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A083058 Number of eigenvalues equal to 1 of n X n matrix A(i,j)=1 if j=1 or i divides j. 8
1, 0, 1, 1, 2, 3, 4, 4, 5, 6, 7, 8, 9, 10, 11, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

All numbers occur at least once, but the A000295: Eulerian numbers > 0 appear twice. - Robert G. Wilson v, Apr 19 2006

It appears that a(n)=sum{k=0..n-1, (1+(-1)^A000108(k))/2} (n>1). - Paul Barry, Mar 31 2008

Barry's observation above is true because A000108 obtains odd values only at points (2^j)-1 (A000225) and here the repeated values (A000295) occur precisely at positions given by A000225 and A000079. - Antti Karttunen, Aug 17 2013

a(n)+1 gives a lower bound for nonzero terms of A228086 and A228087. - Antti Karttunen, Aug 17 2013

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..8192

J. B. Conrey, The Riemann Hypothesis, Notices Amer. Math. Soc., 50 (No. 3, March 2003), 341-353. See p. 347.

Will Dana, Eigenvalues of the Redheffer Matrix and their relation to the Mertens function (2015), Theorem 5.

R. Stephan, Some divide-and-conquer sequences ...

R. Stephan, Table of generating functions

FORMULA

a(n) = n - A070939(n), n>1.

a(1)=1, else a(n)=b(n) with b(0)=0, b(2n)=b(n)+n-1, b(2n+1)=b(n)+n. - Ralf Stephan, Oct 11 2003

Except for a(1), a(n) = n - 1 - floor(log(2,n)). - Robert G. Wilson v, Apr 19 2006

It seems that a(n) = A182220(n+1)-1 for all n>1. - Antti Karttunen, Aug 17 2013

MAPLE

A083058 := proc(n)

    if n = 1 then

        1;

    else

        n-floor(log[2](n))-1 ;

    end if;

end proc:

seq(A083058(n), n=1..40) ; # R. J. Mathar, Jul 23 2017

PROG

(PARI) a(n)=if(n<2, n>0, n-floor(log(n)/log(2))-1)

(PARI) a(n)= if(n<1, 0, valuation( subst( charpoly( matrix(n, n, i, j, (j==1) || (0==j%i))), x, x+1), x))

(Scheme) (define (A083058 n) (if (< n 2) n (- n (A070939 n)))) ;; Antti Karttunen, Aug 17 2013

CROSSREFS

Cf. A002321, A070939, A143104.

Sequence in context: A268234 A253188 A105598 * A290082 A127035 A268084

Adjacent sequences:  A083055 A083056 A083057 * A083059 A083060 A083061

KEYWORD

nonn

AUTHOR

Michael Somos, Apr 18 2003

STATUS

approved

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Last modified February 24 04:33 EST 2018. Contains 299595 sequences. (Running on oeis4.)