A182816 -id:A182816 - cp's OEIS frontend

A002997 Carmichael numbers: composite numbers k such that a^(k-1) == 1 (mod k) for every a coprime to k.

561, 1105, 1729, 2465, 2821, 6601, 8911, 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

Offset: 1

N. J. A. Sloane

V. Šimerka found the first 7 terms of this sequence 25 years before Carmichael (see the link and also the remark of K. Conrad). - Peter Luschny, Apr 01 2019
k is composite and squarefree and for p prime, p|k => p-1|k-1.
An odd composite number k is a pseudoprime to base a iff a^(k-1) == 1 (mod k). A Carmichael number is an odd composite number k which is a pseudoprime to base a for every number a prime to k.
A composite odd number k is a Carmichael number if and only if k is squarefree and p-1 divides k-1 for every prime p dividing k. (Korselt, 1899)
Ghatage and Scott prove using Fermat's little theorem that (a+b)^k == a^k + b^k (mod k) (the freshman's dream) exactly when k is a prime (A000040) or a Carmichael number. - Jonathan Vos Post, Aug 31 2005
Alford et al. have constructed a Carmichael number with 10333229505 prime factors, and have also constructed Carmichael numbers with m prime factors for every m between 3 and 19565220. - Jonathan Vos Post, Apr 01 2012
Thomas Wright proved that for any numbers b and M in N with gcd(b,M) = 1, there are infinitely many Carmichael numbers k such that k == b (mod M). - Jonathan Vos Post, Dec 27 2012
Composite numbers k relatively prime to 1^(k-1) + 2^(k-1) + ... + (k-1)^(k-1). - Thomas Ordowski, Oct 09 2013
Composite numbers k such that A063994(k) = A000010(k). - Thomas Ordowski, Dec 17 2013
Odd composite numbers k such that k divides A002445((k-1)/2). - Robert Israel, Oct 02 2015
If k is a Carmichael number and gcd(b-1,k)=1, then (b^k-1)/(b-1) is a pseudoprime to base b by Steuerwald's theorem; see the reference in A005935. - Thomas Ordowski, Apr 17 2016
Composite numbers k such that p^k == p (mod k) for every prime p <= A285512(k). - Max Alekseyev and Thomas Ordowski, Apr 20 2017
If a composite m < A285549(n) and p^m == p (mod m) for every prime p <= prime(n), then m is a Carmichael number. - Thomas Ordowski, Apr 23 2017
The sequence of all Carmichael numbers can be defined as follows: a(1) = 561, a(n+1) = smallest composite k > a(n) such that p^k == p (mod k) for every prime p <= n+2. - Thomas Ordowski, Apr 24 2017
An integer m > 1 is a Carmichael number if and only if m is squarefree and each of its prime divisors p satisfies both s_p(m) >= p and s_p(m) == 1 (mod p-1), where s_p(m) is the sum of the base-p digits of m. Then m is odd and has at least three prime factors. For each prime factor p, the sharp bound p <= a*sqrt(m) holds with a = sqrt(17/33) = 0.7177.... See Kellner and Sondow 2019. - Bernd C. Kellner and Jonathan Sondow, Mar 03 2019
Carmichael numbers are special polygonal numbers A324973. The rank of the n-th Carmichael number is A324975(n). See Kellner and Sondow 2019. - Jonathan Sondow, Mar 26 2019
An odd composite number m is a Carmichael number iff m divides denominator(Bernoulli(m-1)). The quotient is A324977. See Pomerance, Selfridge, & Wagstaff, p. 1006, and Kellner & Sondow, section on Bernoulli numbers. - Jonathan Sondow, Mar 28 2019
This is setwise difference A324050 \ A008578. Many of the same identities apply also to A324050. - Antti Karttunen, Apr 22 2019
If k is a Carmichael number, then A309132(k) = A326690(k). The proof generalizes that of Theorem in A309132. - Jonathan Sondow, Jul 19 2019
Composite numbers k such that A111076(k)^(k-1) == 1 (mod k). Proof: the multiplicative order of A111076(k) mod k is equal to lambda(k), where lambda(k) = A002322(k), so lambda(k) divides k-1, qed. - Thomas Ordowski, Nov 14 2019
For all positive integers m, m^k - m is divisible by k, for all k > 1, iff k is either a Carmichael number or a prime, as is used in the proof by induction for Fermat's Little Theorem. Also related are A182816 and A121707. - Richard R. Forberg, Jul 18 2020
From Amiram Eldar, Dec 04 2020, Apr 21 2024: (Start)
Ore (1948) called these numbers "Numbers with the Fermat property", or, for short, "F numbers".
Also called "absolute pseudoprimes". According to Erdős (1949) this term was coined by D. H. Lehmer.
Named by Beeger (1950) after the American mathematician Robert Daniel Carmichael (1879 - 1967). (End)
For ending digit 1,3,5,7,9 through the first 10000 terms, we see 80.3, 4.1, 7.4, 3.8 and 4.3% apportionment respectively. Why the bias towards ending digit "1"? - Bill McEachen, Jul 16 2021
It seems that for any m > 1, the remainders of Carmichael numbers modulo m are biased towards 1. The number of terms congruent to 1 modulo 4, 6, 8, ..., 24 among the first 10000 terms: 9827, 9854, 8652, 8034, 9682, 5685, 6798, 7820, 7880, 3378 and 8518. - Jianing Song, Nov 08 2021
Alford, Granville and Pomerance conjectured in their 1994 paper that a statement analogous to Bertrand's Postulate could be applied to Carmichael numbers. This has now been proved by Daniel Larsen, see link below. - David James Sycamore, Jan 17 2023

N. G. W. H. Beeger, On composite numbers n for which a^n == 1 (mod n) for every a prime to n, Scripta Mathematica, Vol. 16 (1950), pp. 133-135.
Albert H. Beiler, Recreations in the Theory of Numbers, Dover Publications, Inc. New York, 1966, Table 18, Page 44.
David M. Burton, Elementary Number Theory, 5th ed., McGraw-Hill, 2002.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 142.
CRC Standard Mathematical Tables and Formulae, 30th ed., 1996, p. 87.
Richard K. Guy, Unsolved Problems in Number Theory, A13.
Øystein Ore, Number Theory and Its History, McGraw-Hill, 1948, Reprinted by Dover Publications, 1988, Chapter 14.
Paul Poulet, Tables des nombres composés vérifiant le théorème du Fermat pour le module 2 jusqu'à 100.000.000, Sphinx (Brussels), 8 (1938), 42-45.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 22, 100-103.
Wacław Sierpiński, A Selection of Problems in the Theory of Numbers. Macmillan, NY, 1964, p. 51.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 145-146.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See entry 561 at p. 157.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000 (from the Pinch web site mentioned below)
W. R. Alford, Jon Grantham, Steven Hayman, and Andrew Shallue, Constructing Carmichael numbers through improved subset-product algorithms, Mathematics of Computation, Vol. 83, No. 286 (2014), pp. 899-915, arXiv preprint, arXiv:1203.6664v1 [math.NT], Mar 29 2012.
W. R. Alford, A. Granville, and C. Pomerance, There are infinitely many Carmichael numbers, Ann. of Math. (2) 139 (1994), no. 3, 703-722.
W. R. Alford, A. Granville, and C. Pomerance (1994). "On the difficulty of finding reliable witnesses". Lecture Notes in Computer Science 877, 1994, pp. 1-16.
François Arnault, Thesis.
François Arnault, Constructing Carmichael numbers which are strong pseudoprimes to several bases, Journal of Symbolic Computation, vol. 20, no 2, Aug. 1995, pp. 151-161.
François Arnault, Rabin-Miller primality test: Composite numbers which pass it, Mathematics of Computation, vol. 64, no 209, 1995, pp. 355-361.
François Arnault, The Rabin-Monier theorem for Lucas pseudoprimes, Mathematics of Computation, vol. 66, no 218, April 1997, pp. 869-881.
Joerg Arndt, Matters Computational (The Fxtbook), p. 786.
Eric Bach and Rex Fernando, Infinitely Many Carmichael Numbers for a Modified Miller-Rabin Prime Test, ISSAC '16: Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation, July 2016, pp. 47-54, arXiv preprint, arXiv:1512.00444 [math.NT], 2015.
Sunghan Bae, Su Hu, and Min Sha, On the Carmichael rings, Carmichael ideals and Carmichael polynomials, arXiv:1809.05432 [math.NT], 2018.
Jonathan Bayless and Paul Kinlaw, Explicit Bounds for the Sum of Reciprocals of Pseudoprimes and Carmichael Numbers, Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.4.
Jens Bernheiden, Carmichael numbers (Text in German).
John D. Brillhart, N. J. A. Sloane, and J. D. Swift, Correspondence, 1972.
Ronald Joseph Burthe, Jr. Upper bounds for least witnesses and generating sets, Acta Arith. 80 (1997), no. 4, 311-326.
C. K. Caldwell, The Prime Glossary, Carmichael number.
R. D. Carmichael, Note on a new number theory function, Bull. Amer. Math. Soc. 16 (1910), 232-238.
Keith Conrad, Carmichael numbers and Korselt's criterion, expository paper (2016), 1-3.
K. A. Draziotis, V. Martidis, and S. Tiganourias, Product Subset Problem: Applications to number theory and cryptography, arXiv:2002.07095 [math.NT], 2020. See also Chapter 5, Analysis, Cryptography and Information Science, World Scientific (2023), p. 108.
Harvey Dubner, Carmichael Numbers of the form (6m+1)(12m+1)(18m+1), Journal of Integer Sequences, Vol. 5 (2002) Article 02.2.1.
James Emery, Number Theory, 2013. [Broken link]
Paul Erdős, On the converse of Fermat's theorem, The American Mathematical Monthly, Vol. 56, No. 9 (1949), pp. 623-624; alternative link.
Jan Feitsma and William Galway, Tables of pseudoprimes and related data.
Pratibha Ghatage and Brian Scott, Exactly When is (a+b)^n == a^n + b^n (mod n)?, College Mathematics Journal, Vol. 36, No. 4 (Sep 2005), p. 322.
Claude Goutier, Compressed text file carm10e22.7z containing all the Carmichael numbers up to 10^22. [Local copy, with permission. This is a very large file.]
Andrew Granville, Papers on Carmichael numbers.
Andrew Granville, Primality testing and Carmichael numbers, Notices Amer. Math. Soc., 39 (No. 7, 1992), 696-700.
Andrew Granville and Carl Pomerance, Two contradictory conjectures concerning Carmichael numbers, Math. Comp. 71 (2002), no. 238, 883-908.
Gerhard Jaeschke, The Carmichael numbers to 10^12, Math. Comp., Vol. 55, No. 191 (1990), pp. 383-389.
Eugene Karolinsky and Dmytro Seliutin, Carmichael numbers for GL(m), arXiv:2001.10315 [math.NT], 2020.
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), #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), #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019.
Paul Kinlaw, The reciprocal sums of pseudoprimes and Carmichael numbers, Mathematics of Computation (2023).
A. Korselt, G. Tarry, I. Franel, and G. Vacca, Problème chinois, L'intermédiaire des mathématiciens 6 (1899), 142-144.
Daniel Larsen, Bertrand's Postulate for Carmichael Numbers, arXiv:2111.06963 [math.NT], 2021.
D. H. Lehmer, Errata for Poulet's table, Math. Comp., 25 (1971), 944-945. 25 944 1971.
Max Lewis and Victor Scharaschkin, k-Lehmer and k-Carmichael Numbers, Integers, 16 (2016), #A80.
Renaud Lifchitz, A generalization of the Korselt's criterion - Nested Carmichael numbers.
Romeo Meštrović, Generalizations of Carmichael numbers I, arXiv:1305.1867v1 [math.NT], May 04 2013.
Romeo Meštrović, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.
Yoshio Mimura, Carmichael Numbers up to 10^12. [broken link, WayBackMachine]
Math Reference Project, Carmichael Numbers.
R. G. E. Pinch, The Carmichael numbers up to 10^18, 2008.
R. G. E. Pinch, Carmichael numbers up to 10^15, 10^16, 10^16 to 10^17, 10^17 to 10^18, 10^19, and 10^21.
Carl Pomerance, J. L. Selfridge, and Samuel S. Wagstaff, Jr., The pseudoprimes to 25*10^9, Math. Comp., Vol. 35, No. 151 (1980), pp. 1003-1026.
Carl Pomerance and N. J. A. Sloane, Correspondence, 1991.
Fred Richman, Primality testing with Fermat's little theorem.
Vladimir Shevelev, The number of permutations with prescribed up-down structure as a function of two variables, INTEGERS, 12 (2012), #A1. - From _N. J. A. Sloane_, Feb 07 2013
Václav Šimerka, Zbytky z arithmetické posloupnosti, (On the remainders of an arithmetic progression), Časopis pro pěstování matematiky a fysiky. 14 (1885), 221-225.
Eric Weisstein's World of Mathematics, Carmichael Number, Knoedel Numbers, and Pseudoprime.
Wikipedia, Carmichael number.
Thomas Wright, Infinitely many Carmichael numbers in arithmetic progressions, Bulletin of the London Mathematical Society, 45 (2013) 943-952, arXiv preprint, arXiv:1212.5850 [math.NT], Dec 2012.
Index entries for sequences related to Carmichael numbers.

Cf. A001567, A002445, A002322, A006931, A024556, A027748, A055553, A064238-A064262, A083737, A087441, A087442, A135717, A141711, A153581, A225498, A285512, A285549, A309132, A324290, A324315, A324316, A324973, A324975, A324977, A326690.

Subsequence of A324050.

a002997 n = a002997_list !! (n-1)
a002997_list = [x | x <- a024556_list,
all (== 0) $ map ((mod (x - 1)) . (subtract 1)) $ a027748_row x]
-- Reinhard Zumkeller, Apr 12 2012

[n: n in [3..53*10^4 by 2] | not IsPrime(n) and n mod CarmichaelLambda(n) eq 1]; // Bruno Berselli, Apr 23 2012

filter:= proc(n)
  local q;
  if isprime(n) then return false fi;
  if 2 &^ (n-1) mod n <> 1 then return false fi;
  if not numtheory:-issqrfree(n) then return false fi;
  for q in numtheory:-factorset(n) do
    if (n-1) mod (q-1) <> 0 then return false fi
  od:
  true;
end proc:
select(filter, [seq(2*k+1,k=1..10^6)]); # Robert Israel, Dec 29 2014
isA002997 := n -> 0 = modp(n-1, numtheory:-lambda(n)) and not isprime(n) and n <> 1:
select(isA002997, [$1..10000]); # Peter Luschny, Jul 21 2019

Cases[Range[1,100000,2], n_ /; Mod[n, CarmichaelLambda[n]] == 1 && ! PrimeQ[n]] (* Artur Jasinski, Apr 05 2008; minor edit from Zak Seidov, Feb 16 2011 *)
Select[Range[1,600001,2],CompositeQ[#]&&Mod[#,CarmichaelLambda[#]]==1&] (* Harvey P. Dale, Jul 08 2023 *)

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 \\ Charles R Greathouse IV, Jun 10 2011

is_A002997(n, F=factor(n)~)={ #F>2 && !foreach(F,f,(n%(f[1]-1)==1 && f[2]==1) || return)} \\ No need to check parity: if efficiency is needed, scan only odd numbers. - M. F. Hasler, Aug 24 2012, edited Mar 24 2022

from itertools import islice
from sympy import nextprime, factorint
def A002997_gen(): # generator of terms
    p, q = 3, 5
    while True:
        for n in range(p+2,q,2):
            f = factorint(n)
            if max(f.values()) == 1 and not any((n-1) % (p-1) for p in f):
                yield n
        p, q = q, nextprime(q)
A002997_list = list(islice(A002997_gen(),20)) # Chai Wah Wu, May 11 2022

def isCarmichael(n):
    if n == 1 or is_even(n) or is_prime(n):
        return False
    factors = factor(n)
    for f in factors:
        if f[1] > 1: return False
        if (n - 1) % (f[0] - 1) != 0:
            return False
    return True
print([n for n in (1..20000) if isCarmichael(n)]) # Peter Luschny, Apr 02 2019

Sum_{n>=1} 1/a(n) is in the interval (0.004706, 27.8724) (Bayless and Kinlaw, 2017). The upper bound was reduced to 0.0058 by Kinlaw (2023). - Amiram Eldar, Oct 26 2020, Feb 24 2024

Links for lists of Carmichael numbers updated by Jan Kristian Haugland, Mar 25 2009 and Danny Rorabaugh, May 05 2017

A247074 a(n) = phi(n)/(Product_{primes p dividing n } gcd(p - 1, n - 1)).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 3, 4, 1, 4, 1, 6, 2, 8, 1, 6, 1, 8, 3, 10, 1, 8, 5, 12, 9, 4, 1, 8, 1, 16, 5, 16, 6, 12, 1, 18, 6, 16, 1, 12, 1, 20, 3, 22, 1, 16, 7, 20, 8, 8, 1, 18, 10, 24, 9, 28, 1, 16, 1, 30, 9, 32, 3, 4, 1, 32, 11, 8, 1, 24, 1, 36, 10, 12, 15, 24, 1, 32, 27, 40, 1, 24, 4, 42, 14, 40, 1, 24, 2, 44, 15, 46

Offset: 1

Eric Chen, Nov 16 2014

a(n) = A000010(n)/A063994(n). - Eric Chen, Nov 29 2014
Does every natural number appear in this sequence? If so, do they appear infinitely many times? - Eric Chen, Nov 26 2014
A063994(n) must be a factor of EulerPhi(n). - Eric Chen, Nov 26 2014
Number n is (Fermat) pseudoprimes (or prime) to one in a(n) of its coprime bases. That is, b^(n-1) = 1 (mod n) for one in a(n) of numbers b coprime to n. - Eric Chen, Nov 26 2014
a(n) = 1 if and only if n is 1, prime (A000040), or Carmichael number (A002997). - Eric Chen, Nov 26 2014
a(A191311(n)) = 2. - Eric Chen, Nov 26 2014
a(p^n) = p^(n-1), where p is a prime. - Eric Chen, Nov 26 2014
a(A209211(n)) = EulerPhi(A209211(n)), because A063994(A209211(n)) = 1. - Eric Chen, Nov 26 2014

			EulerPhi(15) = 8, and that 15 is a Fermat pseudoprime in base 1, 4, 11, and 14, the total is 4 bases, so a(15) = 8/4 = 2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Gérard P. Michon, Pseudoprimes

Cf. A063994, A182816, A002997, A191311, A064234, A000010, A063994, A003557, A209211.

a063994[n_] := Times @@ GCD[n - 1, First /@ FactorInteger@ n - 1]; a063994[1] = 1; a247074[n_] := EulerPhi[n]/a063994[n]; Array[a247074, 150]

a(n)=my(f=factor(n));eulerphi(f)/prod(i=1,#f~,gcd(f[i,1]-1,n-1)) \\ Charles R Greathouse IV, Nov 17 2014

A003557(n) <= a(n) <= n, and a(n) is a multiple of A003557(n). - Charles R Greathouse IV, Nov 17 2014

A334006 Triangle read by rows: T(n,k) = (the number of nonnegative bases m < n such that m^k == m (mod n))/(the number of nonnegative bases m < n such that -m^k == m (mod n)) for nonnegative k < n, n >= 1.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 2, 1, 3, 1, 5, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 7, 1, 3, 1, 3, 1, 1, 4, 1, 5, 1, 5, 1, 5, 1, 9, 1, 3, 1, 3, 1, 7, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 11, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 6, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 13, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 1, 7, 1, 3, 1, 3

Offset: 1

Juri-Stepan Gerasimov, Apr 12 2020

If the sum of proper divisors of q in row q <= q, then q are 1, 2, 3, 4, 5, 8, 16, 17, 32, 64, 128, 256, 257, ...(union of Fermat primes and powers of 2).

			Triangle T(n,k) begins:
  n\k| 0   1  2  3  4   5  6  7  8   9 10 11 12  13 14 15 16
  ---+------------------------------------------------------
   1 | 1;
   2 | 1,  1;
   3 | 1,  3, 1;
   4 | 1,  2, 1, 3;
   5 | 1,  5, 1, 1, 1;
   6 | 1,  3, 1, 3, 1,  3;
   7 | 1,  7, 1, 3, 1,  3, 1;
   8 | 1,  4, 1, 5, 1,  5, 1, 5;
   9 | 1,  9, 1, 3, 1,  3, 1, 7, 1;
  10 | 1,  5, 1, 1, 1,  5, 1, 1, 1,  5;
  11 | 1, 11, 1, 3, 1,  3, 1, 3, 1,  3, 1;
  12 | 1,  6, 1, 9, 1,  9, 1, 9, 1,  9, 1, 9;
  13 | 1, 13, 1, 1, 1,  5, 1, 1, 1,  5, 1, 1, 1;
  14 | 1,  7, 1, 3, 1,  3, 1, 7, 1,  3, 1, 3, 1, 7;
  15 | 1, 15, 1, 3, 1, 15, 1, 3, 1, 15, 1, 3, 1, 15, 1;
  16 | 1,  8, 1, 5, 1,  9, 1, 5, 1,  9, 1, 5, 1,  9, 1, 5;
  17 | 1, 17, 1, 1, 1,  1, 1, 1, 1,  1, 1, 1, 1,  1, 1, 1, 1;
  ...
For (n, k) = (7, 3), there are three nonnegative values of m < n such that m^3 == m (mod 7) (namely 0, 1, and 6) and one nonnegative value of m < n such that -m^3 == m (mod 7) (namely 0), so T(7,3) = 3/1 = 3.

Cf. A000012, A000079, A002322, A010684, A019434, A023506, A026741, A182816, A333570, A337454, A337632, A337633, A337820, A337910.

[[#[m: m in [0..n-1] | m^k mod n eq m]/#[m: m in [0..n-1] | -m^k mod n eq m]: k in [0..n-1]]: n in [1..17]];

T(n, k) = sum(m=0, n-1, Mod(m, n)^k == m)/sum(m=0, n-1, -Mod(m, n)^k == m);
matrix(7, 7, n, k, k--; if (k>=n, 0, T(n,k))) \\ to see the triangle \\ Michel Marcus, Apr 17 2020

Name corrected by Peter Kagey, Sep 12 2020

A371883 a(n) is the number of divisors d of n such that d^n mod n = d.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 1, 4, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 3, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 1, 1, 3, 4, 1, 2, 2, 2, 1, 2, 1, 2, 2, 1, 1, 3, 1, 2, 1, 2, 1, 3, 2, 2, 2, 1, 1, 3, 2, 1, 2, 2, 2, 1, 1, 2, 1, 2

Offset: 1

Juri-Stepan Gerasimov, Apr 10 2024

nonn

1 <= a(n) < A000005(n) for n >= 2.

			a(1) = 0: 1 divides 1, but 1^1 mod 1 = 0 (not 1).
a(2) = 1: 1 divides 2, and 1^2 mod 2 = 1;
          2 divides 2, but 2^2 mod 2 = 0 (not 2).
a(6) = 2: 1 divides 6, and 1^6 mod 6 = 1;
          2 divides 6, but 2^6 mod 6 = 4 (not 2);
          3 divides 6, and 3^6 mod 6 = 3;
          6 divides 6, but 6^6 mod 6 = 0 (not 6).

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000005, A182816, A279024, A371513, A371884.

[#[d: d in Divisors(n) | d^n mod n eq d]: n in [1..100]];

a[n_] := DivisorSum[n, 1 &, PowerMod[#, n, n] == # &]; Array[a, 100] (* Amiram Eldar, Apr 11 2024 *)

a(n) = sumdiv(n, d, d^n % n == d); \\ Michel Marcus, Apr 20 2024

from sympy import divisors
def a(n): return sum(1 for d in divisors(n)[:-1] if pow(d, n, n) == d)
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Apr 10 2024

A333570 Number of nonnegative values c such that c^n == -c (mod n).

Original entry on oeis.org

1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 4, 1, 4, 3, 2, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 8, 1, 8, 1, 2, 1, 4, 3, 4, 1, 4, 3, 4, 1, 8, 1, 4, 1, 4, 1, 4, 1, 4, 3, 8, 1, 4, 3, 4, 1, 4, 1, 8, 1, 4, 1, 2, 1, 24, 1, 4, 1, 16, 1, 4, 1, 4, 3, 8, 1, 8, 1, 4, 1, 4, 1, 8, 5, 4, 3, 4, 1, 8, 7, 4, 1, 4, 3

Offset: 1

Juri-Stepan Gerasimov, Mar 27 2020

nonn

a(n) is the number of nonnegative bases c < n such that c^n + c == 0 (mod n).
a(2^k) = 2 for k > 0.
a(p^m) = 1 for odd prime p with m >= 0.
Let fy(n) = (the number of values b in Z/nZ such that b^y = b)/(the number of values c in Z/nZ such that -c^y = c) for nonnegative y, then:
f0(n) = A000012(n),
f1(n) = A026741(n),
f2(n) = A000012(n),
1 <= f3(n) <= n,
f4(n) = A000012(n), ...,
1 <= fn(n) = A182816(n)/a(n) <= n, where fn(n) = n for odd noncomposite numbers A006005 and Carmichael numbers A002997.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000012, A000079, A000961, A002997, A006005, A026741, A182816.

[#[c: c in [0..n-1] | -c^n mod n eq c]: n in [1..95]];

a(n) = sum(c=1, n, Mod(c, n)^n == -c); \\ Michel Marcus, Mar 27 2020

a(n) = A182816(n)/r for some odd r.

A371513 a(n) is the smallest number m with n divisors d such that d^m mod m = d.

Original entry on oeis.org

1, 2, 6, 42, 30, 105, 910, 561, 1365, 5005, 5565, 11305, 36465, 140505, 239785, 41041, 682465, 873145, 185185, 418285, 1683969, 2113665, 5503785, 1242241, 6697405, 8549905, 31932901, 11996985, 31260405, 30534805, 47031061, 825265, 27265161, 32306365, 55336645, 21662641

Offset: 0

Juri-Stepan Gerasimov, Apr 10 2024

nonn

			a(0) = 1 with divisors {};
a(1) = 2 with divisor {1};
a(2) = 6 with divisors {1, 3};
a(3) = 42 with divisors {1, 7, 21};
a(4) = 30 with divisors {1, 6, 10, 15};
a(5) = 105 with divisors {1, 7, 15, 21, 35};
a(6) = 910 with divisors {1, 35, 65, 91, 130, 455};
a(7) = 561 with divisors {1, 3, 11, 17, 33, 51, 187};
a(8) = 1365 with divisors {1, 13, 21, 91, 105, 195, 273, 455};
a(9) = 5005 with divisors {1, 11, 55, 65, 77, 143, 385, 715, 1001};
a(10) = 5565 with divisors {1, 7, 15, 21, 35, 105, 265, 371, 1113, 1855};
a(11) = 11305 with divisors {1, 17, 19, 35, 85, 119, 323, 595, 665, 1615, 2261}.

Cf. A182816, A272538, A279024, A371883, A371884.

f[n_] := DivisorSum[n, 1 &, PowerMod[#, n, n] == # &]; seq[max_] := Module[{t = Table[0, {max}], c = 0, n = 1, i}, While[c < max, i = f[n] + 1; If[i <= max && t[[i]] == 0, c++; t[[i]] = n]; n++]; t]; seq[18] (* Amiram Eldar, Apr 11 2024 *)

from sympy import divisors
from itertools import count, islice
def f(n, divs): return sum(1 for d in divs if pow(d, n, n) == d%n)
def agen(verbose=False): # generator of terms
    adict, n = dict(), 0
    for k in count(1):
        divs = divisors(k)[1:]
        if len(divs) < n: continue
        v = f(k, divs)
        if v not in adict:
            adict[v] = k
            if verbose: print("FOUND", v, k)
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 15))) # Michael S. Branicky, Apr 10 2024, updated Apr 17 2024 after Jon E. Schoenfield

a(12)-a(25) from Michael S. Branicky, Apr 10 2024

a(26)-a(35) from Jon E. Schoenfield, Apr 10 2024

A371884 Irregular triangle read by rows in which row n >= 2 lists the divisors d of n such that d^n mod n = d.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 5, 1, 1, 4, 1, 1, 7, 1, 5, 1, 1, 1, 9, 1, 1, 5, 1, 7, 1, 11, 1, 1, 1, 1, 13, 1, 1, 4, 1, 1, 6, 10, 15, 1, 1, 1, 11, 1, 17, 1, 1, 9, 1, 1, 19, 1, 13, 1, 1, 1, 7, 21, 1, 1, 1, 9, 1, 23, 1, 1, 16, 1, 1, 25, 1, 17, 1, 13, 1, 1, 27, 1, 11, 1, 8, 1, 19, 1, 29, 1, 1, 1, 1, 31, 1, 1, 1, 5, 13

Offset: 2

Juri-Stepan Gerasimov, Apr 10 2024

			Triangle begins:
    1;
    1;
    1;
    1;
    1, 3;
    1;
    1;
    1;
    1, 5;
    1;
    1, 4;
    1;
    1, 7;
    1, 5;
    1;
    1;
    1, 9;
    1;
    1, 5;
    1, 7;
    1, 11;
    1;
    1;
    1;
    1, 13;
    1;
    1, 4;
    1;
    1, 6, 10, 15;
    ...

Robert Israel, Table of n, a(n) for n = 2..18962 (rows 2 to 10000, flattened)

Cf. A027750, A182816, A279024, A371513, A371883.

[[d: d in Divisors(n) | d^n mod n eq d]: n in [2..65]];

f:= proc(n) op(sort(convert(select(d -> d^n mod n = d, numtheory:-divisors(n)),list))) end proc:
for n from 2 to 100 do f(n) od; # Robert Israel, May 11 2025

row[n_] := Select[Divisors[n], PowerMod[#, n, n] == # &]; Array[row, 64, 2] // Flatten (* Amiram Eldar, Apr 11 2024 *)

A182817 Number of elements k in Z/mZ such that k^m=k, for nonprime m = A018252(n).

Original entry on oeis.org

1, 2, 4, 2, 3, 4, 4, 4, 9, 2, 4, 4, 9, 4, 4, 5, 4, 3, 8, 8, 2, 9, 4, 9, 4, 4, 9, 4, 8, 4, 15, 4, 4, 7, 4, 9, 8, 4, 9, 4, 9, 4, 8, 4, 9, 2, 25, 24, 4, 9, 16, 4, 4, 9, 8, 9, 8, 4, 3, 4, 8, 25, 4, 9, 4, 8, 49, 4, 9, 4, 9, 4, 4, 9, 4, 8, 4, 45, 4, 4, 8, 9, 8, 8, 9, 4, 15, 4, 9, 8, 11, 4, 9, 8, 5, 8, 2, 9, 16, 8, 49, 4, 9, 4, 8, 8, 9, 4, 9, 4, 25, 4, 9, 8, 8, 4, 27, 16, 9, 8, 4, 9, 4, 9, 4

Offset: 1

M. F. Hasler, Dec 05 2010

nonn

This is a reduced version of A182816, without the "trivial" terms a(n)=n for prime indices n.

Amiram Eldar, Table of n, a(n) for n = 1..10000
"Lunkwill", Carmichael numbers and Fermat primality, Dec 02 2010.

Cf. A018252, A182816, A002997.

f[n_] := Times @@ ((1 + GCD[n-1, #-1]) & /@ FactorInteger[n][[;;,1]]); f /@ Select[Range[165], !PrimeQ[#] &] (* Amiram Eldar, Mar 27 2021 *)

a(n) = A182816(A018252(n)).

A337820 Array read by antidiagonals: T(n,k) (n >= 1, k >= 0) is the ratio (the number of nonnegative bases m < n such that m^k == m (mod n))/(the number of nonnegative bases m < n such that -m^k == m (mod n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 5, 1, 3, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 4, 1, 3, 1, 3, 1, 1, 1, 1, 9, 1, 3, 1, 5, 1, 3, 1, 1, 1, 5, 1, 5, 1, 3, 1, 3, 1, 1, 1, 1, 11, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 1, 6, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 1, 13, 1, 3, 1, 3

Offset: 1

Juri-Stepan Gerasimov, Sep 23 2020

Array read by antidiagonals: T(n,k) (n >=1, k >= 0) is part of n of the form (the number of nonnegative bases m < n such that m^k == m (mod n))/(the number of nonnegative bases m < n such that -m^k == m (mod n)).

			The initial rows of the array are:
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 1, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 13, ...
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 1, 3, 3, 1, 3, 3, 5, 3, 1, 3, 9, 1, ...
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 1, 3, 3, 5, 3, 3, 5, 3, 5, 3, 9, 5, ...
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 1, 3, 3, 1, 3, 7, 5, 7, 1, 3, 9, 1, ...
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 1, 3, 3, 5, 3, 3, 5, 3, 5, 3, 9, 5, ...
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 1, 3, 3, 1, 3, 3, 5, 3, 1, 3, 9, 1, ...
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 1, 3, 3, 5, 3, 7, 5, 7, 5, 3, 9, 5, ...
The initial antidiagonals are:
     1,
     1,  1,
     1,  1, 1,
     1,  3, 1, 1,
     1,  2, 1, 1, 1,
     1,  5, 1, 3, 1, 1,
     1,  3, 1, 3, 1, 1, 1,
     1,  7, 1, 1, 1, 3, 1, 1,
     1,  4, 1, 3, 1, 3, 1, 1, 1,
     1,  9, 1, 3, 1, 5, 1, 3, 1, 1,
     1,  5, 1, 5, 1, 3, 1, 3, 1, 1, 1,
     1, 11, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1,
     1,  6, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1,
     1, 13, 1, 3, 1, 3, 1, 7, 1, 5, 1, 3, 1, 1,
...

Columns 0-2: A000012, A026741, A000012.

Cf. A000010, A000012, A000027, A002322, A182816, A333570, A334006, A334597, A336664.

/* As triangle */ [[#[m: m in [0..n-k-1] | m^k mod (n-k) eq m]/
#[m: m in [0..n-k-1] | -m^k mod (n-k) eq m]: k in [0..n-1]]: n in [1..13]];

T(n, 2*k) = 1; 1 <= T(n, 2*k+1) <= n.

A346669 Numbers r such that the number of nonnegative m < r such that m^k == m (mod r) is equal to k*(the number of nonnegative m < r such that -m^k == m (mod r)), where k = 2^A007814(r-1) + 1.

Original entry on oeis.org

3, 5, 7, 11, 13, 15, 17, 19, 23, 27, 29, 31, 35, 37, 39, 41, 43, 47, 51, 53, 55, 57, 59, 61, 67, 71, 73, 75, 79, 83, 85, 87, 89, 91, 95, 97, 101, 103, 107, 109, 111, 113, 115, 119, 123, 125, 127, 131, 135, 137, 139, 143, 149, 151, 155, 157, 159, 163, 167, 173, 175, 179, 181, 183, 187

Offset: 1

Juri-Stepan Gerasimov, Jul 28 2021

nonn

Conjecture: this sequence contains odd prime numbers A065091, but does not contain Carmichael numbers A002997.
Proof from Jianing Song, Jun 14 2021: (Start)
Conjecture: Let v(n) = A007814(n) be the 2-adic valuation of n. Define
  A(n) = A182816(n) as the number of nonnegative m < n such that m^n == m (mod n),
  B(n) = A333570(n) as the number of nonnegative m < n such that m^n == -m (mod n),
  C(n) as the number of nonnegative m < n such that m^(2^v(n-1)+1) == m (mod n), and
  D(n) as the number of nonnegative m < n such that m^(2^v(n-1)+1) == -m (mod n).
Then A(n)/B(n) = n and C(n)/D(n) = 2^v(n-1) + 1 if and only if n is an odd prime.
The conjecture is correct.
"<=": If n is an odd prime, then A(n) = n, B(n) = 1, C(n) = 2^v(n) + 1, D(n) = 1.
"=>": If A(n)/B(n) = n, since A(n) <= n, we must have A(n) = n, so n is either prime or a Carmichael number.
Let n = (p_1)*(p_2)*...*(p_k) be a Carmichael number. Write d = v(n-1), s = (n-1)/2^d be odd.
If m^(2^d+1) == -m (mod n), then m^(2^d) == -1 (mod n/gcd(m,n)). Since m^(s+1) == 1 (mod n), we have (-1)^s == 1 (mod n/gcd(m,n)), so n/gcd(m,n) = 1, m = 0. This shows that D(n) = 1.
For gcd(m,n) = Product_{i in S} p_i, m^(2^v(n-1)+1) == m (mod n) is equivalent to m == 0 (mod Product_{i in S} p_i), m^(2^d) == 1 (mod Product_{i not in S} p_i). The number of solutions modulo n is Product_{i not in S} gcd(2^d, p_i-1). Hence we have C(n) = Sum_{S subset of {1,2,...,k}} Product_{i not in S} gcd(2^d, p_i - 1) = Product_{i=1..k} (1 + gcd(2^d, p_i-1)).
If n is a Carmichael number such that C(n)/D(n) = 2^d + 1, then Product_{i=1..k} (1 + gcd(2^d, p_i-1)) = 2^d + 1. Hence gcd(2^d, p_i-1) < 2^d for 1 <= i <= k. By Zsigmondy's Theorem, unless d = 1 or 3, 2^d + 1 has a primitive prime factor p such that p does not divide 2^e + 1 for all e < d. So we must have d = 1 or 3 (otherwise p does not divide Product_{i=1..k} (1 + gcd(2^d, p_i-1))), then k = 1 or 2. But a Carmichael number must have at least 3 prime factors, a contradiction! (End)
In the case k = r, this sequence contains odd prime and Carmichael numbers, but does not contain any other numbers.

Cf. A002997, A007814, A065091, A182816, A333570, A340281.

[r: r in [2..190] | #[m: m in [0..r-1] | m^k mod r eq m] eq #[m: m in [0..r-1] | -m^k mod r eq m]*k where k is 2^Valuation(r-1, 2) + 1];

cp's OEIS Frontend

A002997 Carmichael numbers: composite numbers k such that a^(k-1) == 1 (mod k) for every a coprime to k.

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A247074 a(n) = phi(n)/(Product_{primes p dividing n } gcd(p - 1, n - 1)).

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A334006 Triangle read by rows: T(n,k) = (the number of nonnegative bases m < n such that m^k == m (mod n))/(the number of nonnegative bases m < n such that -m^k == m (mod n)) for nonnegative k < n, n >= 1.

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A371883 a(n) is the number of divisors d of n such that d^n mod n = d.

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A333570 Number of nonnegative values c such that c^n == -c (mod n).

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A371513 a(n) is the smallest number m with n divisors d such that d^m mod m = d.

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A371884 Irregular triangle read by rows in which row n >= 2 lists the divisors d of n such that d^n mod n = d.

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