A064086 -id:A064086 - cp's OEIS frontend

A064084 A multiplicative version of 2^n - 1 (A000225).

1, 3, 7, 15, 31, 21, 127, 255, 511, 93, 2047, 105, 8191, 381, 217, 65535, 131071, 1533, 524287, 465, 889, 6141, 8388607, 1785, 33554431, 24573, 134217727, 1905, 536870911, 651, 2147483647, 4294967295, 14329, 393213, 3937, 7665, 137438953471, 1572861, 57337

Offset: 1

Jens Voß, Sep 04 2001

Since n -> 2^n - 1 is an embedding of the ordered structure N = {1, 2, 3, ...} (the order being the "divides" relation) into itself, a(n) always divides A000225(n); the sequence of quotients of A000225 and a is A064085.

			a(6) = (2^2 - 1) * (2^3 - 1) = 21 since 6 = 2 * 3.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000225, A064085, A064086.

a:= n-> mul(2^(i[1]^i[2])-1, i=ifactors(n)[2]):
seq(a(n), n=1..50);  # Alois P. Heinz, Jun 09 2014

a[n_] := Times @@ (2^(Power @@@ FactorInteger[n]) - 1); Array[a, 40] (* Amiram Eldar, Aug 31 2023 *)

a(n) = my(f=factor(n)); for (i=1, #f~, f[i,1] = 2^(f[i,1]^f[i,2])-1; f[i,2]=1); factorback(f); \\ Michel Marcus, Jun 09 2014

a(n) = (2^((p_1)^(e_1)) - 1) * ... * (2^((p_k)^(e_k)) - 1) where (p_1)^(e_1) * ... * (p_k)^(e_k) is the prime factorization of n.

More terms from Michel Marcus, Jun 09 2014

A064085 Quotient of A000225 and A064084.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 11, 1, 39, 1, 43, 151, 1, 1, 171, 1, 2255, 2359, 683, 1, 9399, 1, 2731, 1, 140911, 1, 1649373, 1, 1, 599479, 43691, 8727391, 8965359, 1, 174763, 9588151, 139090655, 1, 1649061309, 1, 572942063, 2221095391, 2796203, 1

Offset: 1

Jens Voß, Sep 04 2001

a(n) is equal to 1 if and only if n is a prime power; the sequence of nontrivial values of a is A064086.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000225, A064084, A064086.

a:= n-> (2^n-1)/mul(2^(i[1]^i[2])-1, i=ifactors(n)[2]):
seq(a(n), n=1..50);  # Alois P. Heinz, Jun 09 2014

a[n_] := (2^n-1)/Times @@ (2^(Power @@@ FactorInteger[n])-1);
Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Jan 13 2025 *)

a(n) = A000225(n) / A064084(n).

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003

cp's OEIS Frontend

A064084 A multiplicative version of 2^n - 1 (A000225).

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