|
| |
|
|
A076078
|
|
a(n) = number of nonempty sets of distinct positive integers that have a least common multiple of n.
|
|
13
| |
|
|
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; 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) 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 (mymontain(AT)yahoo.com), 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). [From Matthew Vandermast (ghodges14(AT)comcast.net), 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. [From Max Alekseyev (maxale(AT)gmail.com), Nov 18 2010]
|
|
|
LINKS
| David Wasserman, Table of n, a(n) for n = 1..1000
Index entries for sequences related to prime signature
|
|
|
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}] (from 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 (dwasserm(AT)earthlink.net), Dec 26 2007
|
|
|
CROSSREFS
| Cf. A076413, A097210-A097218, A097416, A002235.
Sequence in context: A173300 A181236 A087909 * A152061 A103314 A194560
Adjacent sequences: A076075 A076076 A076077 * A076079 A076080 A076081
|
|
|
KEYWORD
| easy,nonn,nice
|
|
|
AUTHOR
| Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Oct 05 2002
|
|
|
EXTENSIONS
| Edited by Dean Hickerson (dean.hickerson(AT)yahoo.com), Oct 08 2002
Definition corrected by David Wasserman (dwasserm(AT)earthlink.net), Dec 26 2007
Edited by Charles R Greathouse IV (charles.greathouse(AT)case.edu), Aug 02 2010
Edited by Max Alekseyev (maxale(AT)gmail.com), Nov 18 2010
|
| |
|
|
