

A352483


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


6



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
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OFFSET

1,5


LINKS



FORMULA

a(n) = n  2 iff n is an odd prime (A065091). (End)
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 p1 } = min {valuation(p1, 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)


MATHEMATICA

a[n_] := Numerator[1/DivisorSigma[0, n]  1/n]; Array[a, 100] (* Amiram Eldar, Apr 13 2022 *)


PROG

(PARI) a(n) = my(d=numdiv(n)); denominator(n*d/(nd));


CROSSREFS



KEYWORD

nonn,frac


AUTHOR



EXTENSIONS

Definition changed to include indices 1 and 2 by M. F. Hasler, Apr 07 2022


STATUS

approved



