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A090780 a(n) = n*Product_{p prime, p|n} (p - 1)/2. 1
1, 1, 3, 2, 10, 3, 21, 4, 9, 10, 55, 6, 78, 21, 30, 8, 136, 9, 171, 20, 63, 55, 253, 12, 50, 78, 27, 42, 406, 30, 465, 16, 165, 136, 210, 18, 666, 171, 234, 40, 820, 63, 903, 110, 90, 253, 1081, 24, 147, 50, 408, 156, 1378, 27, 550, 84, 513, 406, 1711, 60, 1830, 465, 189 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

a(2n+1) is the conjectured value of the length of period of sequence of Genocchi number of first kind read modulo (2n + 1) (cf. A001469).

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = (n/(-2)^omega(n))*(Sum_{d|n} d*mu(d)) = n*A023900(n)/(A076479(n)*A034444(n)).

a(n) = n*A173557(n)/2. - R. J. Mathar, Apr 14 2011

From Jianing Song, Nov 22 2018: (Start)

Multiplicative with a(p^e) = (p - 1)*p^e/2 = A000217(p-1)*p^(e-1).

a(n) = A299822(n)/2^A001221(n).

a(prime(n)) = A034953(n).

a(n) is odd if and only if n = A004614(k) or 2*A004614(k). (End)

MAPLE

A023900 := proc(n) add( d*numtheory[mobius](d), d=numtheory[divisors](n)) ; end proc:

A001221 := proc(n) nops(numtheory[factorset](n)) ; end proc:

A076479 := proc(n) (-1)^A001221(n) ; end proc:

A034444 := proc(n) 2^A001221(n) ; end proc:

A090780 := proc(n) n/A076479(n)/A034444(n) *A023900(n); end proc:

seq(A090780(n), n=1..20) ; # R. J. Mathar, Apr 14 2011

MATHEMATICA

a[n_] := Module[{f, p, e}, fun[p_, e_] := (p - 1)*p^e/2;

If[n == 1, 1, Times @@ (fun @@@ FactorInteger[n])]]; Array[a, 50] (* Amiram Eldar, Nov 23 2018 *)

CROSSREFS

Cf. A023900, A034444. - R. J. Mathar, Feb 08 2011

Sequence in context: A114486 A176743 A220466 * A184174 A277821 A318280

Adjacent sequences:  A090777 A090778 A090779 * A090781 A090782 A090783

KEYWORD

nonn,mult

AUTHOR

Benoit Cloitre, Feb 12 2004

STATUS

approved

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Last modified May 25 15:24 EDT 2019. Contains 323572 sequences. (Running on oeis4.)