A352482 -id:A352482 - cp's OEIS frontend

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

Michel Marcus, Mar 18 2022

M. F. Hasler, Table of n, a(n) for n = 1..10000 (a(3..10^4) from Michel Marcus), Apr 13 2022

Cf. A000005, A049820, A065091, A146566, A352482 (denominator).

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

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

apply( {A352483(n)=numerator(1/numdiv(n)-1/n)}, [3..99]) \\ M. F. Hasler, Apr 07 2022

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)

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

A353012 Numbers N such that gcd(N - d, N*d) >= d^2, where d = A000005(N) is the number of divisors of N.

Original entry on oeis.org

1, 2, 136, 156, 328, 444, 584, 600, 712, 732, 776, 876, 904, 1096, 1164, 1176, 1308, 1544, 1864, 1884, 1928, 2056, 2172, 2248, 2316, 2504, 2601, 2696, 2748, 2824, 2892, 2904, 3208, 3240, 3249, 3272, 3324, 3464, 3592, 3656, 3756, 4044, 4056, 4168, 4188, 4476, 4552, 4616

Offset: 1

M. F. Hasler, Apr 15 2022

nonn

As d^2 | N-d we have N = k*d^2 + d for some k >= 0 and d > 1. So gcd(k*d^2 + d - d, (N*d^2 + d)*d) = gcd(k*d^2, k*d^3 + d^2) = gcd(k*d^2, d^2) = d^2. So for any N such that d^2 | gcd(N - d, N*d) we have gcd(N - d, N*d) = d^2. - David A. Corneth, Apr 20 2022

Since gcd(N - d, N*d) is never larger than d^2 (if N = n*g, d = f*g with gcd(n,f) = 1, then gcd(N - d, N*d) = g*gcd(n-f,n*f*g) = g*gcd(n-f, f*f*g) <= g*g, since by assumption, no factor of f divides n), so one can also replace "=" by ">=" in the definition.

			N = 1 is in the sequence because d(N) = 1, gcd(1 - 1, 1*1) = 1 = d^2.
N = 2 is in the sequence because d(N) = 2, gcd(2 - 2, 2*2) = 4 = d^2.
N = 136 = 8*17 is in the sequence because d(N) = 4*2 = 8, gcd(8*17 - 8, 8*17*8) = gcd(8*16, 8*8*17) = 8*8 = d^2. Similarly for N = 8*p with any prime p = 8*k + 1.
N = 156 = 2^2*3*13 is in the sequence because d(n) = 3*2*2 = 12, gcd(12*13 - 12, 12*13*12) = gcd(12*12, 12*12*13) = 12*12 = d^2. Similarly for any N = 12*p with prime p = 12*k + 1.
More generally, when N = m*p^k with p^k == 1 (mod m) and m = (k+1)*d(m), then d(N) = d(m)*(k+1) = m and gcd(n - d, n*d) = gcd(m*p^k - m, m*p^k*m) = m*gcd(p^k - 1, p^k*m) = m^2. This holds for m = 8 and 12 with k = 1, for m = 9, 18 and 24 with k = 2, etc: see sequence A033950 for the m-values.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000005 (number of divisors), A352483 (numerator of (n-d)/(n*d)), A352482 (denominator), A049820 (n - d), A146566 (n*d is divisible by n-d), A033950 (refactorable or tau numbers: d(n) | n, supersequence of this).

Select[Range[4650], GCD[#1 - #2, #1 #2] == #2^2 & @@ {#, DivisorSigma[0, #]} &] (* Michael De Vlieger, Apr 21 2022 *)

select( {is(n, d=numdiv(n))=gcd(n-d,d^2)==d^2}, [1..10^4])

For all m in A033950, the sequence contains all numbers m*p^k with k = m/d(m) - 1, and p^k == 1 (mod m), in particular 8*A007519 and 12*A068228 (k = 1, m = 8 and 12), 9*A129805^2, 18*A129805^2 and 24*A215848^2 (k = 2, m = 9, 18 and 24, A^2 = {x^2, x in A}), etc.

cp's OEIS Frontend

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

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A353012 Numbers N such that gcd(N - d, N*d) >= d^2, where d = A000005(N) is the number of divisors of N.

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