A027748 -id:A027748 - cp's OEIS frontend

A003963 Fully multiplicative with a(p) = k if p is the k-th prime.

1, 1, 2, 1, 3, 2, 4, 1, 4, 3, 5, 2, 6, 4, 6, 1, 7, 4, 8, 3, 8, 5, 9, 2, 9, 6, 8, 4, 10, 6, 11, 1, 10, 7, 12, 4, 12, 8, 12, 3, 13, 8, 14, 5, 12, 9, 15, 2, 16, 9, 14, 6, 16, 8, 15, 4, 16, 10, 17, 6, 18, 11, 16, 1, 18, 10, 19, 7, 18, 12, 20, 4, 21, 12, 18, 8, 20, 12, 22, 3, 16, 13, 23, 8, 21, 14, 20, 5

Offset: 1

Marc LeBrun

a(n) is the Matula number of the rooted tree obtained from the rooted tree T having Matula number n, by contracting its edges that emanate from the root. Example: a(49) = 16. Indeed, the rooted tree with Matula number 49 is the tree obtained by merging two copies of the tree Y at their roots. Contracting the two edges that emanate from the root, we obtain the star tree with 4 edges having Matula number 16. - Emeric Deutsch, May 01 2015
The Matula (or Matula-Goebel) number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T. - Emeric Deutsch, May 01 2015
a(n) is the product of the parts of the partition having Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product_{j=1..r} (p_j-th prime) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(75) = 18; indeed, the partition having Heinz number 75 = 3*5*5 is [2,3,3] and 2*3*3 = 18. - Emeric Deutsch, Jun 03 2015
Let T be the free-commutative-monoid monad on the category Set. Then for each set N we have a canonical function m from TTN to TN. If we let N = {1, 2, 3, ...} and enumerate the primes in the usual way (A000040) then unique prime factorization gives a canonical bijection f from N to TN. Then the sequence is given by a(n) = f^-1(m(T(f)(f(n)))). - Oscar Cunningham, Jul 18 2019

T. D. Noe, Table of n, a(n) for n = 1..10000
E. Deutsch, Rooted tree statistics from Matula numbers, Discrete Appl. Math., 160, 2012, 2314-2322.
F. Göbel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to Matula-Göbel numbers
Index entries for sequences computed from indices in prime factorization

Cf. A000720, A001221, A001222, A027748, A049084, A056239, A064553, A124010, A156552, A215366, A227184, A241909, A243354, A243499, A243504.

Product of entries on row n of A112798.

a003963 n = product $
   zipWith (^) (map a049084 $ a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Jun 30 2012

with(numtheory): a := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 1 elif bigomega(n) = 1 then pi(n) else a(r(n))*a(s(n)) end if end proc: seq(a(n), n = 1 .. 88);
# Alternative:
seq(mul(numtheory:-pi(t[1])^t[2], t=ifactors(n)[2]), n=1..100); # Robert Israel, May 01 2015

a[n_] := Times @@ (PrimePi[ #[[1]] ]^#[[2]]& /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 1, 88}]

a(n)=f=factor(n);prod(i=1,#f[,1],primepi(f[i,1])^f[i,2]) \\ Charles R Greathouse IV, Apr 26 2012; corrected by Rémy Sigrist, Jul 18 2019

a(n) = {f = factor(n); for (i=1, #f~, f[i, 1] = primepi(f[i, 1]); ); factorback(f); } \\ Michel Marcus, Feb 08 2015

A003963(n)={n=factor(n); n[,1]=apply(primepi,n[,1]); factorback(n)} \\ M. F. Hasler, May 03 2018

from math import prod
from sympy import primepi, factorint
def A003963(n): return prod(primepi(p)**e for p, e in factorint(n).items()) # Chai Wah Wu, Nov 17 2022

If n = product prime(k)^e(k) then a(n) = product k^e(k).
Multiplicative with a(p^e) = A000720(p)^e. - David W. Wilson, Aug 01 2001
a(n) = Product_{k=1..A001221(n)} A049084(A027748(n,k))^A124010(n,k). - Reinhard Zumkeller, Jun 30 2012
Rec. eq.: a(1)=1, a(k-th prime) = a(k), a(rs)=a(r)a(s). The Maple program is based on this. - Emeric Deutsch, May 01 2015
a(n) = A243504(A241909(n)) = A243499(A156552(n)) = A227184(A243354(n)) - Antti Karttunen, Mar 07 2017

A008472 Sum of the distinct primes dividing n.

Original entry on oeis.org

0, 2, 3, 2, 5, 5, 7, 2, 3, 7, 11, 5, 13, 9, 8, 2, 17, 5, 19, 7, 10, 13, 23, 5, 5, 15, 3, 9, 29, 10, 31, 2, 14, 19, 12, 5, 37, 21, 16, 7, 41, 12, 43, 13, 8, 25, 47, 5, 7, 7, 20, 15, 53, 5, 16, 9, 22, 31, 59, 10, 61, 33, 10, 2, 18, 16, 67, 19, 26, 14, 71, 5, 73

Offset: 1

Olivier Gérard

Sometimes called sopf(n).
Sum of primes dividing n (without repetition) (compare A001414).
Equals A051731 * A061397 = inverse Mobius transform of [0, 2, 3, 0, 5, 0, 7, ...]. - Gary W. Adamson, Feb 14 2008
Equals row sums of triangle A143535. - Gary W. Adamson, Aug 23 2008
a(n) = n if and only if n is prime. - Daniel Forgues, Mar 24 2009
a(n) = n is a new record if and only if n is prime. - Zak Seidov, Jun 27 2009
a(A001043(n)) = A191583(n);
For n > 0: a(A000079(n)) = 2, a(A000244(n)) = 3, a(A000351(n)) = 5, a(A000420(n)) = 7;
a(A006899(n)) <= 3; a(A003586(n)) = 5; a(A033846(n)) = 7; a(A033849(n)) = 8; a(A033847(n)) = 9; a(A033850(n)) = 10; a(A143207(n)) = 10. - Reinhard Zumkeller, Jun 28 2011
For n > 1: a(n) = Sum(A027748(n,k): 1 <= k <= A001221(n)). - Reinhard Zumkeller, Aug 27 2011
If n is the product of twin primes (A037074), a(n) = 2*sqrt(n+1) = sqrt(4n+4). - Wesley Ivan Hurt, Sep 07 2013
From Wilf A. Wilson, Jul 21 2017: (Start)
a(n) + 2, n > 2, is the number of maximal subsemigroups of the monoid of orientation-preserving or -reversing mappings on a set with n elements.
a(n) + 3, n > 2, is the number of maximal subsemigroups of the monoid of orientation-preserving or -reversing partial mappings on a set with n elements.
(End)
The smallest m such that a(m) = n, or 0 if no such number m exists is A064502(n). The only integers that are not in the sequence are 1, 4 and 6. - Bernard Schott, Feb 07 2022

			a(18) = 5 because 18 = 2 * 3^2 and 2 + 3 = 5.
a(19) = 19 because 19 is prime.
a(20) = 7 because 20 = 2^2 * 5 and 2 + 5 = 7.

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from Franklin T. Adams-Watters)
Johann Bartel, R. K. Bhaduri, Matthias Brack, and M. V. N. Murthy, On the asymptotic prime partitions of integers, arXiv:1609.06497 [math-ph], 2017.
James East, Jitender Kumar, James D. Mitchell, and Wilf A. Wilson, Maximal subsemigroups of finite transformation and partition monoids, arXiv:1706.04967 [math.GR], 2017. [_Wilf A. Wilson_, Jul 21 2017]

First difference of A024924.
Cf. A001414 (sopfr), A001222, A051731, A061397, A064502, A085020, A143535.
Sum of the k-th powers of the primes dividing n for k=0..10 : A001221 (k=0), this sequence (k=1), A005063 (k=2), A005064 (k=3), A005065 (k=4), A351193 (k=5), A351194 (k=6), A351195 (k=7), this sequence (k=8), A351197 (k=9), A351198 (k=10).
Cf. A010051.

a008472 = sum . a027748_row  -- Reinhard Zumkeller, Mar 29 2012

[n eq 1 select 0 else &+[p[1]: p in Factorization(n)]: n in [1..100]]; // Vincenzo Librandi, Jun 24 2017

A008472 := n -> add(d, d = select(isprime, numtheory[divisors](n))):
seq(A008472(i), i = 1..40); # Peter Luschny, Jan 31 2012
A008472 := proc(n)
        add( d, d= numtheory[factorset](n)) ;
end proc: # R. J. Mathar, Jul 08 2012

Prepend[Array[Plus @@ First[Transpose[FactorInteger[#]]] &, 100, 2], 0]
Join[{0}, Rest[Total[Transpose[FactorInteger[#]][[1]]]&/@Range[100]]] (* Harvey P. Dale, Jun 18 2012 *)
(* Requires version 7.0+ *) Table[DivisorSum[n, # &, PrimeQ[#] &], {n, 75}] (* Alonso del Arte, Dec 13 2014 *)
Table[Sum[p, {p, Select[Divisors[n], PrimeQ]}], {n, 1, 100}] (* Vaclav Kotesovec, May 20 2020 *)

sopf(n) = local(fac=factor(n)); sum(i=1,matsize(fac)[1],fac[i,1])

vector(100,n,vecsum(factor(n)[,1]~)) \\ Derek Orr, May 13 2015

A008472(n)=vecsum(factor(n)[,1]) \\ M. F. Hasler, Jul 18 2015

from sympy import primefactors
def A008472(n): return sum(primefactors(n)) # Chai Wah Wu, Feb 03 2022

def A008472(n):
    return add(d for d in divisors(n) if is_prime(d))
print([A008472(i) for i in (1..40)]) # Peter Luschny, Jan 31 2012

[sum(prime_factors(n)) for n in range(1,74)] # Giuseppe Coppoletta, Jan 19 2015

Let n = Product_j prime(j)^k(j) where k(j) >= 1, then a(n) = Sum_j prime(j).
Additive with a(p^e) = p.
G.f.: Sum_{k >= 1} prime(k)*x^prime(k)/(1-x^prime(k)). - Franklin T. Adams-Watters, Sep 01 2009
L.g.f.: -log(Product_{k>=1} (1 - x^prime(k))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 06 2017
Dirichlet g.f.: primezeta(s-1)*zeta(s). - Benedict W. J. Irwin, Jul 11 2018
a(n) = Sum_{p|n, p prime} p. - Wesley Ivan Hurt, Feb 04 2022
From Bernard Schott, Feb 07 2022: (Start)
For n > 0: a(A001020(n)) = 11, a(A001022(n)) = 13, a(A001026(n)) = 17, a(A001029(n)) = 19, a(A009967(n)) = 23, a(A009973(n)) = 29, a(A009975(n)) = 31, a(A009981(n)) = 37, a(A009985(n)) = 41, a(A009987(n)) = 43, a(A009991(n)) = 47.
For p odd prime, a(2*p) = p+2 <==> a(A100484(n)) = A052147(n) for n > 1. (End)
a(n) = Sum_{d|n} d * c(d), where c = A010051. - Wesley Ivan Hurt, Jun 22 2024

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

A003557 n divided by largest squarefree divisor of n; if n = Product p(k)^e(k) then a(n) = Product p(k)^(e(k)-1), with a(1) = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 2, 1, 1, 1, 8, 1, 3, 1, 2, 1, 1, 1, 4, 5, 1, 9, 2, 1, 1, 1, 16, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 8, 7, 5, 1, 2, 1, 9, 1, 4, 1, 1, 1, 2, 1, 1, 3, 32, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 5, 2, 1, 1, 1, 8, 27, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 16, 1, 7

Offset: 1

Marc LeBrun

a(n) is the size of the Frattini subgroup of the cyclic group C_n - Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 07 2001.
Also of the Frattini subgroup of the dihedral group with 2*n elements. - Sharon Sela (sharonsela(AT)hotmail.com), Jan 01 2002
Number of solutions to x^m==0 (mod n) provided that n < 2^(m+1), i.e. the sequence of sequences A000188, A000189, A000190, etc. converges to this sequence. - Henry Bottomley, Sep 18 2001
a(n) is the number of nilpotent elements in the ring Z/nZ. - Laszlo Toth, May 22 2009
The sequence of partial products of a(n) is A085056(n). - Peter Luschny, Jun 29 2009
The first occurrence of n in this sequence is at A064549(n). - Franklin T. Adams-Watters, Jul 25 2014
From Hal M. Switkay, Jul 03 2025: (Start)
For n > 1, a(n) is a proper divisor of n. Thus the sequence n, a(n), a(a(n)), ... eventually becomes 1. This yields a minimal factorization of n as a product of squarefree numbers (A005117), each factor dividing all larger factors, in a factorization that is conjugate to the minimal factorization of n as a product of prime powers (A000961), as follows.
Let f(n,0) = n, and let f(n,k) = a(f(n,k-1)) for k > 0. A051903(n) is the minimal value of k such that f(n,k) = 1. A051903(n) <= log(n)/log(2). Since n/a(n) = A007947(n) is always squarefree by definition, n is a product of squarefree factors in the form Product_{i=1..A051903(n)} [f(n,i-1)/f(n,i)].
The two factorizations correspond to conjugate partitions of bigomega(n) = A001222(n). (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative sequences.
Steven R. Finch, Idempotents and Nilpotents Modulo n, arXiv:math/0605019 [math.NT], 2006-2017.

Cf. A007947, A062378, A062379, A064549, A300717 (Möbius transform), A326306 (inv. Möbius transf.), A328572.
Sequences that are multiples of this sequence (the other factor of a pointwise product is given in parentheses): A000010 (A173557), A000027 (A007947), A001615 (A048250), A003415 (A342001), A007434 (A345052), A057521 (A071773).
Cf. A082695 (Dgf at s=2), A065487 (Dgf at s=3).
Cf. A000961, A001222, A005117, A051903.

a003557 n = product $ zipWith (^)
                      (a027748_row n) (map (subtract 1) $ a124010_row n)
-- Reinhard Zumkeller, Dec 20 2013

using Nemo
function A003557(n)
    n < 4 && return 1
    q = prod([p for (p, e) ∈ Nemo.factor(fmpz(n))])
    return n == q ? 1 : div(n, q)
end
[A003557(n) for n in 1:90] |> println  # Peter Luschny, Feb 07 2021

[(&+[(Floor(k^n/n)-Floor((k^n-1)/n)): k in [1..n]]): n in [1..100]]; // G. C. Greubel, Nov 02 2018

A003557 := n -> n/ilcm(op(numtheory[factorset](n))):
seq(A003557(n), n=1..98); # Peter Luschny, Mar 23 2011
seq(n / NumberTheory:-Radical(n), n = 1..98); # Peter Luschny, Jul 20 2021

Prepend[ Array[ #/Times@@(First[ Transpose[ FactorInteger[ # ] ] ])&, 100, 2 ], 1 ] (* Olivier Gérard, Apr 10 1997 *)

a(n)=n/factorback(factor(n)[,1]) \\ Charles R Greathouse IV, Nov 17 2014

for(n=1, 100, print1(direuler(p=2, n, (1 - p*X + X)/(1 - p*X))[n], ", ")) \\ Vaclav Kotesovec, Jun 20 2020

from sympy.ntheory.factor_ import core
from sympy import divisors
def a(n): return n / max(i for i in divisors(n) if core(i) == i)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Apr 16 2017

from math import prod
from sympy import primefactors
def A003557(n): return n//prod(primefactors(n)) # Chai Wah Wu, Nov 04 2022

def A003557(n) : return n*mul(1/p for p in prime_divisors(n))
[A003557(n) for n in (1..98)] # Peter Luschny, Jun 10 2012

Multiplicative with a(p^e) = p^(e-1). - Vladeta Jovovic, Jul 23 2001
a(n) = n/rad(n) = n/A007947(n) = sqrt(J_2(n)/J_2(rad(n))), where J_2(n) is A007434. - Enrique Pérez Herrero, Aug 31 2010
a(n) = (J_k(n)/J_k(rad(n)))^(1/k), where J_k is the k-th Jordan Totient Function: (J_2 is A007434 and J_3 A059376). - Enrique Pérez Herrero, Sep 03 2010
Dirichlet convolution of A000027 and A097945. - R. J. Mathar, Dec 20 2011
a(n) = A000010(n)/|A023900(n)|. - Eric Desbiaux, Nov 15 2013
a(n) = Product_{k = 1..A001221(n)} (A027748(n,k)^(A124010(n,k)-1)). - Reinhard Zumkeller, Dec 20 2013
a(n) = Sum_{k=1..n}(floor(k^n/n)-floor((k^n-1)/n)). - Anthony Browne, May 11 2016
a(n) = e^[Sum_{k=2..n} (floor(n/k)-floor((n-1)/k))*(1-A010051(k))*Mangoldt(k)] where Mangoldt is the Mangoldt function. - Anthony Browne, Jun 16 2016
a(n) = Sum_{d|n} mu(d) * phi(d) * (n/d), where mu(d) is the Moebius function and phi(d) is the Euler totient function (rephrases formula of Dec 2011). - Daniel Suteu, Jun 19 2018
G.f.: Sum_{k>=1} mu(k)*phi(k)*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Nov 02 2018
Dirichlet g.f.: Product_{primes p} (1 + 1/(p^s - p)). - Vaclav Kotesovec, Jun 24 2020
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} mu(n/gcd(n,k))*gcd(n,k).
a(n) = Sum_{k=1..n} mu(gcd(n,k))*(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
a(n) = A001615(n)/A048250(n) = A003415/A342001(n) = A057521(n)/A071773(n). - Antti Karttunen, Jun 08 2021

Secondary definition added to the name by Antti Karttunen, Jun 08 2021

A001615 Dedekind psi function: n * Product_{p|n, p prime} (1 + 1/p).

Original entry on oeis.org

1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, 14, 24, 24, 24, 18, 36, 20, 36, 32, 36, 24, 48, 30, 42, 36, 48, 30, 72, 32, 48, 48, 54, 48, 72, 38, 60, 56, 72, 42, 96, 44, 72, 72, 72, 48, 96, 56, 90, 72, 84, 54, 108, 72, 96, 80, 90, 60, 144, 62, 96, 96, 96, 84, 144, 68, 108, 96

Offset: 1

N. J. A. Sloane

Number of primitive sublattices of index n in generic 2-dimensional lattice; also index of Gamma_0(n) in SL_2(Z).
A generic 2-dimensional lattice L =  consists of all vectors of the form mV + nW, (m,n integers). A sublattice S =  has index |ad-bc| and is primitive if gcd(a,b,c,d) = 1. The generic lattice L has precisely a(2) = 3 sublattices of index 2, namely <2V,W>,  and  (which = ) and so on for other indices.
The sublattices of index n are in 1-to-1 correspondence with matrices [a b; 0 d] with a>0, ad=n, b in [0..d-1]. The number of these is Sum_{d|n} = sigma(n), which is A000203. A sublattice is primitive if gcd(a,b,d) = 1; the number of these is n * product_{p|n} (1+1/p), which is the present sequence.
SL_2(Z) = Gamma is the group of all 2 X 2 matrices [a b; c d] where a,b,c,d are integers with ad-bc = 1 and Gamma_0(N) is usually defined as the subgroup of this for which N|c. But conceptually Gamma is best thought of as the group of (positive) automorphisms of a lattice , its typical element taking V -> aV + bW, W -> cV + dW and then Gamma_0(N) can be defined as the subgroup consisting of the automorphisms that fix the sublattice  of index N. - J. H. Conway, May 05 2001
Dedekind proved that if n = k_i*j_i for i in I represents all the ways to write n as a product, and e_i=gcd(k_i,j_i), then a(n)= sum(k_i / (e_i * phi(e_i)), i in I ) [cf. Dickson, History of the Theory of Numbers, Vol. 1, p. 123].
Also a(n)= number of cyclic subgroups of order n in an Abelian group of order n^2 and type (1,1) (Fricke). - Len Smiley, Dec 04 2001
The polynomial degree of the classical modular equation of degree n relating j(z) and j(nz) is psi(n) (Fricke). - Michael Somos, Nov 10 2006; clarified by Katherine E. Stange, Mar 11 2022
The Mobius transform of this sequence is A063659. - Gary W. Adamson, May 23 2008
The inverse Mobius transform of this sequence is A060648. - Vladeta Jovovic, Apr 05 2009
The Dirichlet inverse of this sequence is A008836(n) * A048250(n). - Álvar Ibeas, Mar 18 2015
The Riemann Hypothesis is true if and only if a(n)/n - e^gamma*log(log(n)) < 0 for any n > 30. - Enrique Pérez Herrero, Jul 12 2011
The Riemann Hypothesis is also equivalent to another inequality, see the Sole and Planat link. - Thomas Ordowski, May 28 2017
An infinitary analog of this sequence is the sum of the infinitary divisors of n (see A049417). - Vladimir Shevelev, Apr 01 2014
Problem: are there composite numbers n such that n+1 divides psi(n)? - Thomas Ordowski, May 21 2017
The sum of divisors d of n such that n/d is squarefree. - Amiram Eldar, Jan 11 2019
Psi(n)/n is a new maximum for each primorial (A002110) [proof in link: Patrick Sole and Michel Planat, Proposition 1 page 2]. - Bernard Schott, May 21 2020
From Jianing Song, Nov 05 2022: (Start)
a(n) is the number of subgroups of C_n X C_n that are isomorphic to C_n, where C_n is the cyclic group of order n. Proof: the number of elements of order n in C_n X C_n is A007434(n) (they are the elements of the form (a,b) in C_n X C_n where gcd(a,b,n) = 1), and each subgroup isomorphic to C_n contains phi(n) generators, so the number of such subgroups is A007434(n)/phi(n) = a(n).
The total number of order-n subgroups of C_n X C_n is A000203(n). (End)

			Let L = <V,W> be a 2-dimensional lattice. The 6 primitive sublattices of index 4 are generated by <4V,W>, <V,4W>, <4V,W+-V>, <2V+W,2W>, <2V,2W+V>. Compare A000203.
G.f. = x + 3*x^2 + 4*x^3 + 6*x^4 + 6*x^5 + 12*x^6 + 8*x^7 + 12*x^8 + 12*x^9 + ...

Tom Apostol, Intro. to Analyt. Number Theory, page 71, Problem 11, where this is called phi_1(n).
David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 228.
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 220.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41, p. 147.
B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 79.
G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (1).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe and N. J. A. Sloane, Table of n, a(n) for n = 1..10000
O. Bordelles and B. Cloitre, An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions, J. Int. Seq. 16 (2013) #13.6.3.
Harriet Fell, Morris Newman, and Edward Ordman, Tables of genera of groups of linear fractional transformations, J. Res. Nat. Bur. Standards Sect. B 67B 1963 61-68.
M. Hampejs, N. Holighaus, L. Toth and C. Wiesmeyr, Representing and counting the subgroups of the group Z_m X Z_n, arXiv:1211.1797 [math.GR], 2012.
W. Hürlimann, Dedekind's arithmetic function and primitive four squares counting functions, Journal of Algebra, Number Theory: Advances and Applications 14:2 (2015), 73-88.
F. A. Lewis et al., Problem 4002, Amer. Math. Monthly, Vol. 49, No. 9, Nov. 1942, pp. 618-619.
E. Pérez Herrero, Recycling Hardy & Wright, Average Order of Dedekind Psi Function, Psychedelic Geometry Blogspot.
Michel Planat, Riemann hypothesis from the Dedekind psi function, arXiv:1010.3239 [math.GM], 2010.
Patrick Sole and Michel Planat, Extreme values of the Dedekind Psi function, to appear in Journal of Combinatorics and Number Theory, arXiv:1011.1825 [math.NT], 2010-2011.
Eric Weisstein's World of Mathematics, Dedekind Function
Wikipedia, Dedekind psi function
Index entries for "core" sequences
Index entries for sequences related to sublattices

Other sequences that count lattices/sublattices: A000203 (with primitive condition removed), A003050 (hexagonal lattice instead), A003051, A054345, A160889, A160891.
Cf. A002110, A027748, A124010, A019269.
Cf. A301594.
Cf. A063659 (Möbius transform), A082020 (average order), A156303 (Euler transform), A173290 (partial sums), A175836 (partial products), A203444 (range).
Cf. A210523 (record values).
Algebraic combinations with other core sequences: A000082, A033196, A175732, A291784, A344695.
Sequences of the form n^k * Product_ {p|n, p prime} (1 + 1/p^k) for k=0..10: A034444 (k=0), this sequence (k=1), A065958 (k=2), A065959 (k=3), A065960 (k=4), A351300 (k=5), A351301 (k=6), A351302 (k=7), A351303 (k=8), A351304 (k=9), A351305 (k=10).
Cf. A082695 (Dgf at s=3), A339925 (Dgf at s=4).

import Data.Ratio (numerator)
a001615 n = numerator (fromIntegral n * (product $
            map ((+ 1) . recip . fromIntegral) $ a027748_row n))
-- Reinhard Zumkeller, Jun 03 2013, Apr 12 2012

m:=75; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[MoebiusMu(k)^2*x^k/(1-x^k)^2: k in [1..2*m]]) )); // G. C. Greubel, Nov 23 2018

A001615 := proc(n) n*mul((1+1/i[1]),i=ifactors(n)[2]) end; # Mark van Hoeij, Apr 18 2012

Join[{1}, Table[n Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]), {n, 2, 100}]] (* T. D. Noe, Jun 11 2006 *)
Table[DirichletConvolve[j, MoebiusMu[j]^2, j, n], {n, 100}] (* Jan Mangaldan, Aug 22 2013 *)
a[n_] := n Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}]; (* Michael Somos, Jan 10 2015 *)
Table[n Product[1 + 1/p, {p, Select[Divisors[n], PrimeQ]}], {n, 1, 100}] (* Vaclav Kotesovec, May 08 2021 *)
Table[n DivisorSum[n, MoebiusMu[#]^2/# &], {n, 20}] (* Eric W. Weisstein, Mar 09 2025 *)

{a(n) = if( n<1, 0, direuler( p=2, n, (1 + X) / (1 - p*X)) [n])};

{a(n) = if( n<1, 0, n * sumdiv( n, d, moebius(d)^2 / d))}; /* Michael Somos, Nov 10 2006 */

a(n)=my(f=factor(n)); prod(i=1,#f~, f[i,1]^f[i,2] + f[i,1]^(f[i,2]-1)) \\ Charles R Greathouse IV, Aug 22 2013

a(n) = n * sumdivmult(n, d, issquarefree(d)/d) \\ Charles R Greathouse IV, Sep 09 2014

from math import prod
from sympy import primefactors
def A001615(n):
    plist = primefactors(n)
    return n*prod(p+1 for p in plist)//prod(plist) # Chai Wah Wu, Jun 03 2021

def A001615(n) : return n*mul(1+1/p for p in prime_divisors(n))
[A001615(n) for n in (1..69)] # Peter Luschny, Jun 10 2012

Dirichlet g.f.: zeta(s) * zeta(s-1) / zeta(2*s). - Michael Somos, May 19 2000
Multiplicative with a(p^e) = (p+1)*p^(e-1). - David W. Wilson, Aug 01 2001
a(n) = A003557(n)*A048250(n) = n*A000203(A007947(n))/A007947(n). - Labos Elemer, Dec 04 2001
a(n) = n*Sum_{d|n} mu(d)^2/d, Dirichlet convolution of A008966 and A000027. - Benoit Cloitre, Apr 07 2002
a(n) = Sum_{d|n} mu(n/d)^2 * d. - Joerg Arndt, Jul 06 2011
From Enrique Pérez Herrero, Aug 22 2010: (Start)
a(n) = J_2(n)/J_1(n) = J_2(n)/phi(n) = A007434(n)/A000010(n), where J_k is the k-th Jordan Totient Function.
a(n) = (1/phi(n))*Sum_{d|n} mu(n/d)*d^(b-1), for b=3. (End)
a(n) = n / Sum_{d|n} mu(d)/a(d). - Enrique Pérez Herrero, Jun 06 2012
a(n^k)= n^(k-1) * a(n). - Enrique Pérez Herrero, Jan 05 2013
If n is squarefree, then a(n) = A049417(n) = A000203(n). - Vladimir Shevelev, Apr 01 2014
a(n) = Sum_{d^2 | n} mu(d) * A000203(n/d^2). - Álvar Ibeas, Dec 20 2014
The average order of a(n) is 15*n/Pi^2. - Enrique Pérez Herrero, Jan 14 2012. See Apostol. - N. J. A. Sloane, Sep 04 2017
G.f.: Sum_{k>=1} mu(k)^2*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Oct 25 2018
a(n) = Sum_{d|n} 2^omega(d) * phi(n/d), Dirichlet convolution of A034444 and A000010. - Daniel Suteu, Mar 09 2019
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} 2^omega(gcd(n,k)).
a(n) = Sum_{k=1..n} 2^omega(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
a(n) = abs(A158523(n)) = A158523(n) * A008836(n). - Enrique Pérez Herrero, Nov 07 2022
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^2). - Ridouane Oudra, Mar 26 2025

More terms from Olivier Gérard, Aug 15 1997

A007913 Squarefree part of n: a(n) is the smallest positive number m such that n/m is a square.

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 2, 1, 10, 11, 3, 13, 14, 15, 1, 17, 2, 19, 5, 21, 22, 23, 6, 1, 26, 3, 7, 29, 30, 31, 2, 33, 34, 35, 1, 37, 38, 39, 10, 41, 42, 43, 11, 5, 46, 47, 3, 1, 2, 51, 13, 53, 6, 55, 14, 57, 58, 59, 15, 61, 62, 7, 1, 65, 66, 67, 17, 69, 70, 71, 2, 73, 74, 3, 19, 77

Offset: 1

R. Muller, Mar 15 1996

Also called core(n). [Not to be confused with the squarefree kernel of n, A007947.]
Sequence read mod 4 gives A065882. - Philippe Deléham, Mar 28 2004
This is an arithmetic function and is undefined if n <= 0.
A note on square roots of numbers: we can write sqrt(n) = b*sqrt(c) where c is squarefree. Then b = A000188(n) is the "inner square root" of n, c = A007913(n), lcm(A007947(b),c) = A007947(n) = "squarefree kernel" of n and bc = A019554(n) = "outer square root" of n. [Corrected by M. F. Hasler, Mar 01 2018]
If n > 1, the quantity f(n) = log(n/core(n))/log(n) satisfies 0 <= f(n) <= 1; f(n) = 0 when n is squarefree and f(n) = 1 when n is a perfect square. One can define n as being "epsilon-almost squarefree" if f(n) < epsilon. - Kurt Foster (drsardonicus(AT)earthlink.net), Jun 28 2008
a(n) is the smallest natural number m such that product of geometric mean of the divisors of n and geometric mean of the divisors of m are integers. Geometric mean of the divisors of number n is real number b(n) = Sqrt(n). a(n) = 1 for infinitely many n. a(n) = 1 for numbers from A000290: a(A000290(n)) = 1. For n = 8; b(8) = sqrt(8), a(n) = 2 because b(2) = sqrt(2); sqrt(8) * sqrt(2) = 4 (integer). - Jaroslav Krizek, Apr 26 2010
Dirichlet convolution of A010052 with the sequence of absolute values of A055615. - R. J. Mathar, Feb 11 2011
Booker, Hiary, & Keating outline a method for bounding (on the GRH) a(n) for large n using L-functions. - Charles R Greathouse IV, Feb 01 2013
According to the formula a(n) = n/A000188(n)^2, the scatterplot exhibits the straight lines y=x, y=x/4, y=x/9, ..., i.e., y=x/k^2 for all k=1,2,3,... - M. F. Hasler, May 08 2014
The Dirichlet inverse of this sequence is A008836(n) * A063659(n). - Álvar Ibeas, Mar 19 2015
a(n) = 1 if n is a square, a(n) = n if n is a product of distinct primes. - Zak Seidov, Jan 30 2016
All solutions of the Diophantine equation n*x=y^2 or, equivalently, G(n,x)=y, with G being the geometric mean, are of the form x=k^2*a(n), y=k*sqrt(n*a(n)), where k is a positive integer. - Stanislav Sykora, Feb 03 2016
If f is a multiplicative function then Sum_{d divides n} f(a(d)) is also multiplicative. For example, A010052(n) = Sum_{d divides n} mu(a(d)) and A046951(n) = Sum_{d divides n} mu(a(d)^2). - Peter Bala, Jan 24 2024

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 1000 terms from T. D. Noe)
Krassimir T. Atanassov, On Some of Smarandache's Problems.
Krassimir T. Atanassov, On the 22nd, 23rd, and the 24th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 2, 80-82.
Andrew Booker, Ghaith Hiary, and Jon Keating, Detecting squarefree numbers, CNTA XII (2012).
Henry Bottomley, Some Smarandache-type multiplicative sequences.
John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv preprint arXiv:1105.3399 [math.GM], 2011.
Vlad Copil and Laurenţiu Panaitopol, Properties of a sequence generated by positive integers, Bulletin mathématique de la Société des Sciences Mathématiques de Roumanie, Nouvelle Série, Vol. 50 (98), No. 2 (2007), pp. 131-137; alternative link.
Florentin Smarandache, Only Problems, Not Solutions!, Xiquan Publ., Phoenix-Chicago, 1993.
Eric Weisstein's World of Mathematics, Squarefree Part.

See A000188, A007947, A008833, A019554, A117811 for related information, specific to n.
See A027746, A027748, A124010 for factorization data for n.
Analogous sequences: A050985, A053165, A055231.
Cf. A002734, A005117 (range of values), A059897, A069891 (partial sums), A090699, A350389.
Related to A006519 via A225546.

a007913 n = product $
            zipWith (^) (a027748_row n) (map (`mod` 2) $ a124010_row n)
-- Reinhard Zumkeller, Jul 06 2012

[ Squarefree(n) : n in [1..256] ]; // N. J. A. Sloane, Dec 23 2006

A007913 := proc(n) local f,a,d; f := ifactors(n)[2] ; a := 1 ; for d in f do if type(op(2,d),'odd') then a := a*op(1,d) ; end if; end do: a; end proc: # R. J. Mathar, Mar 18 2011
# second Maple program:
a:= n-> mul(i[1]^irem(i[2], 2), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 20 2015
seq(n / expand(numtheory:-nthpow(n, 2)), n=1..77);  # Peter Luschny, Jul 12 2022

data = Table[Sqrt[n], {n, 1, 100}]; sp = data /. Sqrt[] -> 1; sfp = data/sp /. Sqrt[x] -> x (* Artur Jasinski, Nov 03 2008 *)
Table[Times@@Power@@@({#[[1]],Mod[ #[[2]],2]}&/@FactorInteger[n]),{n,100}] (* Zak Seidov, Apr 08 2009 *)
Table[{p, e} = Transpose[FactorInteger[n]]; Times @@ (p^Mod[e, 2]), {n, 100}] (* T. D. Noe, May 20 2013 *)
Sqrt[#] /. (c_:1)*a_^(b_:0) -> (c*a^b)^2& /@ Range@100 (* Bill Gosper, Jul 18 2015 *)

PARI
```
a(n)=core(n)
```

from sympy import factorint, prod
def A007913(n):
    return prod(p for p, e in factorint(n).items() if e % 2)
# Chai Wah Wu, Feb 03 2015

[squarefree_part(n) for n in (1..77)] # Peter Luschny, Feb 04 2015

Multiplicative with a(p^k) = p^(k mod 2). - David W. Wilson, Aug 01 2001
a(n) modulo 2 = A035263(n); a(A036554(n)) is even; a(A003159(n)) is odd. - Philippe Deléham, Mar 28 2004
Dirichlet g.f.: zeta(2s)*zeta(s-1)/zeta(2s-2). - R. J. Mathar, Feb 11 2011
a(n) = n/( Sum_{k=1..n} floor(k^2/n)-floor((k^2 -1)/n) )^2. - Anthony Browne, Jun 06 2016
a(n) = rad(n)/a(n/rad(n)), where rad = A007947. This recurrence relation together with a(1) = 1 generate the sequence. - Velin Yanev, Sep 19 2017
From Peter Munn, Nov 18 2019: (Start)
a(k*m) = A059897(a(k), a(m)).
a(n) = n / A008833(n).
(End)
a(A225546(n)) = A225546(A006519(n)). - Peter Munn, Jan 04 2020
From Amiram Eldar, Mar 14 2021: (Start)
Theorems proven by Copil and Panaitopol (2007):
Lim sup_{n->oo} a(n+1)-a(n) = oo.
Lim inf_{n->oo} a(n+1)-a(n) = -oo.
Sum_{k=1..n} 1/a(k) ~ c*sqrt(n) + O(log(n)), where c = zeta(3/2)/zeta(3) (A090699). (End)
a(n) = A019554(n)^2/n. - Jianing Song, May 08 2022
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/30 = 0.328986... . - Amiram Eldar, Oct 25 2022
a(n) = A007947(A350389(n)). - Amiram Eldar, Jan 20 2024

More terms from Michael Somos, Nov 24 2001

Definition reformulated by Daniel Forgues, Mar 24 2009

A002322 Reduced totient function psi(n): least k such that x^k == 1 (mod n) for all x prime to n; also known as the Carmichael lambda function (exponent of unit group mod n); also called the universal exponent of n.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 2, 6, 4, 10, 2, 12, 6, 4, 4, 16, 6, 18, 4, 6, 10, 22, 2, 20, 12, 18, 6, 28, 4, 30, 8, 10, 16, 12, 6, 36, 18, 12, 4, 40, 6, 42, 10, 12, 22, 46, 4, 42, 20, 16, 12, 52, 18, 20, 6, 18, 28, 58, 4, 60, 30, 6, 16, 12, 10, 66, 16, 22, 12, 70, 6, 72, 36, 20, 18, 30, 12, 78, 4, 54

Offset: 1

N. J. A. Sloane

a(n) is the largest order of any element in the multiplicative group modulo n. - Joerg Arndt, Mar 19 2016

Largest period of repeating digits of 1/n written in different bases (i.e., largest value in each row of square array A066799 and least common multiple of each row). - Henry Bottomley, Dec 20 2001

D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10.
W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 53.
Kenneth H. Rosen, Elementary Number Theory and Its Applications, Addison-Wesley, 1984, page 269.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
L. Blum, M. Blum, and M. Shub, A simple unpredictable pseudorandom number generator, SIAM J. Comput. 15 (1986), no. 2, 364-383. see p. 377.
P. J. Cameron and D. A. Preece, Notes on primitive lambda-roots
R. D. Carmichael, Note on a new number theory function, Bull. Amer. Math. Soc. 16 (1909-10), 232-238.
A. Cauchy, Mémoire sur la résolution des équations indéterminées du premier degré en nombres entiers, Oeuvres Complètes. Gauthier-Villars, Paris, 1882-1938, Series (2), Vol. 12, pp. 9-47.
A. de Vries, The prime factors of an integer(along with Euler's phi and Carmichael's lambda functions), Applet
Paul Erdős, Carl Pomerance, and Eric Schmutz, Carmichael's lambda function, Acta Arithmetica 58 (1991), pp. 363-385.
J.-H. Evertse and E. van Heyst, Which new RSA signatures can be computed from some given RSA signatures?, Proceedings of Eurocrypt '90, Lect. Notes Comput. Sci., 473, Springer-Verlag, pp. 84-97, see page 86.
J. M. Grau and A. M. Oller-Marcén, On the congruence sum_{j=1}^{n-1} j^{k(n-1)} == -1 (mod n); k-strong Giuga and k-Carmichael numbers, arXiv preprint arXiv:1311.3522 [math.NT], 2013.
Romeo Meštrović, Generalizations of Carmichael numbers I, arXiv:1305.1867v1 [math.NT], May 04 2013.
P. Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 80.
Benjamin Schreyer, Rigged Horse Numbers and their Modular Periodicity, arXiv:2409.03799 [math.CO], 2024. See p. 12.
Eric Weisstein's World of Mathematics, Carmichael Function
Wikipedia, Carmichael function
Wolfram Research, First 50 values of Carmichael lambda(n)
Index entries for "core" sequences

Cf. A011773, A002174, A002616, A034380, A061258, A062373, A141258, A162578 (partial sums), A218342.

a002322 n = foldl lcm 1 $ map (a207193 . a095874) $
                          zipWith (^) (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Feb 16 2012

[1] cat [ CarmichaelLambda(n) : n in [2..100]];

with(numtheory); A002322 := lambda; [seq(lambda(n), n=1..100)];

Table[CarmichaelLambda[k], {k, 50}] (* Artur Jasinski, Apr 05 2008 *)

A002322(n)= lcm( apply( f -> (f[1]-1)*f[1]^(f[2]-1-(f[1]==2 && f[2]>2)), Vec(factor(n)~))) \\ M. F. Hasler, Jul 05 2009

a(n)=lcm(znstar(n)[2]) \\ Charles R Greathouse IV, Aug 04 2012

from sympy import reduced_totient
def A002322(n): return reduced_totient(n) # Chai Wah Wu, Feb 24 2021

If M = 2^e*P1^e1*P2^e2*...*Pk^ek, lambda(2^e) = 2^(e-1) if e=1 or 2, = 2^(e-2) if e > 2; lambda(M) = lcm(lambda(2^e), (P1-1)*P1^(e1-1), (P2-1)*P2^(e2-1), ..., (Pk-1)*Pk^(ek-1)).

a(n) = lcm_{k=1..A001221(n)} A207193(A095874(A027748(n,k)^A124010(n,k))). - Reinhard Zumkeller, Feb 16 2012

A027746 Irregular triangle in which first row is 1, n-th row (n>1) gives prime factors of n with repetition.

Original entry on oeis.org

1, 2, 3, 2, 2, 5, 2, 3, 7, 2, 2, 2, 3, 3, 2, 5, 11, 2, 2, 3, 13, 2, 7, 3, 5, 2, 2, 2, 2, 17, 2, 3, 3, 19, 2, 2, 5, 3, 7, 2, 11, 23, 2, 2, 2, 3, 5, 5, 2, 13, 3, 3, 3, 2, 2, 7, 29, 2, 3, 5, 31, 2, 2, 2, 2, 2, 3, 11, 2, 17, 5, 7, 2, 2, 3, 3, 37, 2, 19, 3, 13, 2, 2, 2, 5, 41, 2, 3, 7, 43, 2, 2, 11, 3, 3, 5

Offset: 1

Maghraoui Abdelkader

n-th row has length A001222(n) (n>1).

			Triangle begins
  1;
  2;
  3;
  2, 2;
  5;
  2, 3;
  7;
  2, 2, 2;
  3, 3;
  2, 5;
  11;
  2, 2, 3;
  ...

N. J. A. Sloane, First 2048 rows of triangle, flattened
S. von Worley (?), Animated Factorization Diagrams, Oct. 2012.
Brent Yorgey, Factorization diagrams, The Math Less Traveled, Oct 05 2012.

Cf. A000027, A001222, A027748.
a(A022559(A000040(n))+1) = A000040(n).
Column 1 is A020639, columns 2 and 3 correspond to A014673 and A115561.
A281890 measures frequency of each prime in each column, with A281889 giving median values.
Cf. A175943 (partial products), A265110 (partial row products), A265111.

import Data.List (unfoldr)
a027746 n k = a027746_tabl !! (n-1) !! (k-1)
a027746_tabl = map a027746_row [1..]
a027746_row 1 = [1]
a027746_row n = unfoldr fact n where
   fact 1 = Nothing
   fact x = Just (p, x `div` p) where p = a020639 x
-- Reinhard Zumkeller, Aug 27 2011

P:=proc(n) local FM: FM:=ifactors(n)[2]: seq(seq(FM[j][1],k=1..FM[j][2]),j=1..nops(FM)) end: 1; for n from 2 to 45 do P(n) od; # yields sequence in triangular form; Emeric Deutsch, Feb 13 2005

row[n_] := Flatten[ Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]]; Flatten[ Table[ row[n], {n, 1, 45}]] (* Jean-François Alcover, Dec 01 2011 *)

A027746_row(n,o=[1])=if(n>1,concat(apply(t->vector(t[2],i,t[1]), Vec(factor(n)~))),o) \\ Use %(n,[]) if you want the more natural [] for the first row. - M. F. Hasler, Jul 29 2015

def factors(n: int) -> list[int]:
    p = n
    L:list[int] = []
    for f in range(2, p + 1):
        if f * f > p: break
        while True:
            q, r = divmod(p, f)
            if r != 0: break
            L.append(f)
            p = q
            if p == 1: return L
    L.append(p)
    return L  # Peter Luschny, Jul 18 2024

v=[1]
for k in [2..45]: v.extend(p for (p, m) in factor(k) for _ in range(m))
print(v) # Giuseppe Coppoletta, Dec 29 2017

Product_{k=1..A001222(n)} T(n,k) = n.
From Reinhard Zumkeller, Aug 27 2011: (Start)
A001414(n) = Sum_{k=1..A001222(n)} T(n,k), n>1;
A006530(n) = T(n,A001222(n)) = Max_{k=1..A001222(n)} T(n,k);
A020639(n) = T(n,1) = Min_{k=1..A001222(n)} T(n,k). (End)

More terms from James Sellers

A001158 sigma_3(n): sum of cubes of divisors of n.

Original entry on oeis.org

1, 9, 28, 73, 126, 252, 344, 585, 757, 1134, 1332, 2044, 2198, 3096, 3528, 4681, 4914, 6813, 6860, 9198, 9632, 11988, 12168, 16380, 15751, 19782, 20440, 25112, 24390, 31752, 29792, 37449, 37296, 44226, 43344, 55261, 50654, 61740, 61544, 73710, 68922, 86688

Offset: 1

N. J. A. Sloane, R. K. Guy

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).
Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6..24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001
Also the eigenvalues of the Hecke operator T_n for the entire modular normalized Eisenstein form E_4(z) (see A004009): T_n E_4 = a(n) E_4, n >= 1. For the Hecke operator T_n and eigenforms see, e.g., the Koecher-Krieg reference, p. 207, eq. (5) and p. 211, section 4, or the Apostol reference p. 120, eq. (13) and pp. 129 - 133. - Wolfdieter Lang, Jan 28 2016

			G.f. = x + 9*x^2 + 28*x^3 + 73*x^4 + 126*x^5 + 252*x^6 + 344*x^7 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 827.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990, pp. 120, 129 - 133.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 166.
Max Koecher and Aloys Krieg, Elliptische Funktionen und Modulformen, 2. Auflage, Springer, 2007, pp. 207, 211.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Zagier, Don. "Elliptic modular forms and their applications." The 1-2-3 of modular forms. Springer Berlin Heidelberg, 2008. 1-103. See p. 17, G_4(z).

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 827.
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
D. B. Lahiri, Some arithmetical identities for Ramanujan's and divisor functions, Bulletin of the Australian Mathematical Society, Volume 1, Issue 3 December 1969, pp. 307-314. See Theorem 2 p. 308.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Eric Weisstein's World of Mathematics, Divisor Function.
Index entries for sequences related to sigma(n)

Cf. A000005, A000203, A001157.
Cf. A051731, A127093.
Cf. A027748, A124010.
Cf. A004009, A064603 (partial sums).

a001158 n = product $ zipWith (\p e -> (p^(3*e + 3) - 1) `div` (p^3 - 1))
                      (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Jun 30 2013

[DivisorSigma(3,n): n in [1..40]]; // Bruno Berselli, Apr 10 2013

seq(numtheory:-sigma[3](n),n=1..100); # Robert Israel, Feb 05 2016

Table[DivisorSigma[3,n],{n,100}] (* corrected by T. D. Noe, Mar 22 2009 *)

makelist(divsum(n,3),n,1,100); /* Emanuele Munarini, Mar 26 2011 */

N=99; q='q+O('q^N);
Vec(sum(n=1,N,n^3*q^n/(1-q^n))) /* Joerg Arndt, Feb 04 2011 */

{a(n) = if( n<1, 0, sumdiv(n, d, d^3))}; /* Michael Somos, Jan 07 2017 */

from sympy import divisor_sigma
def a(n): return divisor_sigma(n, 3)
print([a(n) for n in range(1, 43)]) # Michael S. Branicky, Jan 09 2021

[sigma(n, 3) for n in range(1, 40)]  # Zerinvary Lajos, Jun 04 2009

Multiplicative with a(p^e) = (p^(3e+3)-1)/(p^3-1). - David W. Wilson, Aug 01 2001
Dirichlet g.f. zeta(s)*zeta(s-3). - R. J. Mathar, Mar 04 2011
G.f.: sum(k>=1, k^3*x^k/(1-x^k)). - Benoit Cloitre, Apr 21 2003
Equals A051731 * [1, 8, 27, 64, 125, ...] = A127093 * [1, 4, 9, 16, 25, ...]. - Gary W. Adamson, Nov 02 2007
L.g.f.: -log(Product_{j>=1} (1-x^j)^(j^2)) = (1/1)*z^1 + (9/2)*z^2 + (28/3)*z^3 + (73/4)*z^4 + ... + (a(n)/n)*z^n + ... - Joerg Arndt, Feb 04 2011
a(n) = Sum{d|n} tau_{-2}^d*J_3(n/d), where tau_{-2} is A007427 and J_3 is A059376. - Enrique Pérez Herrero, Jan 19 2013
a(n) = A004009(n)/240. - Artur Jasinski, Sep 06 2016. See, e.g., Hardy, p. 166, (10.5.6), with Q = E_4, and with present offset 0. - Wolfdieter Lang, Jan 31 2017
8*a(n) = sum of cubes of even divisors of 2*n. - Wolfdieter Lang, Jan 07 2017
G.f.: Sum_{n >= 1} x^n*(1 + 4*x^n + x^(2*n))/(1 - x^n)^4. - Peter Bala, Jan 11 2021
Faster converging g.f.: Sum_{n >= 1} q^(n^2)*( n^3 + ((n + 1)^3 - 3*n^3)*q^n + (4 - 6*n^2)*q^(2*n) + (3*n^3 - (n - 1)^3)*q^(3*n) - n^3*q^(4*n) )/(1 - q^n)^4 - apply the operator x*d/dx three times to equation 5 in Arndt and then set x = 1. - Peter Bala, Jan 21 2021
a(n) = Sum_{1 <= i, j, k <= n} tau(gcd(i, j, k, n)) = Sum_{d divides n} tau(d)* J_3(n/d), where the divisor function tau(n) = A000005(n) and the Jordan totient function J_3(n) = A059376(n). - Peter Bala, Jan 22 2024

A023900 Dirichlet inverse of Euler totient function (A000010).

Original entry on oeis.org

1, -1, -2, -1, -4, 2, -6, -1, -2, 4, -10, 2, -12, 6, 8, -1, -16, 2, -18, 4, 12, 10, -22, 2, -4, 12, -2, 6, -28, -8, -30, -1, 20, 16, 24, 2, -36, 18, 24, 4, -40, -12, -42, 10, 8, 22, -46, 2, -6, 4, 32, 12, -52, 2, 40, 6, 36, 28, -58, -8, -60, 30, 12, -1, 48, -20, -66, 16, 44, -24, -70, 2, -72, 36, 8, 18, 60, -24, -78, 4, -2

Offset: 1

Olivier Gérard

Also called reciprocity balance of n.
Apart from different signs, same as Sum_{d divides n} core(d)*mu(n/d), where core(d) (A007913) is the squarefree part of d. - Benoit Cloitre, Apr 06 2002
Main diagonal of A191898. - Mats Granvik, Jun 19 2011

			x - x^2 - 2*x^3 - x^4 - 4*x^5 + 2*x^6 - 6*x^7 - x^8 - 2*x^9 + 4*x^10 - ...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 37.
D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc. Boston, MA, 1976, p. 125.

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
G. P. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 (516) (2005) 403-408.
K. Dohmen and M. Trinks, An Abstraction of Whitney's Broken Circuit Theorem, arXiv preprint arXiv:1404.5480 [math.CO], 2014.
R. Kemp, On the number of words in the language {w in Sigma* | w = w^R }^2, Discrete Math., 40 (1982), 225-234.
Paul W. Oxby, A Function Based on Chebyshev Polynomials as an Alternative to the Sinc Function in FIR Filter Design, arXiv:2011.10546 [eess.SP], 2020.
László Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv preprint arXiv:1310.7053 [math.NT], 2013.

Cf. A000010, A007913, A023898, A117209, A134842.
Moebius transform is A055615.
Cf. A027748, A173557 (gives the absolute values), A295876.
Cf. A253905 (Dgf at s=3).

a023900 1 = 1
a023900 n = product $ map (1 -) $ a027748_row n
-- Reinhard Zumkeller, Jun 01 2015

A023900 := n -> mul(1-i,i=numtheory[factorset](n)); # Peter Luschny, Oct 26 2010

a[ n_] := If[ n < 1, 0, Sum[ d MoebiusMu @ d, { d, Divisors[n]}]] (* Michael Somos, Jul 18 2011 *)
Array[ Function[ n, 1/Plus @@ Map[ #*MoebiusMu[ # ]/EulerPhi[ # ]&, Divisors[ n ] ] ], 90 ]
nmax = 81; Drop[ CoefficientList[ Series[ Sum[ MoebiusMu[k] k x^k/(1 - x^k), {k, 1, nmax} ], {x, 0, nmax} ], x ], 1 ] (* Stuart Clary, Apr 15 2006 *)
t[n_, 1] = 1; t[1, k_] = 1; t[n_, k_] :=  t[n, k] = If[n < k, If[n > 1 && k > 1, Sum[-t[k - i, n], {i, 1, n - 1}], 0], If[n > 1 && k > 1, Sum[-t[n - i, k], {i, 1, k - 1}], 0]]; Table[t[n, n], {n, 36}] (* Mats Granvik, Robert G. Wilson v, Jun 25 2011 *)
Table[DivisorSum[m, # MoebiusMu[#] &], {m, 90}] (* Jan Mangaldan, Mar 15 2013 *)
f[p_, e_] := (1 - p); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 14 2020 *)

{a(n) = direuler( p=2, n, (1 - p*X) / (1 - X))[n]}

{a(n) = if( n<1, 0, sumdiv( n, d, d * moebius(d)))} /* Michael Somos, Jul 18 2011 */

a(n)=sumdivmult(n,d, d*moebius(d)) \\ Charles R Greathouse IV, Sep 09 2014

from sympy import divisors, mobius
def a(n): return sum([d*mobius(d) for d in divisors(n)]) # Indranil Ghosh, Apr 29 2017

from math import prod
from sympy import primefactors
def A023900(n): return prod(1-p for p in primefactors(n)) # Chai Wah Wu, Sep 08 2023

;; With memoization-macro definec.
(definec (A023900 n) (if (= 1 n) 1 (* (- 1 (A020639 n)) (A023900 (A028234 n))))) ;; Antti Karttunen, Nov 28 2017

a(n) = Sum_{ d divides n } d*mu(d) = Product_{p|n} (1-p).
a(n) = 1 / (Sum_{ d divides n } mu(d)*d/phi(d)).
Dirichlet g.f.: zeta(s)/zeta(s-1). - Michael Somos, Jun 04 2000
a(n+1) = det(n+1)/det(n) where det(n) is the determinant of the n X n matrix M_(i, j) = i/gcd(i, j) = lcm(i, j)/j. - Benoit Cloitre, Aug 19 2003
a(n) = phi(n)*moebius(A007947(n))*A007947(n)/n. Logarithmic g.f.: Sum_{n >= 1} a(n)*x^n/n = log(F(x)) where F(x) is the g.f. of A117209 and satisfies: 1/(1-x) = Product_{n >= 1} F(x^n). - Paul D. Hanna, Mar 03 2006
G.f.: A(x) = Sum_{k >= 1} mu(k) k x^k/(1 - x^k) where mu(k) is the Moebius (Mobius) function, A008683. - Stuart Clary, Apr 15 2006
G.f.: A(x) is x times the logarithmic derivative of A117209(x). - Stuart Clary, Apr 15 2006
Row sums of triangle A134842. - Gary W. Adamson, Nov 12 2007
G.f.: x/(1-x) = Sum_{n >= 1} a(n)*x^n/(1-x^n)^2. - Paul D. Hanna, Aug 16 2008
a(n) = phi(rad(n)) *(-1)^omega(n) = A000010(A007947(n)) *(-1)^A001221(n). - Enrique Pérez Herrero, Aug 24 2010
a(n) = Product_{i = 2..n} (1-i)^( (pi(i)-pi(i-1)) * floor( (cos(n*Pi/i))^2 ) ), where pi = A000720, Pi = A000796. - Wesley Ivan Hurt, May 24 2013
a(n) = -limit of zeta(s)*(Sum_{d divides n} moebius(d)/exp(d)^(s-1)) as s->1 for n>1. - Mats Granvik, Jul 31 2013
a(n) = Sum_{d divides n} mu(d)*rad(d), where rad is A007947. - Enrique Pérez Herrero, May 29 2014
Conjecture for n>1: Let n = 2^(A007814(n))*m = 2^(ruler(n))*odd_part(n), where m = A000265(n), then a(n) = (-1)^(m=n)*(0+Sum_{i=1..m and gcd(i,m)=1} (4*min(i,m-i)-m)) = (-1)^(m1} (4*min(i,m-i)-m)). - I. V. Serov, May 02 2017
a(n) = (-1)^A001221(n) * A173557(n). - R. J. Mathar, Nov 02 2017
a(1) = 1; for n > 1, a(n) = (1-A020639(n)) * a(A028234(n)), because multiplicative with a(p^e) = (1-p). - Antti Karttunen, Nov 28 2017
a(n) = 1 - Sum_{d|n, d > 1} d*a(n/d). - Ilya Gutkovskiy, Apr 26 2019
From Richard L. Ollerton, May 07 2021: (Start)
For n>1, Sum_{k=1..n} a(gcd(n,k)) = 0.
For n>1, Sum_{k=1..n} a(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)) = 0. (End)
a(n) = rad(n)*(-1)^omega(n)*phi(n)/n = A062953(n)*A000010(n)/n. - Amrit Awasthi, Jan 30 2022
a(n) = mu(n)*phi(n) = A008683(n)*A000010(n) whenever n is squarefree. - Amrit Awasthi, Feb 03 2022
From Peter Bala, Jan 24 2024: (Start)
a(n) = Sum_{d divides n} core(d)*mu(d). Cf. Comment by Benoit Cloitre, dated Apr 06 2002.
a(n) = Sum_{d|n, e|n}  n/gcd(d, e) * mu(n/d) * mu(n/e) (the sum is a multiplicative function of n by Tóth, and takes the value 1 - p for n = p^e, a prime power). (End)
From Peter Bala, Feb 01 2024: (Start)
G.f. Sum_{n >= 1} (2*n-1)*moebius(2*n-1)*x^(2*n-1)/(1 + x^(2n-1)).
a(n) = (-1)^(n+1) * Sum_{d divides n, d odd} d*moebius(d). (End)

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A003963 Fully multiplicative with a(p) = k if p is the k-th prime.

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A008472 Sum of the distinct primes dividing n.

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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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A003557 n divided by largest squarefree divisor of n; if n = Product p(k)^e(k) then a(n) = Product p(k)^(e(k)-1), with a(1) = 1.

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A001615 Dedekind psi function: n * Product_{p|n, p prime} (1 + 1/p).

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