A002174 -id:A002174 - cp's OEIS frontend

A143408 Number of numbers k such that the reduced totient function psi(k) = A002174(n).

2, 6, 12, 16, 4, 8, 84, 32, 40, 32, 8, 20, 20, 64, 8, 480, 80, 48, 12, 8, 160, 20, 16, 4, 8, 1216, 8, 64, 16, 872, 24, 160, 8, 532, 52, 120, 12, 424, 100, 24, 4, 8, 944, 24, 144, 12, 1912, 272, 8, 16, 276, 24, 64, 144, 1856, 20, 96, 1276, 40, 112, 12, 8, 116, 20, 16, 96, 8

Offset: 1

T. D. Noe, Aug 13 2008

nonn

a(n) is the number of divisors of A143407(n) that are not divisors of A143407(r) for r

			Because A002174(5)=8 and psi(k)=8 for k=32,96,160,480, we have a(5)=4.

T. D. Noe, Table of n, a(n) for n = 1..500
P. J. Cameron and D. A. Preece, Notes on primitive lambda-roots, 2009. See formula in theorem 5.2 (c) p. 8.

Cf. A002322 (reduced totient function), A079612.

a079612(n) = {if (n%2, 2, res = 1; forprime(p=2, n+1, if (!(n % (p-1)), t = valuation(n, p); if (p==2, if (t, res *= p^(t+2)), res *= p^(t+1)););); res;);}
nb(n) = sumdiv(n, d, moebius(n/d)*numdiv(a079612(d)));
lista(nn) = for (n=1, nn, if (nbs = nb(n), print1(nbs, ", "))); \\ Michel Marcus, May 12 2018

A143417 Irregular triangle in which row n gives the k such that the reduced totient function psi(k) = A002174(n).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 24, 5, 10, 15, 16, 20, 30, 40, 48, 60, 80, 120, 240, 7, 9, 14, 18, 21, 28, 36, 42, 56, 63, 72, 84, 126, 168, 252, 504, 32, 96, 160, 480, 11, 22, 33, 44, 66, 88, 132, 264, 13, 26, 35, 39, 45, 52, 65, 70, 78, 90, 91, 104, 105, 112, 117, 130, 140, 144, 156

Offset: 1

T. D. Noe, Aug 13 2008

Row n consists of the divisors of A143407(n) that are not divisors of A143407(r) for rA143408(n).

			1,2; 3,4,6,8,12,24; 5,10,15,16,20,30,40,48,60,80,120,240

T. D. Noe, Rows n=1..20, flattened

Cf. A002322 (reduced totient function).

A270265 Numbers that are values of the totient function (A002202) but not of the reduced totient function (A002174).

Original entry on oeis.org

1936, 3872, 6348, 7744, 9196, 15004, 15488, 18392, 20812, 21160, 22264, 30008, 35332, 36784, 38416, 41624, 42320, 44528, 51304, 58564, 63916, 65596, 70664, 73084, 73568, 76832, 79420, 81796, 82668, 83248, 84100, 84640, 86152, 96040, 102124, 102608, 109384, 110836

Offset: 1

Michel Marcus, Mar 14 2016

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
William D. Banks, John B. Friedlander, Florian Luca, Francesco Pappalardi, and Igor E. Shparlinski, Coincidences in the values of the Euler and Carmichael functions, Acta Arithmetica 122 (2006), 207-234.

Cf. A000010, A002202.
Cf. A002322, A002174.
Cf. A270266.

isA002174(n) = if(n%2, return(n==1)); my(f=factor(n), pe); for(i=1, #f~, if(n%(f[i, 1]-1)==0, next); pe=f[i, 1]^f[i, 2]; forstep(q=2*pe+1, n+1, 2*pe, if(n%(q-1)==0 && isprime(q), next(2))); return(0)); 1 \\ Charles R Greathouse IV at A002174
is(n) = istotient(n) && !isA002174(n); \\ Amiram Eldar, Nov 30 2024

More terms from Jinyuan Wang, Feb 13 2020

A270266 Numbers that are values of the reduced totient function (A002174) but not of the totient function (A002202).

Original entry on oeis.org

90, 174, 230, 234, 246, 290, 308, 318, 364, 390, 410, 414, 450, 510, 516, 530, 534, 572, 594, 638, 644, 666, 678, 680, 702, 714, 728, 740, 770, 804, 830, 846, 870, 890, 902, 948, 954, 1026, 1036, 1074, 1098, 1100, 1110, 1130, 1134, 1146, 1148, 1164, 1166, 1190, 1204

Offset: 1

Michel Marcus, Mar 14 2016

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
William D. Banks, John B. Friedlander, Florian Luca, Francesco Pappalardi, and Igor E. Shparlinski, Coincidences in the values of the Euler and Carmichael functions, Acta Arithmetica 122 (2006), 207-234.

Cf. A000010, A002202.
Cf. A002322, A002174.
Cf. A270265.

isA002174(n) = if(n%2, return(n==1)); my(f=factor(n), pe); for(i=1, #f~, if(n%(f[i, 1]-1)==0, next); pe=f[i, 1]^f[i, 2]; forstep(q=2*pe+1, n+1, 2*pe, if(n%(q-1)==0 && isprime(q), next(2))); return(0)); 1 \\ Charles R Greathouse IV at A002174
is(n) = !istotient(n) && isA002174(n); \\ Amiram Eldar, Dec 01 2024

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

A049225 Values of totient function applied to squarefree numbers; or numbers of form Product (p_i-1) where p_i are distinct primes.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 28, 30, 32, 36, 40, 42, 44, 46, 48, 52, 56, 58, 60, 64, 66, 70, 72, 78, 80, 82, 84, 88, 92, 96, 100, 102, 104, 106, 108, 112, 116, 120, 126, 128, 130, 132, 136, 138, 140, 144, 148, 150, 156, 160, 162, 164, 166, 168, 172, 176

Offset: 1

Labos Elemer

nonn

Numbers m such that m = phi(k) and |mu(k)| = 1.

			8, 120, 48 are terms as totients of the squarefree numbers 15, 143, 210.
54, 110 are not terms since there are no squarefree numbers k such that phi(k) = 54, 110.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Terms of A049200 after ordering and omitting multiple occurrences.

Cf. A000010, A005117, A008683, A013929, A002174, A002202, A049200.

isok(n) = {my(v = invphi(n)); (#v) && (#select(x->issquarefree(x), v));} \\ Michel Marcus, Feb 25 2019

A340521 List of possible orders of automorphism groups of finite groups.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 28, 30, 32, 36, 40, 42, 44, 46, 48, 52, 54, 56, 58, 60

Offset: 1

Des MacHale, Feb 05 2021

The terms shown here match the initial terms of all of A002174, A002202, A049225, but this is a strictly different sequence since it is known that it contains 3^7 = 2187 (which is the smallest odd term greater than 1), whereas for the other three sequences all terms greater than 1 are even.
This is a supersequence of A002202 since |Aut(Z/nZ)| = phi(n). - Jianing Song, Feb 05 2021
John Bray has produced a group G of order 3^2*19 = 171 such that |Aut G| = 1026 = 2.3^3.19. So 1026 is in the present sequence but is not in A002202. So the present sequence contains both odd and even terms not in A002202. - Des MacHale, Feb 10 2021
For more about this problem, see the references in A137315.

D. MacHale and R. Sheehy, Finite groups with few automorphisms, Mathematical Proceedings of the Royal Irish Academy, Vol. 104A, No. 2 (December 2004), 231-238.

Cf. A002174, A002202, A049225, A137315.

A362401 Numbers in the range of A162296, where A162296(n) is the sum of divisors of n that have a square factor larger than 1.

Original entry on oeis.org

0, 4, 9, 12, 16, 24, 25, 27, 28, 32, 36, 48, 49, 54, 56, 60, 72, 75, 79, 80, 96, 100, 108, 112, 117, 120, 121, 124, 126, 128, 144, 147, 150, 152, 162, 168, 169, 176, 180, 183, 192, 196, 199, 200, 216, 224, 240, 248, 252, 268, 270, 272, 288, 289, 294, 296, 300

Offset: 1

Amiram Eldar, Apr 18 2023

nonn

Possible values of A162296 in increasing order.

			0 is a term since A162296(k) = 0 if k is squarefree (A005117).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A162296.

Similar sequences: A078923, A002191, A002202, A002174, A274790.

s[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1)]; s[1] = 0; m = 300; Select[Union[Array[s, m]], # <= m &]

s(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; ((p^(e + 1) - 1)/(p - 1))) -  prod(i = 1, #f~, f[i, 1] + 1);}
lista(kmax) = select(x -> (x < kmax), Set(vector(kmax, k, s(k))))

A270562 a(n) is the largest number m satisfying lambda(m)=n, or zero if there is no solution, where lambda(m) is Carmichael's lambda function A002322(m).

Original entry on oeis.org

2, 24, 0, 240, 0, 504, 0, 480, 0, 264, 0, 65520, 0, 0, 0, 16320, 0, 28728, 0, 13200, 0, 552, 0, 131040, 0, 0, 0, 6960, 0, 171864, 0, 32640, 0, 0, 0, 138181680, 0, 0, 0, 1082400, 0, 151704, 0, 5520, 0, 1128, 0, 4455360, 0, 0, 0, 12720, 0, 86184, 0, 13920, 0, 1416, 0, 6814407600, 0, 0, 0, 65280

Offset: 1

Joerg Arndt, Mar 19 2016

nonn

a(n) is the largest modulus m such that the largest order of any element in the multiplicative group modulo m is n; a(n) is zero if there is no such group with largest order n.
Omitting the zeros gives A143407.
a(n) = 0 if n is not a term of A002174.

Gheorghe Coserea, Table of n, a(n) for n = 1..50005
R. D. Carmichael, Note on a new number theory function, Bull. Amer. Math. Soc. 16 (1910), 232-238.

Cf. A002322, A002174, A051222, A143407, A270564, A111725.

cp's OEIS Frontend

A143407 Largest number k such that the reduced totient function psi(k) = A002174(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A143408 Number of numbers k such that the reduced totient function psi(k) = A002174(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A143417 Irregular triangle in which row n gives the k such that the reduced totient function psi(k) = A002174(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A270265 Numbers that are values of the totient function (A002202) but not of the reduced totient function (A002174).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Extensions

A270266 Numbers that are values of the reduced totient function (A002174) but not of the totient function (A002202).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

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.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

Python

Formula

A049225 Values of totient function applied to squarefree numbers; or numbers of form Product (p_i-1) where p_i are distinct primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A340521 List of possible orders of automorphism groups of finite groups.

Views

Author

Keywords

Comments

Links

Crossrefs

A362401 Numbers in the range of A162296, where A162296(n) is the sum of divisors of n that have a square factor larger than 1.

Views