A002322 -id:A002322 - cp's OEIS frontend

A336654 Numbers k such that lambda(k) is squarefree, where lambda is the Carmichael lambda function (A002322).

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 18, 21, 22, 23, 24, 28, 31, 33, 36, 42, 43, 44, 46, 47, 49, 56, 59, 62, 63, 66, 67, 69, 71, 72, 77, 79, 83, 84, 86, 88, 92, 93, 94, 98, 99, 103, 107, 118, 121, 124, 126, 129, 131, 132, 134, 138, 139, 141, 142, 147, 154, 158, 161

Offset: 1

Amiram Eldar, Jul 28 2020

nonn

			6 is a term since lambda(6) = 2 is squarefree.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Imre Kátai, Square-free values of the Carmichael function, Mathematica Pannonica, Vol. 16, No. 2 (2005), pp. 199-203.
Francesco Pappalardi, Filip Saidak and Igor E. Shparlinski, Square-free values of the Carmichael function, Journal of Number Theory, Vol. 103, No. 1 (2003), pp. 122-131.

Cf. A002322, A005117, A005596, A049149, A336655, A336656.

Select[Range[160], SquareFreeQ[CarmichaelLambda[#]] &]

The number of terms not exceeding x is (k + o(1)) * x/(log(x)^(1-a)), where a = 0.373955... is Artin's constant (A005596), and k = 0.80328... is another constant (Pappalardi et al., 2003).

A341857 a(n) = psi(n^2)/n, psi = A002322.

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 6, 2, 6, 2, 10, 1, 12, 3, 4, 4, 16, 3, 18, 1, 2, 5, 22, 2, 20, 6, 18, 3, 28, 2, 30, 8, 10, 8, 12, 3, 36, 9, 4, 2, 40, 1, 42, 5, 12, 11, 46, 4, 42, 10, 16, 3, 52, 9, 4, 6, 6, 14, 58, 1, 60, 15, 6, 16, 12, 5, 66, 4, 22, 6, 70, 6, 72, 18, 20, 9

Offset: 1

Jianing Song, Feb 21 2021

nonn

It is easy to show that n divides psi(n) for all n.
For k >= 2, n^k divides psi(n^(k+1)) if and only if n is not of the form 2*(p_1)^(e_1)*(p_2)^(e_2)*...*(p_m)^(e_m), where p_i are distinct odd primes not congruent to 1 modulo 2^k.
It seems that every positive integer occurs in this sequence. The first occurrence of each k is given by A341860.

			psi(220^2) = psi(2^4 * 5^2 * 11^2) = lcm(psi(2^4), psi(5^2), psi(11^2)) = lcm(4, 20, 110) = 220, so a(220) = psi(220^2)/220 = 220/220 = 1.

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

Cf. A002322, A341860.

Indices of 1 are given by A341858.

Array[CarmichaelLambda[#^2]/# &, 100] (* Paolo Xausa, Mar 11 2024 *)

a(n) = A002322(n^2)/n \\ See A002322 for its program

For odd n > 1, a(2n) = a(n)/2.

A353868 Numbers k such that the Carmichael function A002322(k) divides Dedekind psi A001615(k).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 14, 15, 16, 18, 20, 24, 27, 28, 30, 32, 35, 36, 40, 42, 45, 48, 52, 54, 56, 60, 63, 64, 65, 70, 72, 75, 78, 80, 81, 84, 90, 96, 98, 100, 104, 105, 108, 112, 117, 119, 120, 126, 128, 130, 135, 140, 144, 150, 156, 160, 162, 168, 175, 180, 182, 189, 190, 192, 195, 196, 200, 204, 208, 210, 216

Offset: 1

Max Alekseyev, May 08 2022

If coprime s,t are terms, then so is s*t. Also, if t is a term and prime p|t, then p*t is also a term. Squarefree terms are listed in A353869, primitive terms are listed in A353870, and their intersection forms A353871.

user142929, A definition related to pseudoprimes and the Dedekind psi function, MathOverflow, 2022.

Cf. A002322, A001615, A353869, A353870, A353871.

psi[1] = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); Select[Range[216], Divisible[psi[#], CarmichaelLambda[#]] &] (* Amiram Eldar, May 09 2022 *)

A353869 Squarefree numbers k such that the Carmichael function A002322(k) divides the Dedekind psi A001615(k).

Original entry on oeis.org

1, 2, 3, 6, 14, 15, 30, 35, 42, 65, 70, 78, 105, 119, 130, 182, 190, 195, 210, 238, 255, 357, 370, 377, 390, 418, 455, 510, 546, 570, 595, 663, 714, 754, 910, 969, 1045, 1110, 1118, 1131, 1190, 1254, 1326, 1330, 1365, 1547, 1558, 1615, 1785, 1885, 1887, 1938, 2090, 2190, 2261, 2262, 2470, 2590, 2639, 2730

Offset: 1

Max Alekseyev, May 08 2022

If s,t are terms, then so is lcm(s,t); in particular, if s,t are coprime, then s*t is also a term. Primitive squarefree terms are listed in A353871.

Intersection of A005117 and A353868.

Cf. A002322, A001615, A353870, A353871.

psi[1] = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); Select[Range[3000], SquareFreeQ[#] && Divisible[psi[#], CarmichaelLambda[#]] &] (* Amiram Eldar, May 09 2022 *)

A354060 Irregular table read by rows: T(n,k) is the number of solutions to x^k == 1 (mod n), 1 <= k <= psi(n), psi = A002322.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 3, 2, 1, 6, 1, 4, 1, 2, 3, 2, 1, 6, 1, 2, 1, 4, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 4, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12, 1, 2, 3, 2, 1, 6, 1, 4, 1, 8, 1, 4, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 16

Offset: 1

Jianing Song, May 16 2022

Row n and Row n' are the same if and only if (Z/nZ)* = (Z/n'Z)*, where (Z/nZ)* is the multiplicative group of integers modulo m.

Given n, T(n,k) only depends on gcd(k,psi(n)).

			Table starts
n = 1: 1;
n = 2: 1;
n = 3: 1, 2;
n = 4: 1, 2;
n = 5: 1, 2, 1, 4;
n = 6: 1, 2;
n = 7: 1, 2, 3, 2, 1, 6;
n = 8: 1, 4;
n = 9: 1, 2, 3, 2, 1, 6;
n = 10: 1, 2, 1, 4;
n = 11: 1, 2, 1, 2, 5, 2, 1, 2, 1, 10;
n = 12: 1, 4;
n = 13: 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12;
n = 14: 1, 2, 3, 2, 1, 6;
n = 15: 1, 4, 1, 8;
n = 16: 1, 4, 1, 8;
n = 17: 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 16;
n = 18: 1, 2, 3, 2, 1, 6;
n = 19: 1, 2, 3, 2, 1, 6, 1, 2, 9, 2, 1, 6, 1, 2, 3, 2, 1, 18;
n = 20: 1, 4, 1, 8;
...

Jianing Song, Table of n, a(n) for n = 1..8346 (the first 200 rows)

Cf. A354057, A002322, A354061.

T(n,k)=my(Z=znstar(n)[2]); prod(i=1, #Z, gcd(k, Z[i]))

If (Z/nZ)* = C_{k_1} X C_{k_2} X ... X C_{k_r}, then T(n,k) = Product_{i=1..r} gcd(k,k_r).

T(p^e,k) = gcd((p-1)*p^(e-1),k) for odd primes p. T(2,k) = 1, T(2^e,k) = 2*gcd(2^(e-2),k) if k is even and 1 if k is odd.

A066605 Numbers k such that phi(k)/lambda(k) increases to a record value, where phi(k) is the Euler totient function (A000010) and lambda(k) is the Carmichael lambda function (A002322).

Original entry on oeis.org

1, 8, 24, 63, 80, 240, 455, 819, 1365, 2387, 2720, 3276, 3591, 4095, 7280, 9139, 13104, 18981, 21483, 21840, 24605, 32760, 44289, 45695, 63973, 122915, 132867, 172081, 191919, 246753, 319865, 520025, 575757, 860405, 959595, 1233765

Offset: 1

Robert G. Wilson v, Jan 13 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..88

Cf. A000010, A002322.

a = 0; c = {}; Do[ b = EulerPhi[n]/CarmichaelLambda[n]; If[ b > a, a = b; c = Append[c, n]], {n, 1, 10^7}]; c

Title improved by Amiram Eldar, Jul 20 2019

A173694 Arguments n for which the Carmichael lambda function A002322(n) is a perfect square.

Original entry on oeis.org

1, 2, 5, 10, 15, 16, 17, 20, 30, 34, 37, 40, 48, 51, 60, 64, 68, 74, 80, 85, 95, 101, 102, 111, 120, 125, 135, 136, 148, 170, 185, 190, 192, 197, 202, 204, 222, 240, 247, 250, 255, 256, 257, 259, 270, 272, 285, 296, 303, 304, 320, 323, 333, 340, 351, 370, 375, 380, 394, 401

Offset: 1

Michel Lagneau, Nov 25 2010

nonn

			37 is in the sequence because lambda(37) = 36 = 6^2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A002322.

Cf. A010052.

a173694 n = a173694_list !! (n-1)
a173694_list = filter ((== 1) . a010052 . a002322) [1..]
-- Reinhard Zumkeller, Sep 02 2014

for n from 1 to 500 do if issqr(numtheory[lambda](n) ) then printf("%d,",n) ;     end if; end do:

A010052(A002322(a(n))) = 1. - Reinhard Zumkeller, Sep 02 2014

Definition rephrased - R. J. Mathar, Nov 26 2010

A248881 Numbers n such that lambda(sum of even divisors of 2n) = lambda(sum of odd divisors of 2n) where lambda is the Carmichael function (A002322).

Original entry on oeis.org

1, 3, 5, 6, 9, 11, 13, 17, 18, 19, 25, 26, 27, 29, 36, 37, 38, 41, 43, 45, 49, 50, 53, 54, 59, 61, 63, 65, 67, 68, 72, 73, 74, 75, 81, 82, 83, 85, 86, 87, 89, 90, 95, 97, 98, 99, 100, 101, 103, 107, 109, 113, 117, 121, 122, 125, 126, 130, 131, 134, 137, 139

Offset: 1

Michel Lagneau, Mar 05 2015

nonn

Number n such that A002322(A074400(n))= A002322(A000593(n)).
The squares of the form p^2 with p prime are in the sequence because the divisors of 2p^2 are {1,2,p,2p,p^2,2p^2} => sum of even divisors s0 = 2+2p+2p^2 = 2(p^2+p+p^2) and sum of odd divisors s1 = 1+p+p^2 and lambda(s0) = lambda(s1) = lambda(2*s0).
A majority of primes are in the sequence: 3, 5, 11, 13, 17, 19, 29, 37, 41, 43, 53, 59, 61, 67, 73, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 157, 163, 173, 179, 181, 193, 197, ... but the primes 7, 23, 31, 47, 71, 79, 127, 151, 167, 191, 223, 239, 263, 367, 383, 431, ... are not in the sequence.

			18 is in the sequence because A002322(A074400(18))= A002322(78)= 12 and because A002322(A000593(18)) = A002322(13) = 12.

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

Cf. A000593, A002322, A074400, A077591.

lst={};f[x_] := Plus @@ Select[Divisors[x], OddQ[#] &]; g[x_] := Plus @@ Select[Divisors[x], EvenQ[#]&]; Do[If[CarmichaelLambda[f[n]]== CarmichaelLambda[g[n]], AppendTo[lst,n/2]], {n, 1, 500}];lst

a002322(n) = lcm(znstar(n)[2]);
isok(n) = my(sod = sumdiv(2*n, d, d*(d%2))); my(sed = sigma(2*n) - sod); sod && sed && (a002322(sod) == a002322(sed)); \\ Michel Marcus, Mar 07 2015

A252911 Irregular triangular array read by rows: T(n,k) is the number of elements in the multiplicative group of integers modulo n that have order k, n>=1, 1<=k<=A002322(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 2, 0, 0, 2, 1, 3, 1, 1, 2, 0, 0, 2, 1, 1, 0, 2, 1, 1, 0, 0, 4, 0, 0, 0, 0, 4, 1, 3, 1, 1, 2, 2, 0, 2, 0, 0, 0, 0, 0, 4, 1, 1, 2, 0, 0, 2, 1, 3, 0, 4, 1, 3, 0, 4, 1, 1, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 8, 1, 1, 2, 0, 0, 2, 1, 1, 2, 0, 0, 2, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 6, 1, 3, 0, 4

Offset: 1

Geoffrey Critzer, Dec 24 2014

Row sums are A000010.

Column 2 = A155828(n) = A060594(n) - 1.

			1;
1;
1, 1;
1, 1;
1, 1, 0, 2;
1, 1;
1, 1, 2, 0, 0, 2;
1, 3;
1, 1, 2, 0, 0, 2;
1, 1, 0, 2;
1, 1, 0, 0, 4, 0, 0, 0, 0, 4;
1, 3;
1, 1, 2, 2, 0, 2, 0, 0, 0, 0, 0, 4;
1, 1, 2, 0, 0, 2;
1, 3, 0, 4;
T(15,2)=3 because the elements 4, 11, and 14 have order 2 in the modulo multiplication group (Z/15Z)*. We observe that 4^2, 11^2, and 14^2 are congruent to 1 mod 15.

Alois P. Heinz, Rows n = 1..250, flattened
Eric Weisstein's World of Mathematics, Modulo Multiplication Group.

Cf. A000010, A002322, A054522, A060594, A155828.

with(numtheory):
T:= n-> `if`(n=1, 1, (p-> seq(coeff(p, x, j), j=1..degree(p)))(
         add(`if`(igcd(n, i)>1, 0, x^order(i, n)), i=1..n-1))):
seq(T(n), n=1..30);  # Alois P. Heinz, Dec 30 2014

Table[Table[
   Count[Table[
     MultiplicativeOrder[a, n], {a,
      Select[Range[n], GCD[#, n] == 1 &]}], k], {k, 1,
    CarmichaelLambda[n]}], {n, 1, 20}] // Grid

A268177 Numbers m such that Sum_{i=1..q} 1/lambda(d(i)) is an integer, where d(i) are the q divisors of m and lambda is the Carmichael lambda function (A002322).

Original entry on oeis.org

1, 2, 6, 8, 12, 15, 24, 28, 30, 40, 70, 84, 112, 120, 140, 210, 240, 252, 280, 315, 336, 351, 357, 360, 420, 550, 630, 684, 702, 714, 836, 840, 884, 912, 952, 988, 1092, 1100, 1120, 1140, 1364, 1386, 1650, 1710, 1820, 2002, 2040, 2088, 2090, 2200, 2394, 2484

Offset: 1

Michel Lagneau, Jan 28 2016

nonn

The corresponding integers are 1, 2, 3, 3, 4, 2, 5, 3, 4, 4, 3, 5, 4, 7, 4, 5, 8, 6, 5, 3, 7, 2, 2, 8, 7,...
A majority of numbers of the sequence are even, except 1, 15, 315, 351, 357, 2871, 3663,...
Replacing the function lambda(n) by the Euler totient function phi(n) (A000010) gives only the short sequence {1, 2, 6} for n < 10^7.

			6 is in the sequence because the divisors of 6 are {1,2,3,6} => 1/lambda(1)+1/lambda(2)+1/lambda(3)+ 1/lambda(6) = 1/1 + 1/1 + 1/2 + 1/2 = 3 is an integer.

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

Cf. A000010, A002322.

lst={};Do[If[IntegerQ[Total[1/CarmichaelLambda[Divisors[n]]]],AppendTo[lst,n]], {n, 0, 2500}];lst

cp's OEIS Frontend

A336654 Numbers k such that lambda(k) is squarefree, where lambda is the Carmichael lambda function (A002322).

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A341857 a(n) = psi(n^2)/n, psi = A002322.

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A353868 Numbers k such that the Carmichael function A002322(k) divides Dedekind psi A001615(k).

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A353869 Squarefree numbers k such that the Carmichael function A002322(k) divides the Dedekind psi A001615(k).

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A354060 Irregular table read by rows: T(n,k) is the number of solutions to x^k == 1 (mod n), 1 <= k <= psi(n), psi = A002322.

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A066605 Numbers k such that phi(k)/lambda(k) increases to a record value, where phi(k) is the Euler totient function (A000010) and lambda(k) is the Carmichael lambda function (A002322).

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A173694 Arguments n for which the Carmichael lambda function A002322(n) is a perfect square.

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A248881 Numbers n such that lambda(sum of even divisors of 2n) = lambda(sum of odd divisors of 2n) where lambda is the Carmichael function (A002322).

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