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A123709
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a(n) = number of nonzero elements in row n of triangle A123706.
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8
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1, 2, 3, 4, 3, 4, 3, 4, 4, 6, 3, 8, 3, 6, 7, 4, 3, 8, 3, 8, 7, 6, 3, 8, 4, 6, 4, 8, 3, 11, 3, 4, 7, 6, 7, 8, 3, 6, 7, 8, 3, 11, 3, 8, 8, 6, 3, 8, 4, 8, 7, 8, 3, 8, 7, 8, 7, 6, 3, 16, 3, 6, 8, 4, 7, 12, 3, 8, 7, 14, 3, 8, 3, 6, 8, 8
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OFFSET
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1,2
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COMMENTS
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Triangle A123706 is the matrix inverse of triangle A010766, where A010766(n,k) = [n/k]. a(n) = 4 when n is in A123710. a(n) = 8 when n is in A123711. a(n) = 16 when n is in A123712.
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LINKS
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M. F. Hasler, Table of n, a(n) for n = 1..500
P. Luschny, Re: Is the A123706 triangle an extension of the Moebius function?, seqcomp list, Feb 12 2012
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FORMULA
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a(n) = 2^(m+1) - 1 when n is the product of m distinct odd primes. [Corrected by M. F. Hasler, Feb 13 2012]
For any k>1, a(n)=2^k if, and only if, n is a nonsquarefree number with A001221(n) = k-1 (= omega(n), number of distinct prime factors), with the only exception of a(n=6)=2^2. - M. F. Hasler, Feb 12 2012
A123709(n) = 1 + #{ k in 1..n-1 | Moebius(n,k+1) <> Moebius(n,k) }, where Moebius(n,k)={moebius(n/k) if n=0 (mod k), 0 else}, cf. link to message by P. Luschny. - M. F. Hasler, Feb 13 2012
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EXAMPLE
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a(n) = 3 when n is an odd prime.
a(n) = 7 when n is the product of two different odd primes. [Corrected by M. F. Hasler, Feb 13 2012]
a(n) = 15 when n is the product of three different odd primes. [Corrected by M. F. Hasler, Feb 13 2012]
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MATHEMATICA
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Moebius[i_, j_]:=If[Divisible[i, j], MoebiusMu[i/j], 0];
A123709[n_]:=Length[Select[Table[Moebius[n, j]-Moebius[n, j+1], {j, 1, n}], #!=0&]];
Array[A123709, 500] (* Enrique Pérez Herrero, Feb 13 2012 *)
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PROG
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(PARI) {a(n)=local(M=matrix(n, n, r, c, if(r>=c, floor(r/c)))^-1); sum(k=1, n, if(M[n, k]==0, 0, 1))}
(PARI) A123709(n)=#select((matrix(n, n, r, c, r\c)^-1)[n, ], x->x) \\ M. F. Hasler, Feb 12 2012
(PARI) A123709(n)={ my(t=moebius(n)); sum(k=2, n, t+0 != t=if(n%k, 0, moebius(n\k)))+1} /* the "t+0 != ..." is required because of a bug in PARI versions <= 2.4.2, maybe beyond, which seems to be fixed in v. 2.5.1 */ \\ M. F. Hasler, Feb 13 2012
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CROSSREFS
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Cf. A123706, A123707, A123708, A123710, A123711, A123712; A010766.
Sequence in context: A242289 A158515 A285884 * A165482 A323242 A099034
Adjacent sequences: A123706 A123707 A123708 * A123710 A123711 A123712
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna, Oct 09 2006
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STATUS
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approved
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