A002322 -id:A002322 - cp's OEIS frontend

A124240 Numbers n such that lambda(n) divides n, where lambda is Carmichael's function (A002322).

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 36, 40, 42, 48, 54, 60, 64, 72, 80, 84, 96, 100, 108, 120, 126, 128, 144, 156, 160, 162, 168, 180, 192, 200, 216, 220, 240, 252, 256, 272, 288, 294, 300, 312, 320, 324, 336, 342, 360, 378, 384, 400, 420, 432, 440, 468, 480

Offset: 1

Alexander Adamchuk, Oct 22 2006

nonn

Numbers n such that A124239(n) is divisible by n.
If k is in the sequence then 2k is also in the sequence, but if 2m is in the sequence m is not necessarily a term of the sequence.
This sequence is a subsequence of A068563. The first term that is different is A068563(27) = 136. The terms of A068563 that are not the terms of a(n) are listed in A124241.
Also, the sequence of numbers n such that p-1 divides n for all primes p that divide n. - Leroy Quet, Jun 27 2008
Numbers n such that b^n == 1 (mod n) for every b coprime to n. - Thomas Ordowski, Jun 23 2017
Numbers m such that every divisor < m is the difference between two divisors of m. - Michel Lagneau, Aug 11 2017
All terms > 1 in this sequence are even. Furthermore, either 4 or 6 divides a(n) for n > 3. 1806 is the largest squarefree term. - Paul Vanderveen, Apr 24 2022

			a(1) = 1 because 1 divides A124239(1) = 1.
a(2) = 2 because 2 divides A124239(2) = 14.
a(3) = 4 because 4 divides A124239(4) = 3704, but 3 does not divide A124239(3) = 197.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Alexander Kalmynin, On Novák numbers, arXiv:1611.00417 [math.NT], 2016. See Theorem 6 p. 11 where these numbers are called Novák-Carmichael numbers.
Eric Weisstein's World of Mathematics, Carmichael Function

Cf. A002322, A124239, A124241, A068563, A027748, A140470, A141766, A160014, A363523.

a124240 n = a124240_list !! (n-1)
a124240_list = filter
   (\x -> all (== 0) $ map ((mod x) . pred) $ a027748_row x) [1..]
-- Reinhard Zumkeller, Aug 27 2013

a:= proc(n) option remember; local k;
       for k from `if`(n=1, 0, a(n-1))+1 while
       irem(k, numtheory[lambda](k))>0 do od: k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 04 2021
# Using function 'Clausen' from A160014:
aList := m -> select(k -> irem(Clausen(k, 1), Clausen(k, 0)) = 0, [seq(1..m)]):
aList(480); # Peter Luschny, Jun 08 2023

Do[f=n + Sum[ (2k-1)((2k-1)^n-1) / (2(k-1)), {k,2,n} ]; If[IntegerQ[f/n],Print[n]],{n,1,900}]
Flatten[Position[Table[n/CarmichaelLambda[n], {n, 440}], Integer]] (* _T. D. Noe, Sep 11 2008 *)

is(n)=n%lcm(znstar(n)[2])==0 \\ Charles R Greathouse IV, Feb 11 2015

from itertools import islice, count
from sympy.ntheory.factor_ import reduced_totient
def A124240gen(): return filter(lambda n:n % reduced_totient(n) == 0,count(1))
A124240_list = list(islice(A124240gen(),20)) # Chai Wah Wu, Dec 11 2021

k is in a <=> Clausen(k, 0) divides Clausen(k, 1), (Clausen = A160014). - Peter Luschny, Jun 08 2023

New definition from T. D. Noe, Aug 31 2008

Edited by Max Alekseyev, Aug 25 2013

A034380 Ratio of totient to Carmichael's lambda function: a(n) = A000010(n) / A002322(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 4, 1, 2, 1, 2, 2, 1, 1, 4, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 4, 1, 1, 6, 2, 4, 2, 1, 2, 2, 2, 1, 4, 1, 1, 2, 2, 2, 2, 1, 8, 1, 1, 1, 4, 4, 1, 2, 4, 1, 2, 6, 2, 2, 1, 2, 4, 1, 1, 2, 2, 1, 2, 1, 4, 4

Offset: 1

Alex Fink

nonn

a(n)=1 if and only if the multiplicative group modulo n is cyclic (that is, if n is either 1, 2, 4, or of the form p^k or 2*p^k where p is an odd prime). In other words: a(n)=1 if n is a term of A033948, otherwise a(n) > 1 (and n is a term of A033949). - Joerg Arndt, Jul 14 2012

T. D. Noe, Table of n, a(n) for n = 1..10000
W. D. Banks and F. Luca, On integers with a special divisibility property, Archivum Mathematicum (BRNO) 42 (2006) pp 31-42.

Cf. A000010, A002322, A033948, A033949, A062373-A062377, A080400, A289624.

a034380 n = a000010 n `div` a002322 n
-- Reinhard Zumkeller, Sep 02 2014

[1] cat [EulerPhi(n) div CarmichaelLambda(n): n in [2..100]]; // Vincenzo Librandi, Jul 18 2017

Maple
```
A034380 := n-> phi(n) / lambda(n);
```

Table[EulerPhi[n]/CarmichaelLambda[n], {n, 1, 200}] (* Geoffrey Critzer, Dec 23 2014 *)

eulerphi(n)/lcm(znstar(n)[2]) \\ Charles R Greathouse IV, Feb 01 2013

a(n) = A000010(n) / A002322(n).
a(A033948(n)) = 1 [Banks & Luca]. - R. J. Mathar, Jul 29 2007
A002322(n)/A007947(a(n)) = A289624(n). - Antti Karttunen, Jul 17 2017

A061257 Euler transform of reduced totient function psi(n), cf. A002322.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 21, 37, 58, 96, 153, 243, 376, 584, 897, 1353, 2046, 3060, 4552, 6714, 9862, 14386, 20898, 30198, 43427, 62159, 88600, 125804, 177881, 250615, 351819, 492203, 686294, 953954, 1321902, 1826394, 2516364, 3457332, 4737576, 6475332

Offset: 0

Vladeta Jovovic, Apr 21 2001

T. D. Noe, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Cf. A002322, A061258, A006171, A001970, A061255, A061256.

nn = 20; b = Table[CarmichaelLambda[n], {n, nn}]; CoefficientList[Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* T. D. Noe, Jun 19 2012 *)

G.f.: Product_{k=1..infinity} (1 - x^k)^(-psi(k)). a(n)=1/n*Sum_{k=1..n} a(n-k)*b(k), n>1, a(0)=1, b(k)=Sum_{d|k} d*psi(d), cf. A061258.

A303756 Number of values of k, 1 <= k <= n, with A002322(k) = A002322(n), where A002322 is Carmichael lambda.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 1, 4, 2, 2, 1, 5, 1, 3, 3, 4, 1, 4, 1, 5, 5, 2, 1, 6, 1, 2, 2, 6, 1, 6, 1, 1, 3, 2, 3, 7, 1, 3, 4, 7, 1, 8, 1, 4, 5, 2, 1, 8, 2, 2, 3, 6, 1, 4, 3, 9, 5, 2, 1, 9, 1, 2, 10, 4, 7, 5, 1, 5, 3, 8, 1, 11, 1, 2, 4, 6, 3, 9, 1, 10, 1, 2, 1, 12, 6, 3, 3, 6, 1, 10, 11, 4, 4, 2, 3, 2, 1, 4, 5, 5, 1, 7, 1, 12, 13

Offset: 1

Antti Karttunen, Apr 30 2018

nonn

Ordinal transform of A002322.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A002322.

Cf. also A081373, A303755, A303758.

a[n_] := With[{c = CarmichaelLambda[n]}, Select[Range[n], c == CarmichaelLambda[#]&] // Length];
Array[a, 1000] (* Jean-François Alcover, Sep 19 2020 *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A002322(n) = lcm(znstar(n)[2]); \\ From A002322
v303756 = ordinal_transform(vector(up_to,n,A002322(n)));
A303756(n) = v303756[n];

Except for a(2) = 2, a(n) = A303758(n).

A264012 Composite numbers n such that gcd(phi(n), n-1) = lambda(n), where lambda(n) = A002322(n).

Original entry on oeis.org

561, 1105, 2821, 6601, 10585, 29341, 52633, 62745, 63973, 101101, 115921, 126217, 188461, 252601, 278545, 294409, 410041, 512461, 552721, 748657, 825265, 1152271, 1193221, 2100901, 2508013, 2531845, 3146221, 4335241, 4767841, 4909177, 5444489, 5481451, 6049681

Offset: 1

Thomas Ordowski, Nov 01 2015

nonn

Carmichael numbers n such that A049559(n) = A002322(n).
If n is a Carmichael number with n-1 squarefree, then n is in the sequence. The smallest such n = 139952671.
If (n-1)/lambda(n) is a prime (see A174590), then n is in the sequence. - Thomas Ordowski, Oct 17 2016
Numbers n such that gcd(phi(n),n-1) = lambda(n)^2 are 1, 2, 2320690177, ? - Thomas Ordowski and Michel Marcus, Oct 20 2016

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002322, A002997, A049559, A257643.

Select[ Range@ 6100000, CompositeQ@# && GCD[ EulerPhi@#, # - 1] == CarmichaelLambda@# &] (* Michael De Vlieger, Nov 01 2015 *)

forcomposite(n=1, 1e7, if(gcd(eulerphi(n),n-1)==lcm(znstar(n)[2]), print1(n ", "))) \\ Altug Alkan, Nov 01 2015

t(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]>1||(n-1)%(f[i, 1]-1), return(0))); 1;
is(n)=n%2 && !isprime(n) && t(n) && n>1;
c(n)=gcd(eulerphi(n),n-1)/lcm(znstar(n)[2]);
for(n=1, 1e7, if(is(n) && c(n)==1 , print1(n", "))) \\ Altug Alkan, Nov 01 2015

More terms from Altug Alkan, Nov 01 2015

A307437 a(n) is the smallest k such that 2n divides psi(k), psi = A002322.

Original entry on oeis.org

3, 5, 7, 17, 11, 13, 29, 17, 19, 25, 23, 73, 53, 29, 31, 97, 103, 37, 191, 41, 43, 89, 47, 97, 101, 53, 81, 113, 59, 61, 311, 193, 67, 137, 71, 73, 149, 229, 79, 187, 83, 203, 173, 89, 181, 235, 283, 97, 197, 101, 103, 313, 107, 109, 121, 113, 229, 233, 709, 241

Offset: 1

Jianing Song, Apr 08 2019

a(n) exists for all n: by Dirichlet's theorem on arithmetic progressions, there exists a prime p congruent to 1 modulo 2n, so 2n divides psi(p) = p - 1.
a(n) is the smallest k such that (Z/kZ)* contains C_(2n) as a subgroup, where (Z/kZ)* is the multiplicative group of integers modulo n.
a(n) is the smallest k such that there exists some x such that ord(x,k) = 2n, where ord(x,k) is the multiplicative order of x modulo k.
Record values of a(n)/n occur at n = 1, 4, 12, 19, 59, 167, 196, 197, 227, 317, 457, 521, 706, ... (A341888).
From Jianing Song, Feb 21 2021: (Start)
a(n) is bounded above by (2n)^2 since n divides psi(n^2).
a(n) is usually odd. There are only 7 values <= 10^4 for n such that a(n) is even, namely n = 256, 512, 1024, 2816, 4096, 5632 and 8192 (A341887). (End)
a(n) is odd or divisible by 16, since psi(k) = psi(2k) = psi(4k) = psi(8k) for odd k > 1. - Jianing Song, Feb 22 2021

			For n = 7, psi(29) = 28 and 29 is the smallest k such that 14 divides psi(k), so a(7) = 29.
For n = 27, psi(81) = 54 and 81 is the smallest k such that 54 divides psi(k), so a(27) = 81.
For n = 40, psi(187) = 80 and 187 is the smallest k such that 80 divides psi(k), so a(40) = 187.
For n = 42, psi(203) = 84 and 203 is the smallest k such that 84 divides psi(k), so a(42) = 203.

Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Multiplicative group of integers modulo n

Cf. A002322, A307436, A341887, A341888.

N:= 100: # for a(1)..a(N)
V:= Vector(N): count:= 0:
for k from 3 while count < N do
  S:= select(t -> t <= N and V[t]=0, numtheory:-divisors(numtheory:-lambda(k)/2));
  if nops(S) > 0 then count:= count + nops(S); V[convert(S,list)]:= k fi
od:
convert(V,list); # Robert Israel, Jul 10 2019

a(n) = my(i=1); while(A002322(i)%(2*n), i++); i \\ See A002322 for its program

from sympy import reduced_totient
def A307437(n):
    k = 1
    while reduced_totient(k) % (2*n):
        k += 1
    return k # Chai Wah Wu, Feb 24 2021

From Jianing Song, Feb 26 2021: (Start)
For odd prime p, a((p-1)/2*p^e) = p^(e+1) if (p-1)*p^e+1 is composite, (p-1)*p^e+1 otherwise. Proof: suppose a((p-1)/2*p^e) = p^a*r < p^(e+1), p does not divide r, then (p-1)*p^e | lcm((p-1)*p^(a-1), psi(r)) => p^e | lcm(p^(a-1), psi(r)).
If p^e | p^(a-1), then a((p-1)/2*p^e) >= p^a >= p^(e+1).
If p^e does not divide p^(a-1), then p^e | psi(r). r must have a prime factor of the form q = 2*t*p^e+1. If a >= 1, then a((p-1)/2*p^e) >= p*(2*p^e+1) > p^(e+1). So we must have a = 0. Write r = r'*q^b, then p-1 | lcm(psi(r'), 2*t*p^e*q^(b-1)) => p-1 | lcm(psi(r'), 2*t), hence 2*t*r' >= 2*t*psi(r') >= lcm(psi(r'), 2*t) => p-1. If 2*t*r' > p-1, then a((p-1)/2*p^e) >= r'*q = r'*(2*t*p^e+1) > p^(e+1). If 2*t*r' = p-1, then r' = psi(r') => r' = 1, 2*t = p-1, hence (p-1)*p^e+1 is prime. (End)

A268336 a(n) = A174824(n)/n, where A174824(n) = lcm(A002322(n), n) and A002322(n) is the Carmichael lambda function (also known as the reduced totient function or the universal exponent of n).

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 6, 1, 2, 2, 10, 1, 12, 3, 4, 1, 16, 1, 18, 1, 2, 5, 22, 1, 4, 6, 2, 3, 28, 2, 30, 1, 10, 8, 12, 1, 36, 9, 4, 1, 40, 1, 42, 5, 4, 11, 46, 1, 6, 2, 16, 3, 52, 1, 4, 3, 6, 14, 58, 1, 60, 15, 2, 1, 12, 5, 66, 4, 22, 6, 70, 1, 72, 18, 4, 9, 30, 2, 78, 1, 2

Offset: 1

Juri-Stepan Gerasimov, Feb 01 2016

Cf. A002322, A068563, A124240, A174824.

[1] cat [Lcm(n, CarmichaelLambda(n))/n: n in [2..100]]: // Feb 03 2016

Table[LCM[n, CarmichaelLambda@ n]/n, {n, 100}] (* Michael De Vlieger, Feb 03 2016, after T. D. Noe at A174824 *)

a(n)=my(ps); ps=factor(n)[, 1]~; m = n; for(k=1, #ps, m=lcm(m, ps[k]-1)); m/n \\ Michel Marcus, Feb 21 2016

apply( {A268336(n)=lcm(lcm([p-1|p<-factor(n)[,1]]),n)/n}, [1..99]) \\ [...] = znstar(n)[2], but 3x faster. - M. F. Hasler, Nov 13 2019

a(n) = A174824(n)/n.

a(A124240(n)) = 1. - Michel Marcus, Feb 21 2016

More terms from Vincenzo Librandi, Feb 03 2016

A277127 a(n) = n - lambda(n), where lambda(n) = A002322(n).

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 6, 3, 6, 1, 10, 1, 8, 11, 12, 1, 12, 1, 16, 15, 12, 1, 22, 5, 14, 9, 22, 1, 26, 1, 24, 23, 18, 23, 30, 1, 20, 27, 36, 1, 36, 1, 34, 33, 24, 1, 44, 7, 30, 35, 40, 1, 36, 35, 50, 39, 30, 1, 56, 1, 32, 57, 48, 53, 56, 1, 52, 47, 58, 1, 66, 1, 38, 55, 58, 47, 66, 1, 76, 27, 42, 1

Offset: 1

Thomas Ordowski, Oct 01 2016

nonn

Largest m < n such that b^m == b^n (mod n) for every integer b.

Altug Alkan, Table of n, a(n) for n = 1..10000

Cf. A002322, A033948, A051953, A276976.

Table[n - CarmichaelLambda@ n, {n, 83}] (* Michael De Vlieger, Oct 01 2016 *)

a(n) = n - lcm(znstar(n)[2]); \\ Altug Alkan, Oct 01 2016

a(p) = 1 for prime p.
a(p^2) = p prime.
a(n) = A051953(n) for n in A033948.

More terms from Altug Alkan, Oct 01 2016

A296077 a(n) = 1 if 1 + A002322(n) is prime, 0 otherwise, where A002322 is Carmichael lambda.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1

Offset: 1

Antti Karttunen, Dec 05 2017

nonn

Out of the first 65537 values, 39743 are 1's (indicating primes), and 25794 are 0's, indicating nonprimes.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for characteristic functions

Characteristic function for A263028.

Cf. A002322, A010051, A263027, A263029 (positions of zeros), A296076, A296079.

Table[If[PrimeQ[CarmichaelLambda[n]+1],1,0],{n,120}] (* Harvey P. Dale, Sep 23 2020 *)

A296077(n) = isprime(1+lcm(znstar(n)[2]));

a(n) = A010051(A263027(n)) = A010051(1+A002322(n)).

A376849 Numbers k such that psi(k + psi(k)) = psi(k) + psi(psi(k)), where psi = A002322.

Original entry on oeis.org

2, 5, 10, 34, 80, 111, 196, 204, 222, 328, 351, 646, 654, 704, 837, 876, 935, 943, 969, 1053, 1100, 1140, 1220, 1224, 1372, 1408, 1526, 1824, 1864, 2368, 2496, 2511, 2715, 2816, 2842, 3159, 3294, 3528, 3648, 3672, 4332, 4473, 4600, 4736, 4992, 5500, 5632, 6325, 6528, 7296, 7412, 7512, 7533, 7832, 7959

Offset: 1

Robert Israel, Oct 06 2024

nonn

			a(3) = 10 is a term because psi(10) = 4, psi(4) = 2, and psi(10 + 4) = psi(14) = 6 = 4 + 2.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A002322, A376830, A376831, A376843, A376844, A376848, A376851.

filter:= proc(k) uses numtheory; local s;
 s:= lambda(k);
 lambda(k+s) = s + lambda(s)
end proc:
select(filter, [$1..10000]);

Select[Range[8000], CarmichaelLambda[ #+CarmichaelLambda[#]] == CarmichaelLambda[#]+CarmichaelLambda[CarmichaelLambda[#]] &] (* Stefano Spezia, Oct 07 2024 *)

cp's OEIS Frontend

A124240 Numbers n such that lambda(n) divides n, where lambda is Carmichael's function (A002322).

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A034380 Ratio of totient to Carmichael's lambda function: a(n) = A000010(n) / A002322(n).

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A061257 Euler transform of reduced totient function psi(n), cf. A002322.

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A303756 Number of values of k, 1 <= k <= n, with A002322(k) = A002322(n), where A002322 is Carmichael lambda.

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A264012 Composite numbers n such that gcd(phi(n), n-1) = lambda(n), where lambda(n) = A002322(n).

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A307437 a(n) is the smallest k such that 2n divides psi(k), psi = A002322.

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A268336 a(n) = A174824(n)/n, where A174824(n) = lcm(A002322(n), n) and A002322(n) is the Carmichael lambda function (also known as the reduced totient function or the universal exponent of n).

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