A049149 -id:A049149 - cp's OEIS frontend

A065299 Numbers k such that sigma(k)*phi(k) is squarefree.

1, 2, 4, 9, 121, 242, 529, 1058, 2209, 3481, 4418, 5041, 6889, 6962, 10082, 11449, 13778, 17161, 22898, 27889, 32041, 34322, 51529, 55778, 57121, 64082, 96721, 103058, 114242, 120409, 128881, 146689, 175561, 185761, 193442, 196249, 218089

Offset: 1

Labos Elemer, Oct 29 2001

nonn

			All solutions are either squares or twice squares. Proper subset of A055008 or A028982. Several squares (of primes) and 2*squares are not here. E.g., 242 is here because phi(242) = 110, sigma(242) = 399, 2*5*11*3*7*19 is squarefree; 18 is not here, since 2*3*3*13 is not squarefree.

Harry J. Smith and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (terms 1..500 from Smith)

A062354(m) is squarefree.

Cf. A000010, A000203, A008683, A055008, A028982, A049195-A049199, A049149.

a[x_] := Abs[MoebiusMu[DivisorSigma[1, x]*EulerPhi[x]]] Do[s=as[n]; If[Equal[s, 1], Print[{n, Sqrt[n]}]], {n, 1, 1000000}]
Select[Range[250000],SquareFreeQ[DivisorSigma[1,#]*EulerPhi[#]]&] (* Harvey P. Dale, Jul 15 2015 *)

n=0; for (m = 1, 10^9, s=abs(moebius(sigma(m)*eulerphi(m))); if (s==1, write("b065299.txt", n++, " ", m); if (n==500, return))) \\ Harry J. Smith, Oct 15 2009

is(f)=my(n=#f~, v=List()); for(i=1,n, if(f[i,1]>2, listput(v,f[i,1]-1)); if(f[i,2]>2, return(0), f[i,2]>1, listput(v,f[i,1])); listput(v, (f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1))); for(i=2,#v, for(j=1,i-1, if(gcd(v[i],v[j])>1, return(0)))); for(i=1,#v, if(!issquarefree(v[i]), return(0))); 1
sq(f)=f[,2]*=2; f
double(f)=if(#f~ && f[1,1]==2, f[1,2]++, f=concat([2,1],f)); f
list(lim)=my(v=List()); forsquarefree(n=1,sqrtint(lim\1), if(is(sq(n[2])), listput(v,n[1]^2))); forsquarefree(n=1,sqrtint(lim\2), if(is(double(sq(n[2]))), listput(v,2*n[1]^2))); Set(v) \\ Charles R Greathouse IV, Feb 05 2018

Solutions to abs(A008683(A000203(x)*A000010(x))) = 1.

A049195 Numbers k such that the Euler totient function phi(k) is divisible by a square.

Original entry on oeis.org

5, 8, 10, 12, 13, 15, 16, 17, 19, 20, 21, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 63, 64, 65, 66, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 87, 88, 89, 90, 91, 92, 93, 95

Offset: 1

Labos Elemer

nonn

For k > 2 and p a prime: p^k is in this sequence. - Enrique Pérez Herrero, Apr 25 2013

The asymptotic density of this sequence is 1 since its complement (A049149) has a density 0. - Amiram Eldar, Mar 02 2021

			72 and 73 belong because neither phi(72) = 24 nor phi(73) = 72 are squarefree.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..2500

Complement of A049149.

Cf. A000010, A005117, A013929.

Select[Range[100], MoebiusMu[EulerPhi[#]] == 0 &]
Select[Range[100],!SquareFreeQ[EulerPhi[#]]&] (* Harvey P. Dale, Apr 30 2018 *)

A049198 Numbers that are not squarefree and whose Euler totient function is squarefree.

Original entry on oeis.org

4, 9, 18, 49, 98, 121, 242, 529, 961, 1058, 1849, 1922, 2209, 3481, 3698, 4418, 4489, 5041, 6241, 6889, 6962, 8978, 10082, 10609, 11449, 12482, 13778, 17161, 19321, 21218, 22898, 27889, 32041, 34322, 36481, 38642, 44521, 49729, 51529, 55778, 57121, 64082, 69169

Offset: 1

Labos Elemer

nonn

Numbers k such that abs(mu(phi(k))) = 1 and abs(mu(k)) = 0.

Contains all the squares p^2 of primes p such that p-1 is squarefree (A039787). - Amiram Eldar, Mar 18 2025

			a(27) = 13778 = 2*83*83 is divisible by a square, but phi(13778) = 6806 = 2*41*83 is squarefree.

Donovan Johnson, Table of n, a(n) for n = 1..5000

Intersection of A049149 and A013929.

Cf. A000010, A005117, A008683, A039787, A049199.

Select[Range[70000], Abs[ MoebiusMu[ EulerPhi[ # ] ] ] == 1 && Abs[ MoebiusMu[ # ] ] == 0 &]

isok(k)=!issquarefree(k) && issquarefree(eulerphi(k)) \\ Donovan Johnson, Jun 20 2012

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

Original entry on oeis.org

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).

A336655 Odd numbers k such that the multiplicative order of 2 modulo k is squarefree.

Original entry on oeis.org

1, 3, 7, 9, 11, 21, 23, 31, 33, 43, 47, 49, 59, 63, 67, 69, 71, 77, 79, 83, 89, 93, 99, 103, 107, 121, 127, 129, 131, 139, 141, 147, 151, 161, 167, 177, 179, 191, 201, 207, 211, 213, 217, 223, 227, 231, 233, 237, 239, 249, 253, 263, 267, 279, 281, 283, 301, 307

Offset: 1

Amiram Eldar, Jul 28 2020

nonn

			3 is a term since the multiplicative order of 2 modulo 3 is 2 which is squarefree.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Francesco Pappalardi, Square free values of the order function, New York J. Math., Vol. 9 (2003), pp. 331-344.

Cf. A002326, A005117, A049149, A336654, A336656.

Select[2 * Range[160] - 1, SquareFreeQ[MultiplicativeOrder[2, #]] &]

isok(k) = (k % 2) && issquarefree(znorder(Mod(2,k))); \\ Michel Marcus, Jul 29 2020

The number of terms not exceeding x is (a + o(1))* x * log(x)^(b-1), where a and b (~ 0.46437) are constants (Pappalardi, 2003).

A336656 Numbers k not divisible by 3 such that the multiplicative order of 3 modulo k is squarefree.

Original entry on oeis.org

1, 2, 4, 7, 8, 11, 13, 14, 22, 23, 26, 28, 31, 43, 44, 46, 47, 49, 52, 56, 59, 61, 62, 67, 71, 77, 79, 83, 86, 88, 91, 92, 94, 98, 103, 104, 107, 118, 121, 122, 124, 131, 134, 139, 142, 143, 154, 157, 158, 161, 166, 167, 169, 172, 179, 182, 184, 188, 191, 196

Offset: 1

Amiram Eldar, Jul 28 2020

nonn

			2 is a term since the multiplicative order of 3 modulo 2 is 1 which is squarefree.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Francesco Pappalardi, Square free values of the order function, New York J. Math., Vol. 9 (2003), pp. 331-344.

Cf. A007734, A005117, A049149, A336654, A336655.

Select[Range[200], !Divisible[#, 3] && SquareFreeQ[MultiplicativeOrder[3, #]] &]

isok(k) = (k % 3) && issquarefree(znorder(Mod(3,k))); \\ Michel Marcus, Jul 29 2020

The number of terms not exceeding x is (a + o(1))* x * log(x)^(b-1), where a and b (~ 0.51175) are constants (Pappalardi, 2003).

cp's OEIS Frontend

A065299 Numbers k such that sigma(k)*phi(k) is squarefree.

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A049195 Numbers k such that the Euler totient function phi(k) is divisible by a square.

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A049198 Numbers that are not squarefree and whose Euler totient function is squarefree.

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

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A336655 Odd numbers k such that the multiplicative order of 2 modulo k is squarefree.

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A336656 Numbers k not divisible by 3 such that the multiplicative order of 3 modulo k is squarefree.

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