A000001 -id:A000001 - 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

A072410 Number of iterations of the map k -> A000001(k) needed to reach 1 starting at n, or -1 if no such number exists.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 03 2008

nonn

The old entry with this sequence number was a duplicate of A052409.
It appears that a(n) is the number of times n appears in A142978, excluding the first column of infinitely many 1's. - Ron Wolf, Dec 16 2020
Preceding comment is incorrect. The first counterexample is a(19) = 1, whereas 19 appears twice in A142978. - Eric M. Schmidt, Mar 22 2021

			Conway et al. remark that every number less than 2048 reaches 1 after at most 5 steps and give the following examples:
672 -> 1280 -> 1116461 -> 1
1024 -> 49487367289 -> 1
720 -> 840 -> 186 -> 6 -> 2 -> 1
320 -> 1640 -> 68 -> 5 -> 1
384 -> 20169 -> 67 -> 1
128 -> 2328 -> 64 -> 267 -> 1
960 -> 11394 -> 60 -> 13 -> 1
864 -> 4725 -> 51 -> 1
1344 -> 11720 -> 49 -> 2 -> 1
1440 -> 5958 -> 16 -> 14 -> 2 -> 1
1248 -> 1460 -> 15 -> 1
256 -> 56092 -> 11 -> 1
1728 -> 47937 -> 6 -> 2 -> 1
512 -> 10494213 -> 5 -> 1
1536 -> 408641062 -> 4 -> 2 -> 1
1664 -> 21507 -> 2 -> 1
1280 -> 1116461 -> 1

Eric M. Schmidt, Table of n, a(n) for n = 1..2047 [using data from example and A000001. a(1024) corrected by _Jinyuan Wang_, Jun 26 2022]
John H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica.

Cf. A000001, A066952 (indices of records).

A090052 Group-abundant numbers: n such that the number of groups of order n (A000001) exceeds n.

Original entry on oeis.org

32, 48, 64, 96, 128, 144, 160, 192, 256, 288, 320, 384, 432, 448, 480, 512, 576, 640, 648, 672, 704, 720, 768, 800, 832, 864, 896, 960, 1024, 1088, 1152, 1216, 1248, 1280, 1296, 1344, 1408, 1440, 1458, 1536, 1600, 1664, 1728, 1792, 1920, 1944, 2016, 2048, 2112, 2160, 2176, 2187, 2240, 2304, 2400, 2432, 2496, 2560, 2592, 2688, 2816, 2880, 2916, 2944

Offset: 1

J. H. Conway, Jan 21 2004

nonn

It seems fairly certain that 1 is the only group-perfect number and that almost all numbers are group-deficient. However, all that is known at present is that all squarefree numbers except 1 are group-deficient.

			32 is in the sequence because A000001(32) = 51 > 32, 48 is in the sequence because A000001(48) = 52 > 48 and since the exact number of groups of order 2048 that have exponent-2 class 2 is 1774274116992170 then 2048 is in the sequence because A000001(2048) > 1774274116992170 > 2048. - _Muniru A Asiru_, Nov 26 2017

Alex Meiburg, Table of n, a(n) for n = 1..178
J. H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica.

Cf. A000001.

A090052 := Filtered([1..2015], n -> NumberSmallGroups(n) > n); # Muniru A Asiru, Nov 30 2017

1944, 2016, and 2048 added by Eric M. Schmidt, Aug 02 2012
a(49)-a(52) from Muniru A Asiru, Nov 26 2017
a(53)-a(178) from Alex Meiburg, Dec 30 2017, partially using https://github.com/olexandr-konovalov/gnu/

A185291 Dirichlet convolution of A000001 with itself.

Original entry on oeis.org

1, 2, 2, 5, 2, 6, 2, 14, 5, 6, 2, 18, 2, 6, 4, 42, 2, 18, 2, 18, 6, 6, 2, 58, 5, 6, 14, 16, 2, 18, 2, 150, 4, 6, 4, 60, 2, 6, 6, 56, 2, 24, 2, 16, 10, 6, 2, 202, 5, 18, 4, 18, 2, 58, 6, 52, 6, 6, 2, 66, 2, 6, 16, 717, 4, 18, 2, 18, 4, 18, 2, 218, 2, 6, 12

Offset: 1

Ben Branman, Jan 25 2012

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Dirichlet Series Generating Function.
Eric Weisstein's World of Mathematics, Finite Group.

con[f_, g_, x_] := Sum[f[k] g[x/k], {k, Divisors[x]}];Table[con[FiniteGroupCount, FiniteGroupCount, x], {x, 1, 100}]

DC(a,b)=vector(min(#a,#b),n,sumdiv(n,d,a[d]*b[n/d]))
A185291=DC(A1,A1) /* where A1 is a vector of the first values of A000001 */ - M. F. Hasler, Jan 26 2012

Data and Mmca code corrected by M. F. Hasler, Jan 26 2012

A118712 a(n) = Determinant of n X n circulant matrix whose first row is A000001(1), A000001(2), ..., A000001(n) where A000001(n) = number of groups of order n.

Original entry on oeis.org

1, 0, 0, -5, 6, -16, 9, -134400, 647248, -1711908, 6076067, -85248000, 116477425, -1764364437, 909276004, -522319050599375232, 14313181351994538493, -165893335414907083200, 2939566160282258664451, -5007637771411479278976, 75399747694572065660672

Offset: 1

Jonathan Vos Post, May 20 2006

sign

			a(4) = -5 because of the determinant -5 =
|1,1,1,2|
|2,1,1,1|
|1,2,1,1|
|1,1,2,1|.
a(11) = 6076067 = determinant
|1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1|
|1, 1, 1, 1, 2, 1, 2, 1, 5, 2, 2|
|2, 1, 1, 1, 1, 2, 1, 2, 1, 5, 2|
|2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 5|
|5, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1|
|1, 5, 2, 2, 1, 1, 1, 1, 2, 1, 2|
|2, 1, 5, 2, 2, 1, 1, 1, 1, 2, 1|
|1, 2, 1, 5, 2, 2, 1, 1, 1, 1, 2|
|2, 1, 2, 1, 5, 2, 2, 1, 1, 1, 1|
|1, 2, 1, 2, 1, 5, 2, 2, 1, 1, 1|
|1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 1|.

Eric Weisstein's World of Mathematics, Circulant Matrix.

Cf. A000001, A048954, A052182, A066933, A086459, A086569.

A118712 := n -> DeterminantMat(List([0..n-1], i->List([0..n-1], j->NrSmallGroups(((j-i) mod n)+1)))); # Eric M. Schmidt, Nov 17 2013

a(1) corrected by and more terms from Eric M. Schmidt, Nov 17 2013

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

A142863 Semiprimes n (A001358) for which A000001(n) is 2.

Original entry on oeis.org

4, 6, 9, 10, 14, 21, 22, 25, 26, 34, 38, 39, 46, 49, 55, 57, 58, 62, 74, 82, 86, 93, 94, 106, 111, 118, 121, 122, 129, 134, 142, 146, 155, 158, 166, 169, 178, 183, 194, 201, 202, 203, 205, 206, 214, 218, 219, 226, 237, 253, 254, 262, 274, 278, 289, 291, 298, 301

Offset: 1

N. J. A. Sloane, Oct 03 2008

nonn

Semiprimes p^2 or pq with pT. D. Noe, Oct 08 2008

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. A000001, A001358.

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

A173930 Decimal expansion of the number whose continued fraction expansion is A000001.

Original entry on oeis.org

1, 5, 7, 7, 1, 6, 3, 3, 7, 0, 3, 4, 3, 2, 2, 1, 3, 4, 9, 8, 6, 1, 0, 8, 9, 3, 4, 5, 9, 6, 8, 8, 0, 8, 6, 0, 4, 8, 0, 1, 2, 0, 8, 6, 1, 4, 2, 9, 0, 5, 4, 3, 0, 3, 4, 9, 0, 8, 9, 3, 4, 5, 0, 2, 0, 0, 0, 9, 8, 8, 5, 3, 1, 2, 2, 8, 7, 6

Offset: 1

Schmieding-Forland (Kerranti(AT)gmail.com), Mar 02 2010

			1.577163370... = 1/(1+1/(1+1/(1+1/(2+... +1/(A000001(i)+...

Charles R Greathouse IV, Table of n, a(n) for n = 1..2500 (first 400 terms from Muniru A Asiru)

Cf. A000001.

Digits := 80 ; read("transforms3") ;
L := BFILETOLIST("b000001.txt") ; for n from 80 to 200 do x := numtheory[nthconver](L,n) ; x := evalf(x) ; print(x) ; end do : # R. J. Mathar, Mar 05 2010

FiniteGroupCount[Range[80]] // FromContinuedFraction // N[#, 80]& // RealDigits // First (* Jean-François Alcover, Apr 06 2020 *)

Keyword:cons added, more digits appended by R. J. Mathar, Mar 05 2010

A184335 Numbers k such that A000001(k) > 1 and A000001(k) | k.

Original entry on oeis.org

4, 6, 10, 14, 20, 22, 26, 28, 34, 38, 42, 44, 46, 50, 58, 62, 74, 75, 76, 78, 82, 86, 90, 92, 94, 106, 114, 118, 122, 124, 125, 134, 135, 142, 146, 158, 166, 172, 178, 186, 188, 194, 202, 204, 206, 214, 218, 222, 226, 236, 254, 258

Offset: 1

Jonathan Vos Post, Feb 13 2011

Numbers k such that the number of groups of order k (when greater than 1) divides the group order k. I require a proper divisor > 1 because trivially for any p there is 1 group (the cyclic group) of order p, and 1 | p. Even semiprimes A100484 are a proper subset, because when k = p*q for primes p and q, then A000001(k) = 1 if gcd(p,q-1) = 1, 2 if gcd(p,q-1) = p, and (p < q).

			a(11) = 42 is in the sequence because there are 6 nonisomorphic groups of order 42, and 42/6 = 7.
a(18) = 75 is the first odd value, because there are 5 nonisomorphic groups of order 75, and 75/5 = 15. The next odd value is 125.

Nathaniel Johnston, Table of n, a(n) for n = 1..324

Cf. A000001, A100484.

a(44) - a(52) from Nathaniel Johnston, Apr 26 2011

A208769 Dirichlet inverse of the finite group count (A000001).

Original entry on oeis.org

1, -1, -1, -1, -1, 0, -1, -2, -1, 0, -1, 0, -1, 0, 1, -5, -1, 0, -1, 0, 0, 0, -1, -1, -1, 0, -2, 1, -1, 0, -1, -23, 1, 0, 1, 0, -1, 0, 0, 0, -1, 0, -1, 1, 1, 0, -1, -8, -1, 0, 1, 0, -1, -1, 0, -1, 0, 0, -1, 0, -1, 0, 1, -159, 1, 0, -1, 0, 1, 0, -1, -6, -1, 0, 0, 1, 1, 0, -1, -10, -6, 0, -1, 1, 1, 0, 1, 0, -1, 0, 1, 1, 0, 0, 1, -60, -1, 0, 1, -2, -1, 0, -1, 0

Offset: 1

Ben Branman, Mar 01 2012

Antti Karttunen, Table of n, a(n) for n = 1..2047 (computed from the b-file of A000001; a(1024) corrected by Andrey Zabolotskiy)
Wikipedia, Dirichlet convolution (Dirichlet inverse)

Cf. A129667 (abelian version), A000688, A000001, A185291.

a[1] = 1; a[n_] := a[n] = -Sum[FiniteGroupCount[n/k] a[k], {k, Drop[Divisors[n], -1]}]; Table[a[n], {n, 100}]

v000001 = readvec("b000001_to.txt"); \\ Prepared with gawk ' { print $2 } ' from the b-file of A000001.
A000001(n) = v000001[1+n];
A208769(n) = if(1==n,1,-sumdiv(n,d,if(dA000001(n/d)*A208769(d),0))); \\ Antti Karttunen, Jun 13 2018, after Mathematica-code

a(1) = 1; for n > 1, a(n) = -Sum_{d|n, dA000001(n/d)*a(d). - Antti Karttunen, Jun 13 2018

More terms from Antti Karttunen, Jun 13 2018

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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A072410 Number of iterations of the map k -> A000001(k) needed to reach 1 starting at n, or -1 if no such number exists.

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A090052 Group-abundant numbers: n such that the number of groups of order n (A000001) exceeds n.

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A185291 Dirichlet convolution of A000001 with itself.

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A118712 a(n) = Determinant of n X n circulant matrix whose first row is A000001(1), A000001(2), ..., A000001(n) where A000001(n) = number of groups of order n.

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

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A142863 Semiprimes n (A001358) for which A000001(n) is 2.

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A173930 Decimal expansion of the number whose continued fraction expansion is A000001.

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