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A090780 a(n) = n*Product_{p prime, p|n} (p - 1)/2. 1

%I #40 Aug 11 2023 09:36:06

%S 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,

%T 78,27,42,406,30,465,16,165,136,210,18,666,171,234,40,820,63,903,110,

%U 90,253,1081,24,147,50,408,156,1378,27,550,84,513,406,1711,60,1830,465,189

%N a(n) = n*Product_{p prime, p|n} (p - 1)/2.

%C 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).

%H Robert Israel, <a href="/A090780/b090780.txt">Table of n, a(n) for n = 1..10000</a>

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

%F a(n) = n*A173557(n)/2. - _R. J. Mathar_, Apr 14 2011

%F From _Jianing Song_, Nov 22 2018: (Start)

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

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

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

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

%F Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 2/(p-1)^2) = 5.72671092223951683002237367406848393189560038246828458038126468772919585... - _Vaclav Kotesovec_, Sep 20 2020

%F From _Jianing Song_, Aug 11 2023: (Start)

%F a(n) = phi(n) * Product_{p|n, p prime} (p/2), where phi = A000010.

%F Equals A000010(n)*A007947(n)/2^A001221(n). (End)

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

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

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

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

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

%p seq(A090780(n),n=1..20) ; # _R. J. Mathar_, Apr 14 2011

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

%t If[n == 1, 1, Times @@ (fun @@@ FactorInteger[n])]]; Array[a, 50] (* _Amiram Eldar_, Nov 23 2018 *)

%o (PARI) a(n) = my(f=factor(n)[,1]); n*prod(k=1, #f, (f[k]-1)/2); \\ _Michel Marcus_, May 26 2019

%o (PARI) a(n) = eulerphi(n)*factorback(factorint(n)[, 1]/2) \\ _Jianing Song_, Aug 11 2023

%Y Cf. A023900, A034444. - _R. J. Mathar_, Feb 08 2011

%Y Cf. A299822, A000010, A007947, A001221.

%K nonn,mult

%O 1,3

%A _Benoit Cloitre_, Feb 12 2004

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Last modified March 28 08:02 EDT 2024. Contains 371236 sequences. (Running on oeis4.)