A352483 Numerator of 1/d - 1/n = (n-d)/(n*d) where d is the number of divisors of n (A000005).
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, 27, 11, 29, 13, 29, 15, 31, 1, 35, 17, 35, 1, 39, 17, 41, 19, 13, 21, 45, 19, 46, 11, 47, 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
Offset: 1
Links
- M. F. Hasler, Table of n, a(n) for n = 1..10000 (a(3..10^4) from Michel Marcus), Apr 13 2022
Programs
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Mathematica
a[n_] := Numerator[1/DivisorSigma[0, n] - 1/n]; Array[a, 100] (* Amiram Eldar, Apr 13 2022 *)
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PARI
a(n) = my(d=numdiv(n)); denominator(n*d/(n-d));
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PARI
apply( {A352483(n)=numerator(1/numdiv(n)-1/n)}, [3..99]) \\ M. F. Hasler, Apr 07 2022
Formula
From Bernard Schott, Mar 23 2022: (Start)
a(n) = 1 iff n is in A146566.
a(n) = n - 2 iff n is an odd prime (A065091). (End)
From M. F. Hasler, Apr 06 2022: (Start)
More generally, explaining the "rays" visible in the graph:
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.
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.
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.
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).
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.
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}.
a(n) = (n - 12)/9 if n = 3*p^2*q, p and q distinct primes > 3 and q == 1 (mod 3). (End)
Extensions
Definition changed to include indices 1 and 2 by M. F. Hasler, Apr 07 2022