OFFSET
1,2
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
LINKS
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
EXAMPLE
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}.
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 *)
PROG
(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
CROSSREFS
KEYWORD
easy,nonn,nice
AUTHOR
Amarnath Murthy, Oct 05 2002
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
STATUS
approved