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Search: id:A076078
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| A076078 |
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a(n) = number of nonempty sets of distinct positive integers that have a least common multiple of n. |
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+0
12
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| 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; listen)
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OFFSET
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1,2
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COMMENT
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a(n)=1 iff n=1, a(p^k)=2^k, a(p*q)=10; where p & q are unique primes. a(n) can not equal an odd number >1. - Robert G. Wilson v.
If m has more divisors than n, then a(m) > a(n). - Matthew Vandermast (ghodges14(AT)comcast.net), 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 distict primes and r_1,r_2,...,r_k are natural numbers then 2^(r_1*r_2*...*r_k)||a(n). - Farideh Firoozbakht (f.firoozbakht(AT)math.ui.ac.ir), Aug 06 2005
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LINKS
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David Wasserman, Table of n, a(n) for n = 1..1000
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FORMULA
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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).
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EXAMPLE
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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}.
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MATHEMATICA
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f[n_] := Block[{d = Divisors[n]}, Plus @@ (MoebiusMu[n/d](2^DivisorSigma[0, d] - 1))]; Table[ f[n], {n, 75}] (from Robert G. Wilson v)
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PROGRAM
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(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 (dwasserm(AT)earthlink.net), Dec 26 2007
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CROSSREFS
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Cf. A076413, A097210-A097218, A097416.
Cf. A002235.
Adjacent sequences: A076075 A076076 A076077 this_sequence A076079 A076080 A076081
Sequence in context: A131999 A103178 A087909 this_sequence A103314 A111741 A111793
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KEYWORD
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easy,nonn,nice
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AUTHOR
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Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Oct 05 2002
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EXTENSIONS
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Edited by Dean Hickerson (dean(AT)math.ucdavis.edu), Oct 08 2002
Definition corrected by David Wasserman (dwasserm(AT)earthlink.net), Dec 26 2007
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