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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.
(Formerly M0650)

+20
77

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

REFERENCES

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).

LINKS

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).

Keith Conrad, When are all groups of order n cyclic?, University of Connecticut, 2019.

P. J. Dukes and J. 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.

J. M. Grau, A. M. Oller-Marcen, M. Rodríguez, and D. 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)

J. Pakianathan and K. 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

FORMULA

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

MAPLE

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

MATHEMATICA

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 *)

PROG

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

(Haskell)

import Data.List (elemIndices)

a003277 n = a003277_list !! (n-1)

a003277_list = map (+ 1) $ elemIndices 1 a009195_list

-- Reinhard Zumkeller, Feb 27 2012

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

(Sage) # 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

CROSSREFS

Subsequence of A051532.

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.

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane and Richard Stanley

EXTENSIONS

More terms from Christian G. Bower

STATUS

approved

A072410

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

+20
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

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

LINKS

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.

EXAMPLE

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

CROSSREFS

Cf. A000001, A066952 (indices of records).

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Oct 03 2008

STATUS

approved

A090052

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

+20
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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.

LINKS

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.

EXAMPLE

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

PROG

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

CROSSREFS

Cf. A000001.

KEYWORD

nonn

AUTHOR

J. H. Conway, Jan 21 2004

EXTENSIONS

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/

STATUS

approved

A185291

Dirichlet convolution of A000001 with itself.

+20
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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.

MATHEMATICA

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

PROG

(PARI) 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

KEYWORD

nonn

AUTHOR

Ben Branman, Jan 25 2012

EXTENSIONS

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

STATUS

approved

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.

+20
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

Table of n, a(n) for n=1..21.

Eric Weisstein's World of Mathematics, Circulant Matrix.

EXAMPLE

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|.

PROG

(GAP) 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

CROSSREFS

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

KEYWORD

sign

AUTHOR

Jonathan Vos Post, May 20 2006

EXTENSIONS

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

STATUS

approved

A142862

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

+20
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Semiprimes pq with p<q and gcd(p,q-1)=1. - T. D. Noe, Oct 08 2008

REFERENCES

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

LINKS

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.

MATHEMATICA

Select[Select[Range[1000], FactorInteger[#][[All, 2]] == {1, 1} &], !

Divisible[FactorInteger[#][[2, 1]] - 1, FactorInteger[#][[1, 1]]] &] (* Geoffrey Critzer, Nov 07 2015 *)

CROSSREFS

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

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Oct 03 2008

EXTENSIONS

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

STATUS

approved

A142863

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

+20
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Semiprimes p^2 or pq with p<q and gcd(p,q-1)=p. - T. D. Noe, Oct 08 2008

LINKS

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.

CROSSREFS

Cf. A000001, A001358.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Oct 03 2008

EXTENSIONS

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

STATUS

approved

A173930

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

+20
1

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

EXAMPLE

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

MAPLE

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

MATHEMATICA

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

CROSSREFS

Cf. A000001.

KEYWORD

nonn,cons

AUTHOR

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

EXTENSIONS

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

STATUS

approved

A184335

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

+20
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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).

LINKS

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

EXAMPLE

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.

CROSSREFS

Cf. A000001, A100484.

KEYWORD

nonn,easy

AUTHOR

Jonathan Vos Post, Feb 13 2011

EXTENSIONS

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

STATUS

approved

A208769

Dirichlet inverse of the finite group count (A000001).

+20
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,8

LINKS

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)

FORMULA

a(1) = 1; for n > 1, a(n) = -Sum_{d|n, d<n} A000001(n/d)*a(d). - Antti Karttunen, Jun 13 2018

MATHEMATICA

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

PROG

(PARI)

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(d<n, A000001(n/d)*A208769(d), 0))); \\ Antti Karttunen, Jun 13 2018, after Mathematica-code

CROSSREFS

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

KEYWORD

sign,hard

AUTHOR

Ben Branman, Mar 01 2012

EXTENSIONS

More terms from Antti Karttunen, Jun 13 2018

STATUS

approved

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