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 A076078 a(n) = number of nonempty sets of distinct positive integers that have a least common multiple of n. 38
 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 (list; graph; refs; listen; history; text; internal format)
 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 nonnegaive 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 A000255(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 David Wasserman, Table of n, a(n) for n = 1..1000 FORMULA 2^d(n) - 1 = sum(a(m), m divides n), where d(n)=A000005(n) is the number of divisors of n, so a(n) = sum(mu(n/m)*(2^d(m)-1), m divides n). 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}. 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); 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 Cf. A076413, A097210-A097218, A097416, A002235. Sequence in context: A280684 A322671 A087909 * A292786 A326486 A053204 Adjacent sequences:  A076075 A076076 A076077 * A076079 A076080 A076081 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

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Last modified January 19 04:17 EST 2020. Contains 331031 sequences. (Running on oeis4.)