A050384 -id:A050384 - cp's OEIS frontend

A003277 Cyclic numbers: k such that k and phi(k) are relatively prime; also k such that there is just one group of order k, i.e., A000001(k) = 1.

1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 123, 127, 131, 133, 137, 139, 141, 143, 145, 149, 151, 157, 159, 161, 163, 167, 173

Offset: 1

N. J. A. Sloane and Richard Stanley

Except for a(2)=2, all the terms in the sequence are odd. This is because of the existence of a non-cyclic dihedral group of order 2n for each n>1. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 09 2001
Also gcd(n, A051953(n)) = 1. - Labos Elemer
n such that x^n == 1 (mod n) has no solution 2 <= x <= n. - Benoit Cloitre, May 10 2002
There is only one group (the cyclic group of order n) whose order is n. - Gerard P. Michon, Jan 08 2008  [This is a 1947 result of Tibor Szele. - Charles R Greathouse IV, Nov 23 2011]
Any divisor of a Carmichael number (A002997) must be odd and cyclic. Conversely, G. P. Michon conjectured (c. 1980) that any odd cyclic number has at least one Carmichael multiple (if the conjecture is true, each of them has infinitely many such multiples). In 2007, Michon & Crump produced explicit Carmichael multiples of all odd cyclic numbers below 10000 (see link, cf. A253595). - Gerard P. Michon, Jan 08 2008
Numbers n such that phi(n)^phi(n) == 1 (mod n). - Michel Lagneau, Nov 18 2012
Contains A000040, and all members of A006094 except 6. - Robert Israel, Jul 08 2015
Number m such that n^n == r (mod m) is solvable for any r. - David W. Wilson, Oct 01 2015
Numbers m such that A074792(m) = m + 1. - Thomas Ordowski, Jul 16 2017
Squarefree terms of A056867 (see McCarthy link p. 592 and similar comment with "cubefree" in A051532). - Bernard Schott, Mar 24 2022

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
J. S. Rose, A Course on Group Theory, Camb. Univ. Press, 1978, see p. 7.
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
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Max Alekseyev, Michon's conjecture (Open Problem Garden, Aug. 2007).
Joel E. Cohen, Conjectures about Primes and Cyclic Numbers, arXiv:2508.08335 [math.NT], 2025. See pp. 1, 2, 30.
Keith Conrad, When are all groups of order n cyclic?, University of Connecticut, 2019.
Peter J. Dukes and Joanna Niezen, Pairwise balanced designs of dimension three, Australasian Journal Of Combinatorics, Volume 61(1) (2015), pages 98-113.
Paul Erdős, Some asymptotic formulas in number theory, J. Indian Math. Soc. (N.S.) 12 (1948), pp. 75-78.
Jose M. Grau, Manuel Rodríguez, A. Oller-Marcen, and Daniel Sadornil, Fermat test with gaussian base and Gaussian pseudoprimes, arXiv preprint arXiv:1401.4708 [math.NT], 2014.
Donald J. McCarthy, A survey of partial converses to Lagrange's theorem on finite groups, Transactions of the New York Academy of Sciences, Vol. 33, No. 6, Series II (1971), pp. 586-594. See page 592.
Gérard P. Michon, Carmichael Divisors
Gérard P. Michon and J. K. Crump, Carmichael Multiples of Odd Cyclic Numbers (up to 10000)
Jonathan Pakianathan and Krishnan Shankar, Nilpotent Numbers, Amer. Math. Monthly, 107, August-September 2000, 631-634.
R. P. Stanley, Letter to N. J. A. Sloane, c. 1991
T. Szele, Über die endlichen Ordnungszahlen, zu denen nur eine Gruppe gehört, Commentarii Mathematici Helvetici 20 (1947), pp. 265-267.
Index entries for sequences related to groups

Subsequence of A051532. Intersection of A056867 and A005117.
Cf. A000010, A008966, A009195, A050384 (the same sequence but with the primes removed). Also A000001(a(n)) = 1.
Cf. A002997, A006094, A054395, A055561, A054396, A054397, A056867, A135850, A249550, A249551, A249552, A249553, A249554, A249555, A036537, A051953, A253595.

import Data.List (elemIndices)
a003277 n = a003277_list !! (n-1)
a003277_list = map (+ 1) $ elemIndices 1 a009195_list
-- Reinhard Zumkeller, Feb 27 2012

[n: n in [1..200] | Gcd(n, EulerPhi(n)) eq 1]; // Vincenzo Librandi, Jul 09 2015

select(t -> igcd(t, numtheory:-phi(t))=1, [$1..1000]); # Robert Israel, Jul 08 2015

Select[Range[175], GCD[#, EulerPhi[#]] == 1 &] (* Jean-François Alcover, Apr 04 2011 *)
Select[Range@175, FiniteGroupCount@# == 1 &] (* Robert G. Wilson v, Feb 16 2017 *)
Select[Range[200],CoprimeQ[#,EulerPhi[#]]&] (* Harvey P. Dale, Apr 10 2022 *)

isA003277(n) = gcd(n,eulerphi(n))==1 \\ Michael B. Porter, Feb 21 2010

# Compare A050384.
def isPrimeTo(n, m): return gcd(n, m) == 1
def isCyclic(n): return isPrimeTo(n, euler_phi(n))
[n for n in (1..173) if isCyclic(n)] # Peter Luschny, Nov 14 2018

n = p_1*p_2*...*p_k (for some k >= 0), where the p_i are distinct primes and no p_j-1 is divisible by any p_i.
A000001(a(n)) = 1.
Erdős proved that a(n) ~ e^gamma n log log log n, where e^gamma is A073004. - Charles R Greathouse IV, Nov 23 2011
A000005(a(n)) = 2^k. - Carlos Eduardo Olivieri, Jul 07 2015
A008966(a(n)) = 1. - Bernard Schott, Mar 24 2022

More terms from Christian G. Bower

A181830 The number of positive integers <= n that are strongly prime to n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 1, 6, 2, 6, 4, 4, 4, 11, 4, 12, 6, 6, 6, 18, 6, 12, 9, 14, 8, 22, 6, 22, 14, 14, 12, 20, 8, 27, 16, 20, 12, 32, 10, 34, 18, 18, 16, 42, 14, 32, 17, 26, 20, 46, 16, 32, 20, 28, 24, 54, 14, 48, 28, 32, 26, 41, 16

Offset: 0

Peter Luschny, Nov 17 2010

k is strongly prime to n if and only if k is relatively prime to n and k does not divide n - 1.
It is conjectured (see Scroggs link) that a(n) is also the number of cardboard braids that work with n slots. - Matthew Scroggs, Sep 23 2017
a(n) is odd if and only if n is in A002522 but n <> 2. - Robert Israel, Jun 20 2018

			a(11) = card({1,2,3,4,5,6,7,8,9,10} - {1,2,5,10}) = card({3,4,6,7,8,9}) = 6.

Robert Israel, Table of n, a(n) for n = 0..10000
Peter Luschny, Strong coprimality
Matthew Scroggs, Braiding, pt. 2. Two results and a conjecture

Cf. A000005, A000010, A002522, A050384, A181831, A181832, A181833, A181834, A181835, A181836.

a[0]=0; a[1]=0; a[n_ /; n > 1] := Select[Range[n], CoprimeQ[#, n] && !Divisible[n-1, #] &] // Length; Table[a[n], {n, 0, 66}] (* Jean-François Alcover, Jun 26 2013 *)

a(n)=if(n<2, 0, eulerphi(n)-numdiv(n-1));
for (i=0, 66, print1(a(i), ", ")) \\ Michel Marcus, May 22 2017

def isstrongprimeto(k, n): return not(k.divides(n - 1)) and gcd(k, n) == 1
print([sum(int(isstrongprimeto(k, n)) for k in srange(n+1)) for n in srange(67)])
# Peter Luschny, Dec 03 2023

a(n) = phi(n) - tau(n-1) for n > 1, where phi(n) = A000010(n) and tau(n) = A000005(n).

Corrected a(1) to 0 by Peter Luschny, Dec 03 2023

A074792 Least k > 1 such that k^n == 1 (mod n).

Original entry on oeis.org

2, 3, 4, 3, 6, 5, 8, 3, 4, 9, 12, 5, 14, 13, 16, 3, 18, 5, 20, 3, 4, 21, 24, 5, 6, 25, 4, 13, 30, 11, 32, 3, 34, 33, 36, 5, 38, 37, 16, 3, 42, 5, 44, 21, 16, 45, 48, 5, 8, 9, 52, 5, 54, 5, 16, 13, 7, 57, 60, 7, 62, 61, 4, 3, 66, 23, 68, 13, 70, 29, 72, 5, 74, 73, 16, 37, 78, 17, 80, 3

Offset: 1

Benoit Cloitre, Sep 07 2002

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..10000

a(n) = {A076944(n)}^(1/n).

Cf. A076942, A076943, A076944.

Do[k = 2; While[ !IntegerQ[(k^n - 1)/n], k++ ]; Print[k], {n, 1, 80}] (* Robert G. Wilson v *)

a(n)=if(n<0,0,s=2; while((s^n-1)%n>0,s++); s)

a(n)=my(s=2); while(Mod(s,n)^n-1!=0, s++); return(s) \\ Rémy Sigrist, Apr 02 2017

If p is prime a(p)=p+1 and a(2p)=2p-1; if n is in A050384 a(n)=n+1; if n is in A067945 a(n)=3 etc. It seems that sum(k=1, n, a(k)) is asymptotic to c*n^2 with c=0.2...

A163671 Expansion of Sum_( x^k / (1 - x^(k^2)) ).

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 3, 3, 3, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 2, 4, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 3, 3, 3, 3, 2, 2, 2, 3, 2, 2, 2, 4, 2, 3, 2, 3, 2, 3, 2, 3, 3, 2, 2, 4, 2, 3, 2, 3, 2, 4, 2, 3, 2, 2, 2, 4, 2, 2, 3, 3, 2, 2, 2, 3, 2, 3, 2, 4, 2, 2, 4

Offset: 1

Franklin T. Adams-Watters, Aug 02 2009

nonn

Number of divisors d of n such that n/d == 1 (mod d). Such divisors must be unitary divisors.
Up to n = 10000, the largest value in this sequence is 7; the first occurrence of each integer up to 7 in the sequence is 1, 2, 6, 42, 30, 210, 2310; which except for 42 is the primorial numbers. However, a(30030) = 7.
If n is the product of two distinct primes then a(n) = 2 if and only if there are no nontrivial groups of order n. This relation does not hold if n is the product of 3 or more distinct primes or is not squarefree.
First occurrences: a(1) = 1, a(2) = 2, a(6) = 3, a(42) = 4, a(30) = 5, a(210) = 6, a(2310) = 7, a(87780) = 8, a(53130) = 9, a(7375830) = 10, a(172320330) = 11. - Charles R Greathouse IV, Jun 01 2016

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Unitary Divisor.

Cf. A000005, A001227, A002110, A003277, A050384, A006881, A034444.

Table[Sum[(Floor[(k^2 - k)/n] - Floor[(k^2 - k - 1)/n])*(Floor[n/k] - Floor[(n - 1)/k]), {k, 1, n}], {n, 1, 50}] (* G. C. Greubel, Aug 02 2017 *)

al(n)=local(m);m=sqrtint(n);Vec(sum(k=1,m,(x^k+x*O(x^n))/(1-x^(k^2))+x^(m+1)/(1-x)))

a(n)=local(d,r);r=0;d=divisors(n);for(k=1,#d,if((n-d[k])%d[k]^2==0,r++));r

a(n)=sumdiv(n,d,(n-d)%d^2==0) \\ Charles R Greathouse IV, Jun 01 2016

a(n) = Sum_{k=1..n} (floor((k^2-k)/n) - floor((k^2-k-1)/n))*(floor(n/k) - floor((n-1)/k)). - Anthony Browne, Jun 01 2016

2 <= a(n) <= 2*omega(n) for n > 1. In particular a(p^e) = 2 for each prime p and each e > 0. - Charles R Greathouse IV, Jun 01 2016

A350344 Composite k such that k^2 is an abelian order.

Original entry on oeis.org

35, 65, 77, 85, 115, 119, 133, 143, 161, 185, 187, 209, 215, 217, 221, 235, 247, 259, 265, 299, 319, 323, 329, 335, 341, 365, 371, 377, 391, 403, 407, 413, 415, 427, 437, 451, 469, 481, 485, 493, 511, 515, 517, 527, 533, 535, 551, 553, 559, 565, 583, 589, 595

Offset: 1

Jianing Song, Dec 25 2021

nonn

Numbers k such that k^2 is an abelian order with at least 4 groups.
Number of the form p_1*p_2*...*p_r where r > 1, the p_i are distinct primes and no (p_j)^2-1 is divisible by any p_i.
The smallest number k such that k^2 is an abelian order with at least 8 groups is A350340(3) = 595.
No term can be divisible by 2 or 3.

			For primes p, q, if p^2 !== 1 (mod q), q^2 !== 1 (mod p), then p*q is a term since every group of that order is abelian. Such group is isomorphic to C_{p^2*q^2}, C_p X C_{p*q^2}, C_q X C_{p^2*q} or C_{p*q} X C_{p*q}.
95 is not a term since 95^2 = 5^2 * 19^2 is not an abelian order. Note that 95 itself is a cyclic number.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A051532 (abelian orders), A050384, A350340.

Equals A350342 \ ({1} U A000040).

isA051532(n) = my(f=factor(n), v=vector(#f[, 1])); for(i=1, #v, if(f[i, 2]>2, return(0), v[i]=f[i, 1]^f[i, 2])); for(i=1, #v, for(j=i+1, #v, if(v[i]%f[j, 1]==1 || v[j]%f[i, 1]==1, return(0)))); 1 \\ Charles R Greathouse IV's program for A051532
isA350344(n) = (n>1) && !isprime(n) && isA051532(n^2)

A350345(n) = a(n)^2.

A350345 Squares of composite numbers k that are abelian orders.

Original entry on oeis.org

1225, 4225, 5929, 7225, 13225, 14161, 17689, 20449, 25921, 34225, 34969, 43681, 46225, 47089, 48841, 55225, 61009, 67081, 70225, 89401, 101761, 104329, 108241, 112225, 116281, 133225, 137641, 142129, 152881, 162409, 165649, 170569, 172225, 182329, 190969

Offset: 1

Jianing Song, Dec 25 2021

nonn

Square numbers k that are abelian orders with at least 4 groups.
Number of the form (p_1*p_2*...*p_r)^2 where r > 1, the p_i are distinct primes and no (p_j)^2-1 is divisible by any p_i.
The smallest square number k that is an abelian order with at least 8 groups is A350341(3) = 354025.
No term can be divisible by 2 or 3.

			For primes p, q, if p^2 !== 1 (mod q), q^2 !== 1 (mod p), then p^2*q^2 is a term since every group of that order is abelian. Such group is isomorphic to C_{p^2*q^2}, C_p X C_{p*q^2}, C_q X C_{p^2*q} or C_{p*q} X C_{p*q}.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A051532 (abelian orders), A050384, A350341.
Equals A350343 \ ({1} U A001248).
A350323 is a subsequence.

isA051532(n) = my(f=factor(n), v=vector(#f[, 1])); for(i=1, #v, if(f[i, 2]>2, return(0), v[i]=f[i, 1]^f[i, 2])); for(i=1, #v, for(j=i+1, #v, if(v[i]%f[j, 1]==1 || v[j]%f[i, 1]==1, return(0)))); 1 \\ Charles R Greathouse IV's program for A051532
isA350345(n) = issquare(n) && (n>1) && !isprime(sqrtint(n)) && isA051532(n^2)

a(n) = A350344(n)^2.

A386234 Number of good involutions of all nontrivial core quandles of order n.

Original entry on oeis.org

1, 4, 1, 3, 1, 72, 2, 3, 1, 31, 1, 3, 1, 10856, 1, 7, 1, 47, 2, 3, 1

Offset: 3

Luc Ta, Jul 21 2025

A good involution f of a quandle Q is an involution that commutes with all inner automorphisms and satisfies the identity f(y)(x) = y^-1(x). We call the pair (Q,f) a symmetric quandle. A symmetric quandle isomorphism is a quandle isomorphism that intertwines good involutions.

A core quandle Core(G) is a group G viewed as a kei (i.e., involutory quandle) under the operation g(h) = g*h^-1*g. Note that Core(G) is nontrivial if and only if exp(G) > 2.

			For n = 4 the only nontrivial core quandle is the dihedral quandle R4 = Core(Z/4Z) of order 4. It is well-known (see Thm. 3.2 of Kamada and Oshiro) that R4 has exactly four good involutions. Hence a(4) = 4.
For n = 6 the only nontrivial core quandles are Core(S3) and R6 = Core(Z/6Z), which have one and two good involutions, respectively. Hence a(6) = 3.

Seiichi Kamada, Quandles with good involutions, their homologies and knot invariants, Intelligence of Low Dimensional Topology 2006, World Scientific Publishing Co. Pte. Ltd., 2007, 101-108.

Seiichi Kamada and Kanako Oshiro, Homology groups of symmetric quandles and cocycle invariants of links and surface-links, Trans. Amer. Math. Soc., 362 (2010), no. 10, 5501-5527.
Lực Ta, Good involutions of conjugation subquandles, arXiv:2505.08090 [math.GT], 2025. See Table 3.
Lực Ta, Symmetric-Rack-Classification, GitHub, 2025.
Index entries for sequences related to quandles and racks

Cf. A000001, A386233, A181770, A178432, A386231, A386232.

GAP
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See Ta, GitHub link
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Let n > 2. Then Ta, Cor. 7.17 implies the following. If n appears in A000040 or A050384, then a(n) = 1. If n appears in A221048, then a(n) = 2. If n > 4 and n appears in A100484, then a(n) = 3.

A052106 a(n) = lcm(n, n - phi(n)).

Original entry on oeis.org

0, 2, 3, 4, 5, 12, 7, 8, 9, 30, 11, 24, 13, 56, 105, 16, 17, 36, 19, 60, 63, 132, 23, 48, 25, 182, 27, 112, 29, 330, 31, 32, 429, 306, 385, 72, 37, 380, 195, 120, 41, 210, 43, 264, 315, 552, 47, 96, 49, 150, 969, 364, 53, 108, 165, 224, 399, 870, 59, 660, 61, 992, 189

Offset: 1

Labos Elemer, Jan 20 2000

nonn

See also A009195, A003277, A050384 when totient and cototient give results identical to each other. This sequence is not identical to A009262.

a(n) = n iff n is in A246655. - Ivan Neretin, May 29 2016

			For n=255, phi(n)=128, cototient(255) = 255 - 128 = 127, a(255) = lcm(255,127) = 32385, while A009262(255) = lcm(255,phi(255)) = 128*255 = 32640;
for n=72, phi(72)=24, A051953(72) = 72 - 24 = 48, a(72) = lcm(72,48) = 144, while A009262(72) = lcm(72,24) = 72.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000010, A009262, A051953.

Table[LCM[n, n - EulerPhi[n]], {n, 63}] (* Ivan Neretin, May 29 2016 *)

a(n) = lcm(n, A051953(n)).

A142862 Semiprimes n (A001358) for which A000001(n) is 1.

Original entry on oeis.org

15, 33, 35, 51, 65, 69, 77, 85, 87, 91, 95, 115, 119, 123, 133, 141, 143, 145, 159, 161, 177, 185, 187, 209, 213, 215, 217, 221, 235, 247, 249, 259, 265, 267, 287, 295, 299, 303, 319, 321, 323, 329, 335, 339, 341, 365, 371, 377, 391, 393, 395, 403, 407, 411

Offset: 1

N. J. A. Sloane, Oct 03 2008

nonn

Semiprimes pq with pT. D. Noe, Oct 08 2008

D. S. Dummit and R. M. Foote, Abstract Algebra, Wiley, 3rd Edition, 2003, page 135.

T. D. Noe, Table of n, a(n) for n=1..1000
John H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica.

Cf. A050384, A003277. [Franklin T. Adams-Watters, Feb 27 2009]

Select[Select[Range[1000],FactorInteger[#][[All, 2]] == {1, 1} &], !
Divisible[FactorInteger[#][[2, 1]] - 1, FactorInteger[#][[1, 1]]] &] (* Geoffrey Critzer, Nov 07 2015 *)

More terms from R. J. Mathar, Oct 04 2008

A259172 Numbers in A259145 that are neither prime nor semiprime.

Original entry on oeis.org

561, 595, 1105, 1235, 1245, 1495, 1547, 1885, 2405, 2555, 2717, 2849, 3115, 3495, 3655, 3657, 3689, 3815, 4521, 4795, 4945, 5035, 5385, 5395, 5453, 5457, 5709, 5865, 6083, 6141, 6251, 6285, 6365, 6391, 6501, 6695, 6755, 6969, 7021, 7887, 8113, 8255, 8355

Offset: 1

Carlos Eduardo Olivieri, Jun 19 2015

Regarding the distribution: Let K be the union of primes and semiprimes in A259145. Let S be the set of other terms. The growth rate of the cardinality of S with respect to the cardinality of K is significantly slower. For instance, if we take the first 50000 terms of A259145, about 32.5 percent are contained in S. If we take the first 350000 terms, about 38.2 percent are contained in S.
a(n) that are in A002997 (Carmichael numbers) for a(n) <= 10^6 are 561, 1105, 8911, 10585, 29341, 825265.
a(n) that are in A051015 (Zeisel numbers) for a(n) <= 3*10^6 are 1885, 353977, 2953711.

Carlos Eduardo Olivieri, Table of n, a(n) for n = 1..8906
Eric Naslund, Integers with a predetermined prime factorization, arXiv:1203.2363 [math.NT], 2012.

Cf. A001221, A001358, A002997, A007053, A051015, A125527, A259145.

Subsequence of A000469, A033942, A050384 (conjuctered).

Select[Range[25000], PrimeQ[#^2 - EulerPhi[#]] && PrimeNu[#] > 2 &]

A001221(a(n)) > 2.

A000005(a(n)) = 2^k, k >= 3.

cp's OEIS Frontend

A003277 Cyclic numbers: k such that k and phi(k) are relatively prime; also k such that there is just one group of order k, i.e., A000001(k) = 1.

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A181830 The number of positive integers <= n that are strongly prime to n.

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A074792 Least k > 1 such that k^n == 1 (mod n).

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A163671 Expansion of Sum_( x^k / (1 - x^(k^2)) ).

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A350344 Composite k such that k^2 is an abelian order.

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A350345 Squares of composite numbers k that are abelian orders.

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A386234 Number of good involutions of all nontrivial core quandles of order n.

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