A069209 -id:A069209 - cp's OEIS frontend

A009262 a(n) = lcm(n, phi(n)).

1, 2, 6, 4, 20, 6, 42, 8, 18, 20, 110, 12, 156, 42, 120, 16, 272, 18, 342, 40, 84, 110, 506, 24, 100, 156, 54, 84, 812, 120, 930, 32, 660, 272, 840, 36, 1332, 342, 312, 80, 1640, 84, 1806, 220, 360, 506, 2162, 48, 294, 100, 1632, 312, 2756, 54, 440, 168, 684, 812, 3422

Offset: 1

David W. Wilson

nonn

This is a divisibility sequence: if n divides m, a(n) divides a(m). - Franklin T. Adams-Watters, Mar 30 2010
a(n) = n iff n is in A007694.
a(n) is a divisor of A299822(n). It is a proper divisor iff n is in A069209. - Max Alekseyev, Oct 11 2024

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
Index to divisibility sequences

Cf. A000010, A007694, A007947, A052106, A069209, A076512, A109395, A174824.

with(numtheory); A009262:=n->lcm(n, phi(n)); seq(A009262(n), n=1..100); # Wesley Ivan Hurt, Jan 29 2014

Table[LCM[n, EulerPhi[n]], {n, 100}] (* Wesley Ivan Hurt, Jan 29 2014 *)

a(n)=lcm(eulerphi(n), n) \\ Charles R Greathouse IV, May 29 2016

a(n) = A000010(n) * A109395(n) = n * A076512(n) = A299822(n) / gcd(A007947(n),phi(A007947(n))). - Max Alekseyev, Oct 11 2024

A247371 Number of groups of order n for which all Sylow subgroups are cyclic.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 2, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 1, 6, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 2, 2, 1, 6, 1, 2, 2, 1, 1, 4, 1, 3, 1, 4, 1, 2, 1, 2, 1, 2, 1, 6, 1, 3, 1, 2, 1, 6, 1, 2, 1

Offset: 1

Eric M. Schmidt, Sep 15 2014

nonn

For squarefree n this gives the total number of groups of order n.

Eric M. Schmidt, Table of n, a(n) for n = 1..10000
M. Ram Murty and V. Kumar Murty, On groups of squarefree order, Math. Ann. 267, no. 3, 299-309, 1984.

Cf. A000001, A069209.

def pnu(pp, m) : return prod(gcd(pp, q-1) for q in prime_divisors(m))
def a(n) : s = n.radical(); return sum(prod(sum((pnu(p^(k+1), s//prod(c)) - pnu(p^k, s//prod(c))) // (p^k*(p-1)) for k in range(n.valuation(p))) for p in c) for c in powerset(prime_divisors(n)))

a(A005117(n)) = A000001(A005117(n)). - Michel Marcus, Sep 15 2014

A295126 Denominator of Sum_{d|n} mu(n/d)/d, where mu is the Möbius function A008683.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 6, 13, 7, 15, 16, 17, 9, 19, 5, 7, 11, 23, 12, 25, 13, 27, 14, 29, 15, 31, 32, 33, 17, 35, 18, 37, 19, 13, 10, 41, 7, 43, 22, 45, 23, 47, 24, 49, 25, 51, 13, 53, 27, 11, 28, 19, 29, 59, 15, 61, 31, 21, 64, 65, 33, 67, 17, 69, 35

Offset: 1

Mats Granvik and Robert G. Wilson v, Nov 15 2017

a(n) <= n.
a(n) <> n when n is in A069209.
n == 0 (mod a(n)).
First occurrence of k: 1, 2, 3, 4, 5, 12, 7, 8, 9, 40, 11, 24, 13, 28, 15, 16, 17, 36, 19, 80, 63, 44, 23, 48, 25, ..., ;
First occurrence of k = a(n)/n: 1, 6, 21, 20, 55, 42, 203, 120, 171, 110, 253, 84, 689, 406, 465, 272, 1751, 342, ..., .

			a(6) = 1 since mu(6)/1 + mu(3)/2 + mu(2)/3 + mu(1)/6 = 1 - 1/2 - 1/3 + 1/6 = 1/3.

Mats Granvik and Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Cf. A008683, A069209, A191898, A007913, A023900, A173557, A295127 (numerator).

f:= n -> denom(add(numtheory:-mobius(n/k)/k, k=numtheory:-divisors(n))):
map(f, [$1..100]); # Robert Israel, Nov 16 2017

f[n_] := Block[{d = Divisors@ n}, Plus @@ (MoebiusMu[d]/Reverse@ d)]; Denominator@ Array[f, 70]
f[p_, e_] := -(p-1)/p^e; a[1] = 1; a[n_] := Denominator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Jun 06 2025 *)

a(n) = denominator(sumdiv(n, d, moebius(n/d)/d)); \\ Michel Marcus, Nov 17 2017

cp's OEIS Frontend

A009262 a(n) = lcm(n, phi(n)).

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A247371 Number of groups of order n for which all Sylow subgroups are cyclic.

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A295126 Denominator of Sum_{d|n} mu(n/d)/d, where mu is the Möbius function A008683.

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Programs

Maple

Mathematica

PARI