A076078 a(n) is the number of nonempty sets of distinct positive integers that have a least common multiple of n.

1, 2, 2, 4, 2, 10, 2, 8, 4, 10, 2, 44, 2, 10, 10, 16, 2, 44, 2, 44, 10, 10, 2, 184, 4, 10, 8, 44, 2, 218, 2, 32, 10, 10, 10, 400, 2, 10, 10, 184, 2, 218, 2, 44, 44, 10, 2, 752, 4, 44, 10, 44, 2, 184, 10, 184, 10, 10, 2, 3748, 2, 10, 44, 64, 10, 218, 2, 44, 10, 218, 2, 3392, 2, 10

Offset: 1

Author

Amarnath Murthy, Oct 05 2002

Keywords

• easy
• nonn
• nice

Comments

a(n)=1 iff n=1, a(p^k)=2^k, a(p*q)=10; where p & q are unique primes. a(n) cannot equal an odd number >1. - Robert G. Wilson v

If m has more divisors than n, then a(m) > a(n). - Matthew Vandermast, Aug 22 2004

If n is of the form p^r*q^s where p & q are distinct primes and r & s are nonnegative integers then a(n)=2^(rs)*(2^(r+s+1) -2^r-2^s+1); for example f(1400846643)=f(3^5*7^8)=2^(5*8)*(2^ (5+8+1)-2^5-2^8+1)=17698838672310272. Also if n=p_1^r_1*p_2^r_2*...*p_k^r_k where p_1,p_2,...,p_k are distinct primes and r_1,r_2,...,r_k are natural numbers then 2^(r_1*r_2*...*r_k)||a(n). - Farideh Firoozbakht, Aug 06 2005

None of terms is divisible by Mersenne numbers 3 or 7. For any n, a(n) is congruent to A008836(n) mod 3. Since A008836(n) is always 1 or -1, this implies that A000225(2)=3 never divides a(n). - Matthew Vandermast, Oct 12 2010

There are terms divisible by larger Mersenne numbers. For example, a(2*3*5*7*11*13*19*23^3) is divisible by 31. - Max Alekseyev, Nov 18 2010

Examples

			a(6) = 10. The sets with LCM 6 are {6}, {1,6}, {2,3}, {2,6}, {3,6}, {1,2,3}, {1,2,6}, {1,3,6}, {2,3,6}, {1,2,3,6}.
		

Links

• David Wasserman, Table of n, a(n) for n = 1..1000
• Index entries for sequences related to prime signature

Crossrefs

Cf. A076413, A097210-A097218, A097416, A002235.

Cf. A069626.

Programs

Maple

with(numtheory): seq(add(mobius(n/d)*(2^tau(d)-1), d in divisors(n)), n=1..80); # Ridouane Oudra, Mar 12 2024

Mathematica

f[n_] := Block[{d = Divisors[n]}, Plus @@ (MoebiusMu[n/d](2^DivisorSigma[0, d] - 1))]; Table[ f[n], {n, 75}] (* Robert G. Wilson v *)

PARI

a(n) = local(f, l, s, t, q); f = factor(n); l = matsize(f)[1]; s = 0; forvec(v = vector(l, i, [0, 1]), q = sum(i = 1, l, v[i]); t = (-1)^(l - q)*2^prod(i = 1, l, f[i, 2] + v[i]); s += t); s; \\ Definition corrected by David Wasserman, Dec 26 2007

Formula

2^d(n) - 1 = Sum_{m|n} a(m), where d(n) = A000005(n) is the number of divisors of n, so a(n) = Sum_{m|n} mu(n/m)*(2^d(m) - 1).

a(n) = 2*A069626(n), for n > 1. - Ridouane Oudra, Mar 12 2024

Extensions

Edited by Dean Hickerson, Oct 08 2002

Definition corrected by David Wasserman, Dec 26 2007

Edited by Charles R Greathouse IV, Aug 02 2010

Edited by Max Alekseyev, Nov 18 2010