A063994 -id:A063994 - cp's OEIS frontend

A064234 The least k such that A063994(k) = Product_{primes p dividing k} gcd(p-1, k-1) = n, or 0 if there's no such k.

1, 3, 28, 5, 66, 7, 232, 45, 190, 11, 276, 13, 1106, 0, 286, 17, 1854, 19, 3820, 891, 2752, 23, 1128, 595, 2046, 0, 532, 29, 1770, 31, 9952, 425, 1288, 0, 2486, 37, 8474, 0, 742, 41, 3486, 43, 7612, 5589, 2356, 47, 13584, 325, 9850, 0, 20554, 53, 5778, 0, 12926, 435, 13282, 59, 42540, 61

Offset: 1

Robert G. Wilson v, Sep 22 2001

nonn

From Richard N. Smith, Jul 15 2019: (Start)
The comment in the "Mathematica" section is not true: A063994(65513) = 76 (thus a(76) = 65513 instead of 0), but 76 is an even nontotient (in the sequence A005277).
The first counterexample of the comment is A063994(1541) = 484, which is an even nontotient, for the counterexamples <= 2^20, see the link.
Also A063994(1072871) = 68. (thus a(68) = 1072871).
Conjecture: a(n) = 0 iff n == 2 mod 4 and n+1 is composite, if this conjecture is true, then a(54), a(110), a(294), etc. would be 0.
Another conjecture: If A063994(k) = n and n == 2 mod 4, then n+1 is prime and k is a power of n+1. (End)

Richard N. Smith, Table of n, a(n) for n = 1..187
Richard N. Smith, Numbers n such that A063994(n) is an even nontotient (format: n,A063994(n))

Cf. A063994, A005277, A002202, A049225, A254139.

f[ n_ ] := If[ n == 1, 1, Apply[ Times, GCD[ n - 1, Transpose[ FactorInteger[ n ] ] [ [ 1 ] ] - 1 ] ] ]; a = Table[ 0, {100} ]; Do[ m = f[ n ]; If[ m < 101 && a[ [ m ] ] == 0, a[ [ m ] ] = n ], {n, 1, 10^7} ]; a a(54) > 2*10^7. The zeros appear at positions that are the values in the sequence A005277, the nontotients: even n such that phi(m) = n has no solution. [Warning: This is wrong, see the "comment" section]

a063994(n)=my(f=factor(n)[, 1]); prod(i=1, #f, gcd(f[i]-1, n-1))
a(n)=if(n%4==2 && !isprime(n+1), 0, for(k=1, 2^30, if(a063994(k)==n,return(k)))) \\ Richard N. Smith, Jul 15 2019, after Charles R Greathouse IV in A063994

a(54) - a(60) from Richard N. Smith, Jul 15 2019

A330756 Number of values of k, 1 <= k <= n, with A063994(k) = A063994(n), where A063994(n) = Product_{primes p dividing n} gcd(p-1, n-1).

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 1, 5, 2, 6, 1, 7, 1, 8, 2, 9, 1, 10, 1, 11, 3, 12, 1, 13, 4, 14, 3, 1, 1, 15, 1, 16, 5, 17, 6, 18, 1, 19, 7, 20, 1, 21, 1, 22, 1, 23, 1, 24, 2, 25, 8, 2, 1, 26, 9, 27, 10, 28, 1, 29, 1, 30, 11, 31, 2, 1, 1, 32, 12, 3, 1, 33, 1, 34, 13, 4, 14, 35, 1, 36, 4, 37, 1, 38, 3, 39, 15, 40, 1, 41, 2, 42, 16

Offset: 1

Antti Karttunen, Dec 30 2019

nonn

Ordinal transform of A063994.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A063994, A209211.

Cf. also A081373, A303756, A330747.

A063994[n_] := If[n==1, 1, Times @@ GCD[n-1, First /@ FactorInteger[n]-1]];
Module[{b}, b[_] = 0;
a[n_] := With[{t = A063994[n]}, b[t] = b[t]+1]];
Array[a, 105] (* Jean-François Alcover, Jan 12 2022 *)

up_to = 65537;
A063994(n) = { my(f=factor(n)[, 1]); prod(i=1, #f, gcd(f[i]-1, n-1)); }; \\ From A063994
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
v330756 = ordinal_transform(vector(up_to, n, A063994(n)));
A330756(n) = v330756[n];

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

Original entry on oeis.org

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

A049559 a(n) = gcd(n - 1, phi(n)).

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 6, 1, 2, 1, 10, 1, 12, 1, 2, 1, 16, 1, 18, 1, 4, 1, 22, 1, 4, 1, 2, 3, 28, 1, 30, 1, 4, 1, 2, 1, 36, 1, 2, 1, 40, 1, 42, 1, 4, 1, 46, 1, 6, 1, 2, 3, 52, 1, 2, 1, 4, 1, 58, 1, 60, 1, 2, 1, 16, 5, 66, 1, 4, 3, 70, 1, 72, 1, 2, 3, 4, 1, 78, 1, 2, 1, 82, 1, 4, 1, 2, 1, 88, 1, 18, 1, 4

Offset: 1

Labos Elemer, Dec 28 2000

nonn

For prime n, a(n) = n - 1. Question: do nonprimes exist with this property?
Answer: No. If n is composite then a(n) < n - 1. - Charles R Greathouse IV, Dec 09 2013
Lehmer's totient problem (1932): are there composite numbers n such that a(n) = phi(n)? - Thomas Ordowski, Nov 08 2015
a(n) = 1 for n in A209211. - Robert Israel, Nov 09 2015

			a(9) = 2 because phi(9) = 6 and gcd(8, 6) = 2.
a(10) = 1 because phi(10) = 4 and gcd(9, 4) = 1.

Richard K. Guy, Unsolved Problems in Number Theory, B37.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Lehmer's Totient Problem

Cf. A000010, A002322, A039766, A063994, A160595, A209211, A219428, A264012, A264024, A280262, A283656, A283872, A284089, A284440, A318827, A318829, A318830, A330747 (ordinal transform), A340195.

Cf. also A009195, A058515, A058663, A187730, A258409, A339964, A340071, A340081, A340087 for more or less analogous sequences.

[Gcd(n-1, EulerPhi(n)): n in [1..80]]; // Vincenzo Librandi, Oct 13 2018

seq(igcd(n-1, numtheory:-phi(n)), n=1..100); # Robert Israel, Nov 09 2015

Table[GCD[n - 1, EulerPhi[n]], {n, 93}] (* Michael De Vlieger, Nov 09 2015 *)

a(n)=gcd(eulerphi(n),n-1) \\ Charles R Greathouse IV, Dec 09 2013

from sympy import totient, gcd
print([gcd(totient(n), n - 1) for n in range(1, 101)]) # Indranil Ghosh, Mar 27 2017

a(p^m) = a(p) = p - 1 for prime p and m > 0. - Thomas Ordowski, Dec 10 2013
From Antti Karttunen, Sep 09 2018: (Start)
a(n) = A000010(n) / A160595(n) = A063994(n) / A318829(n).
a(n) = n - A318827(n) = A000010(n) - A318830(n).
(End)
a(n) = gcd(A000010(n), A219428(n)) = gcd(A000010(n), A318830(n)). - Antti Karttunen, Jan 05 2021

A258409 Greatest common divisor of all (d-1)'s, where the d's are the positive divisors of n.

Original entry on oeis.org

1, 2, 1, 4, 1, 6, 1, 2, 1, 10, 1, 12, 1, 2, 1, 16, 1, 18, 1, 2, 1, 22, 1, 4, 1, 2, 1, 28, 1, 30, 1, 2, 1, 2, 1, 36, 1, 2, 1, 40, 1, 42, 1, 2, 1, 46, 1, 6, 1, 2, 1, 52, 1, 2, 1, 2, 1, 58, 1, 60, 1, 2, 1, 4, 1, 66, 1, 2, 1, 70, 1, 72, 1, 2, 1, 2, 1, 78, 1

Offset: 2

Ivan Neretin, May 29 2015

nonn

a(n) = 1 for even n; a(p) = p-1 for prime p.
a(n) is even for odd n (since all divisors of n are odd).
It appears that a(n) = A052409(A005179(n)), i.e., it is the largest integer power of the smallest number with exactly n divisors. - Michel Marcus, Nov 10 2015
Conjecture: GCD of all (p-1) for prime p|n. - Thomas Ordowski, Sep 14 2016
Conjecture is true, because the set of numbers == 1 (mod g) is closed under multiplication. - Robert Israel, Sep 14 2016
Conjecture: a(n) = A289508(A328023(n)) = GCD of the differences between consecutive divisors of n. See A328163 and A328164. - Gus Wiseman, Oct 16 2019

			65 has divisors 1, 5, 13, and 65, hence a(65) = gcd(1-1,5-1,13-1,65-1) = gcd(0,4,12,64) = 4.

Ivan Neretin, Table of n, a(n) for n = 2..10000

Cf. A049559, A057237, A060680, A063994, A187730, A027750.
Cf. A084190 (similar but with LCM).
Looking at prime indices instead of divisors gives A328167.
Partitions whose parts minus 1 are relatively prime are A328170.
Cf. A000005, A060681, A060683, A193829, A289508.

a258409 n = foldl1 gcd $ map (subtract 1) $ tail $ a027750_row' n
-- Reinhard Zumkeller, Jun 25 2015

f:= n -> igcd(op(map(`-`,numtheory:-factorset(n),-1))):
map(f, [$2..100]); # Robert Israel, Sep 14 2016

Table[GCD @@ (Divisors[n] - 1), {n, 2, 100}]

a(n) = my(g=0); fordiv(n, d, g = gcd(g, d-1)); g; \\ Michel Marcus, May 29 2015

a(n) = gcd(apply(x->x-1, divisors(n))); \\ Michel Marcus, Nov 10 2015

a(n)=if(n%2==0, return(1)); if(n%3==0, return(2)); if(n%5==0 && n%4 != 1, return(2)); gcd(apply(p->p-1, factor(n)[,1])) \\ Charles R Greathouse IV, Sep 19 2016

A182816 Number of values b in Z/nZ such that b^n = b.

Original entry on oeis.org

1, 2, 3, 2, 5, 4, 7, 2, 3, 4, 11, 4, 13, 4, 9, 2, 17, 4, 19, 4, 9, 4, 23, 4, 5, 4, 3, 8, 29, 8, 31, 2, 9, 4, 9, 4, 37, 4, 9, 4, 41, 8, 43, 4, 15, 4, 47, 4, 7, 4, 9, 8, 53, 4, 9, 4, 9, 4, 59, 8, 61, 4, 9, 2, 25, 24, 67, 4, 9, 16, 71, 4, 73, 4, 9, 8, 9, 8, 79, 4, 3, 4, 83, 8, 25, 4, 9, 4, 89, 8, 49, 4, 9, 4, 9, 4, 97, 4, 9, 4, 101, 8, 103, 4, 45, 4, 107, 4, 109, 8, 9, 8, 113

Offset: 1

M. F. Hasler, Dec 05 2010

nonn

a(n) is the number of nonnegative bases b < n such that b^n == b (mod n).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A063994.

f:= n -> mul(1+igcd(n-1,p[1]-1), p = ifactors(n)[2]):
map(f, [$1..200]); # Robert Israel, Sep 05 2018

Table[Times @@ Map[(1 + GCD[n - 1, # - 1]) &, FactorInteger[n][[All, 1]] ], {n, 113}] (* Michael De Vlieger, Sep 01 2020 *)

PARI
```
A182816(n)=sum(a=1,n,Mod(a,n)^n==a);
```

{ A182816(n) = my(p=factor(n)[,1]); prod(j=1,#p,1+gcd(n-1,p[j]-1)); } \\ Max Alekseyev, Dec 06 2010

a(n) = n for primes A000040 and Carmichael numbers A002997.
a(n) = Product_{i=1..m} (1 + gcd(n-1, p_i-1)), where p_1, p_2, ..., p_m are all distinct primes dividing n. - Max Alekseyev, Dec 06 2010
a(p^k) = p for prime p with k > 0. - Thomas Ordowski, Sep 05 2018

A209211 Numbers n such that n-1 and phi(n) are relatively prime.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 54, 56, 58, 60, 62, 64, 68, 72, 74, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 114, 116, 118, 120, 122, 126, 128, 132, 134, 136, 138

Offset: 1

José María Grau Ribas, Mar 06 2012

nonn

A063994(a(n)) = 1. - Reinhard Zumkeller, Mar 02 2013
a(n) = A111305(n-2) for n >= 3. - Emmanuel Vantieghem, Jul 03 2013
n such that A049559(n) = 1.  Includes A100484 and A000079. - Robert Israel, Nov 09 2015
All terms except the first one are even. Missing even terms are A039772. - Robert G. Wilson v, Sep 26 2016
Numbers n such that A187730(n) = 1. - Thomas Ordowski, Dec 29 2016

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000010, A000079, A003277, A039772, A049559, A063994, A100484, A111305.

a209211 n = a209211_list !! (n-1)
a209211_list = filter (\x -> (x - 1) `gcd` a000010 x == 1) [1..]
-- Reinhard Zumkeller, Mar 02 2013

select(n -> igcd(n-1, numtheory:-phi(n)) = 1, [$1..1000]); # Robert Israel, Nov 09 2015

Select[Range[200], GCD[# - 1, EulerPhi[#]] == 1 &]

isok(n) = gcd(n-1, eulerphi(n)) == 1; \\ Michel Marcus, Sep 26 2016

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

A381319 Order of linear recurrence with constant coefficients of solutions of k satisfying k^(n-1) == 1 (mod n^2) for a given n.

Original entry on oeis.org

2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 5, 2, 17, 2, 19, 2, 5, 2, 23, 2, 5, 2, 3, 4, 29, 2, 31, 2, 5, 2, 5, 2, 37, 2, 5, 2, 41, 2, 43, 2, 9, 2, 47, 2, 7, 2, 5, 4, 53, 2, 5, 2, 5, 2, 59, 2, 61, 2, 5, 2, 17, 6, 67, 2, 5, 4, 71, 2, 73, 2, 5, 4, 5, 2, 79, 2, 3, 2, 83, 2, 17, 2, 5, 2, 89

Offset: 2

Mike Sheppard, Feb 20 2025

nonn

For a given n, the solutions for k have the linear recurrence with constant coefficients k(m) = k(m-1) + k(m-(a(n)-1)) - k(m-a(n)), with order a(n). If a(n)=2 then the term k(m-1) appears twice and is k(m) = 2*k(m-1) - k(m-2).

Also, k(m) - k(m-(a(n)-1)) = n^2 = k(m-1) - k(m-a(n)), so all have nonhomogeneous linear recurrence of k(m) = k(m-(a(n)-1)) + n^2. Equivalently, k(m) = k(m-A063994(n)) + n^2, with order A063994(n). Thus, k(m) ~ (n^2 / A063994(n)) * m = (n^2 / (a(n)-1)) * m.

			For n=5 the congruence equation k^4 ==1 mod (5^2) has solutions of k (A056021) which satisfy k(m) = k(m-1) + k(m-4) - k(m-5), the order being 5, a(5)=5.
For n=9, k^8==1 mod (9^2) has solutions of k with recurrence k(m) = k(m-1) + k(m-2) - k(m-3), order 3, a(9)=3.

Paolo Xausa, Table of n, a(n) for n = 2..10000

Cf. A063994, A056020 (n=3), A056021 (n=5), A056022 (n=7), A056024 (n=11), A056025 (n=13), A056028 (n=19), A056031 (n=23), A056034 (n=29), A056035 (n=31).

A381319[n_] := Times @@ GCD[FactorInteger[n][[All, 1]] - 1, n - 1] + 1;
Array[A381319, 100, 2] (* Paolo Xausa, Mar 05 2025 *)

a(n) = 1 + A063994(n).

a(p) = p if p is prime.

A071294 Number of witnesses for strong pseudoprimality of 2n+1, i.e., number of bases b, 1 <= b <= 2n, in which 2n+1 is a strong pseudoprime.

Original entry on oeis.org

2, 4, 6, 2, 10, 12, 2, 16, 18, 2, 22, 4, 2, 28, 30, 2, 2, 36, 2, 40, 42, 2, 46, 6, 2, 52, 2, 2, 58, 60, 2, 6, 66, 2, 70, 72, 2, 2, 78, 2, 82, 6, 2, 88, 18, 2, 2, 96, 2, 100, 102, 2, 106, 108, 2, 112, 2, 2, 2, 10, 2, 4, 126, 2, 130, 18, 2, 136, 138, 2, 2, 6, 2, 148, 150, 2, 2, 156, 2, 2

Offset: 1

J.-F. Guiffes (guiffes.jean-francois(AT)wanadoo.fr), Jun 11 2002

nonn

Number of integers b, 1 <= b <= 2n, such that if 2n = 2^k*m with odd m, then the sequence (b^m, b^(2*m), ..., b^(2^k*m)) modulo 2n+1 satisfies the Rabin-Miller test.
Comments from R. J. Mathar, Jul 03 2012 (Start)
The subsequence related to composite 2n+1 is characterized with records in A195328 and associated 2n+1 tabulated in A141768.
Let N = 2n+1 = product_{i=1..s} p_i^r_i be the prime factorization of the odd 2n+1. Related odd parts q and q_i are defined by N-1=2^k*q and p_i-1 = 2^(k_i)*q_i, with sorting such that k_1 <= k_2 <=k_3... Then a(n) = (1+sum_{j=0..k1-1} 2^(j*s)) *product_{i=1..s} gcd(q,qi).
Reduces to A006093 if 2n+1 is prime.
This might be correlated with 2*A195508(n). (End)

Paulo Ribenboim, The Little Book of Bigger Primes, 2nd ed., Springer-Verlag, New York, 2004, p. 98.

Amiram Eldar, Table of n, a(n) for n = 1..10000
F. Arnault, The Rabin-Monier theorem for Lucas pseudoprimes, Mathematics of Computation, vol. 66, no 218, April 1997, pp. 869-881.
Louis Monier, Evaluation and comparison of two efficient primality testing algorithms, Theoretical Computer Science, Vol. 11 (1980), pp. 97-108.
Wikipedia, Strong pseudoprime
Index entries for sequences related to pseudoprimes

Cf. A000265, A001221, A006093, A007814, A063994, A141768, A195328.
Cf. also A001262, A020229, A020231, A020233.
Different from A060684.

rabinmiller := proc(n,a); k := 0; mu := n-1; while irem(mu,2)=0 do k := k+1; mu := mu/2 od; G := a&^mu mod(n); h := 0; if G=1 then RETURN(1) else while h1 do h := h+1; G := G&^2 mod n; od; if h n-1 then RETURN(0) else RETURN(1) fi; if G=1 then RETURN(1); fi; fi; end; compte := proc(n) local l; RETURN(sum('rabinmiller(2*n+1,l)','l'=1..2*n)); end;
Maple code from R. J. Mathar, Jul 03 2012 (Start)
A000265 := proc(n)
     n/2^padic[ordp](n,2) ;
end proc:
a := proc(n)
     q := A000265(n-1) ;
     B := 1;
     s := 0 ;
     k1 := 10000000000000 ;
     for pf in ifactors(n)[2] do
         pi := op(1,pf) ;
         qi := A000265(pi-1) ;
         ki := ilog2((pi-1)/qi) ;
         k1 := min(k1,ki) ;
         B := B*igcd(q,qi) ;
         s := s+1 ;
     end do:
     1+add(2^(j*s),j=0..k1-1) ;
     return B*% ;
end proc:
seq(a(2*n+1),n=1..60) ;

o[n_] := (n-1)/2^IntegerExponent[n-1, 2]; a[n_?PrimeQ] := n-1; a[n_] := Module[{p = FactorInteger[n][[;;, 1]]}, om = Length[p]; Product[GCD[o[n], o[p[[k]]]], {k, 1, om}] * (1 + (2^(om * Min[IntegerExponent[#, 2]& /@ (p - 1)]) - 1)/(2^om - 1))]; Table[a[n], {n, 3, 121, 2}] (* Amiram Eldar, Nov 08 2019 *)

For k = 2*n+1, a(k) = k - 1 if k is prime, otherwise, a(k) = (1 + 2^(omega(k)*nu(k)) - 1)/(2^omega(k)-1)) * Product_{p|k} gcd(od(k-1), od(p-1)), where omega(m) is the number of distinct prime factors of m (A001221), od(m) is the largest odd divisor of m (A000265) and nu(m) = min_{p|m} A007814(p-1). - Amiram Eldar, Nov 08 2019

Edited by Max Alekseyev, Sep 20 2018

Edited by N. J. A. Sloane, Nov 15 2019, merging R. J. Mathar's A182291 with this entry.

cp's OEIS Frontend

A064234 The least k such that A063994(k) = Product_{primes p dividing k} gcd(p-1, k-1) = n, or 0 if there's no such k.

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A330756 Number of values of k, 1 <= k <= n, with A063994(k) = A063994(n), where A063994(n) = Product_{primes p dividing n} gcd(p-1, n-1).

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

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A049559 a(n) = gcd(n - 1, phi(n)).

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A258409 Greatest common divisor of all (d-1)'s, where the d's are the positive divisors of n.

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A182816 Number of values b in Z/nZ such that b^n = b.

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