Radically weakening the Lehmer and Carmichael conditions, International Journal of Number Theory, Vol. 9, No. 5 (2013), pp. 1215-1224; arXiv preprint, arXiv:1210.2001 [math.NT], 2012.

Cf. A000010, A002997 (Carmichael numbers), A173703.

is_c(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(k, n) = ((k-1)^n % eulerphi(k)) != 0;
a(n) = my(k=1); while (!(is_c(k) && isok(k,n)), k++); k; \\ Michel Marcus, Dec 25 2020

a(7)-a(9) from Richard Pinch, Feb 18 2012

a(10) calculated using data from Claude Goutier and added by Amiram Eldar, Apr 20 2024

A225005 Number of Carmichael numbers (A002997) less than 2^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 5, 6, 9, 10, 15, 19, 23, 33, 45, 55, 69, 95, 130, 162, 214, 290, 375, 483, 656, 864, 1118, 1446, 1874, 2437, 3130, 4058, 5188, 6642, 8521, 11002, 14236, 18400, 23631, 30521, 39376, 50685, 65590, 84817, 109857, 141892, 183507, 237217, 307278, 398506, 517446, 672105, 873109, 1136472, 1479525, 1927138, 2513234, 3278553, 4279356

Offset: 1

Max Alekseyev, Apr 23 2013

nonn

Amiram Eldar, Table of n, a(n) for n = 1..73 (terms 65..69 added using A182490; terms 70..73 calculated using data from Claude Goutier)
Jan Feitsma, The pseudoprimes below 2^64 - statistics. [Wayback Machine link]
Jan Feitsma and William Galway, Tables of pseudoprimes and related data.
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.

Cf. A002997, A055553, A208276, A108797.

Partial sums of A182490.

A253595 Least Carmichael number that is divisible by the n-th cyclic number A003277(n), or 0 if no such number exists.

Original entry on oeis.org

561, 1105, 1729, 561, 1105, 62745, 561, 1729, 6601, 2465, 2821, 561, 825265, 29341, 6601, 334153, 62745, 561, 2433601, 74165065, 29341, 1105, 8911, 116150434401, 10024561, 10585, 41041, 2508013, 55462177, 1105, 11921001

Offset: 3

Tim Johannes Ohrtmann, Jan 05 2015

nonn

Has any odd cyclic number at least one Carmichael multiple?

			a(8) = 62745 because this is the least Carmichael number which is divisible by 15 (the 8th cyclic number).

Amiram Eldar, Table of n, a(n) for n = 3..2747 (calculated using data from Claude Goutier; terms 3..291 from Tim Johannes Ohrtmann, terms 292..853 from Max Alekseyev)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Gérard P. Michon, Carmichael Multiples of Odd Cyclic Numbers.
Open Problem Garden, Michon's conjecture.
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A003277, A135721.

Korselt(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]>1||(n-1)%(f[i, 1]-1), return(0))); 1
isA002997(n)=n%2 && !isprime(n) && Korselt(n) && n>1
a(n) = {on = odd cyclic number(n); cn = 1; until (isA002997(cn) && (cn % on == 0), cn++); cn; }

a(292)-a(853) from Max Alekseyev, Apr 26 2015

Escape clause added by Jianing Song, Dec 12 2021

cp's OEIS Frontend

A135719 a(n) is the index of the smallest Carmichael number (A002997) with n prime divisors, or 0 if no such number exists.

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A175531 Carmichael numbers of order 2.

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A290793 Carmichael numbers k such that Euler totient function of k (phi(k)) is a cube.

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Mathematica

A290945 Triangular Carmichael numbers.

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Mathematica

A291612 Carmichael numbers k that satisfy 2^d == 2^(k/d) (mod k) for all d|k and are not Super-Poulet numbers (A050217).

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A329417 Carmichael numbers m that have at least 3 prime factors p such that (p-1)*p^2 divides m-p.

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Perl

A202562 Carmichael numbers whose prime factors do not divide any smaller Carmichael number.

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A207080 The smallest Carmichael number k such that phi(k) does not divide (k-1)^n, where phi is the Euler totient function.

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A225005 Number of Carmichael numbers (A002997) less than 2^n.

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