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A352483 Numerator of 1/d - 1/n = (n-d)/(n*d) where d is the number of divisors of n (A000005).

%I A352483 #45 Apr 18 2022 09:47:50
%S A352483 0,0,1,1,3,1,5,1,2,3,9,1,11,5,11,11,15,1,17,7,17,9,21,1,22,11,23,11,
%T A352483 27,11,29,13,29,15,31,1,35,17,35,1,39,17,41,19,13,21,45,19,46,11,47,
%U A352483 23,51,23,51,3,53,27,57,1,59,29,19,57,61,29,65,31,65,31,69,5,71,35,23,35,73
%N A352483 Numerator of 1/d - 1/n = (n-d)/(n*d) where d is the number of divisors of n (A000005).
%H A352483 M. F. Hasler, <a href="/A352483/b352483.txt">Table of n, a(n) for n = 1..10000</a> (a(3..10^4) from Michel Marcus), Apr 13 2022
%F A352483 From _Bernard Schott_, Mar 23 2022: (Start)
%F A352483 a(n) = 1 iff n is in A146566.
%F A352483 a(n) = n - 2 iff n is an odd prime (A065091). (End)
%F A352483 From _M. F. Hasler_, Apr 06 2022: (Start)
%F A352483 More generally, explaining the "rays" visible in the graph:
%F A352483 a(n) = n - d with d = 2^w if n is the product of w distinct odd primes, and with d = e+1 if n = p^e, prime p not dividing e+1.
%F A352483 a(n) = n/2 - d with d = 3 if n = 4*p, prime p > 3, and with d = 2^w if n = 2*k where k is the product of w distinct odd primes.
%F A352483 a(n) = n/3 - 2^w if n = 3*p^2 with prime p > 3, w = 1, or if n = 9*k where k is the product of w distinct primes > 3.
%F A352483 a(n) = n/5 - d with d = 2 if n = 5^4*p, odd prime p <> 5, or with d = 4 if n = 3^4*5*p, prime p > 5, not p == 4 (mod 5).
%F A352483 a(n) = n/6 - d with d = 2 if n = 18*p, or with d = 4 if n = 18*p^3 or 18*p*q, primes q > p > 3.
%F A352483 a(n) = (p - 1)/2^m if n = 8*p, where m = max { m <= 3 : 2^m divides p-1 } = min {valuation(p-1, 2), 3}.
%F A352483 a(n) = (n - 12)/9 if n = 3*p^2*q, p and q distinct primes > 3 and q == 1 (mod 3). (End)
%t A352483 a[n_] := Numerator[1/DivisorSigma[0, n] - 1/n]; Array[a, 100] (* _Amiram Eldar_, Apr 13 2022 *)
%o A352483 (PARI) a(n) = my(d=numdiv(n)); denominator(n*d/(n-d));
%o A352483 (PARI) apply( {A352483(n)=numerator(1/numdiv(n)-1/n)}, [3..99]) \\ _M. F. Hasler_, Apr 07 2022
%Y A352483 Cf. A000005, A049820, A065091, A146566, A352482 (denominator).
%K A352483 nonn,frac
%O A352483 1,5
%A A352483 _Michel Marcus_, Mar 18 2022
%E A352483 Definition changed to include indices 1 and 2 by _M. F. Hasler_, Apr 07 2022