A097702 - OEIS

%I #24 Sep 01 2024 09:37:02

%S 0,2,3,5,6,8,9,11,14,15,17,18,20,21,23,26,27,29,30,32,33,35,36,38,39,

%T 41,42,44,45,47,48,50,51,53,54,56,57,59,63,65,66,68,69,71,72,74,75,77,

%U 78,80,81,83,86,89,90,92,93,95,96,98,99,101,102,104,105,107,108,110

%N a(n) = (A063880(n) - 108)/216.

%C Conjecture: k is a term iff 6*k+3 is squarefree. - _Vladeta Jovovic_, Aug 27 2004

%C It is only a conjecture that all terms are integers (confirmed up to 10^6 by _Robert G. Wilson v_).

%C From _Amiram Eldar_, Aug 31 2024: (Start)

%C The first conjecture is true. If m = 216*k + 108 = 108 * (2*k + 1) is a term of A063880, then 2*k+1 is a squarefree number coprime to 6. This is because sigma(n)/usigma(n) is multiplicative, equals 1 if and only if n is squarefree and larger than 1 otherwise, sigma(108)/usigma(108) = 2 and sigma(3^k)/usigma(3^k) increases with k. 6*k+3 = 3*(2*k+1) is squarefree because 2*k+1 is a squarefree coprime to 6.

%C Assuming that (A063880(n) - 108)/216 is an integer for all n, we have a(n) = (A276378(n) - 1)/2. (End)

%H Amiram Eldar, <a href="/A097702/b097702.txt">Table of n, a(n) for n = 1..10000</a>

%t usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[ #, n/# ] == 1 &]]; (Select[ Range[ 24500], DivisorSigma[1, # ] == 2usigma[ # ] &] - 108)/216 (* _Robert G. Wilson v_, Aug 28 2004 *)

%o (PARI) {u(n)=sumdiv(n,d,if(gcd(d,n/d)==1,d))}

%o n=2; while(n<50000,n++; if(sigma(n)==2*u(n),print1((n-108)/216", ")))

%Y Cf. A000203, A005117, A034448, A063880, A097703, A276378.

%K nonn

%O 1,2

%A _Ralf Stephan_, Aug 26 2004