A146566 -id:A146566 - 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

A352482 Denominator of (n-d)/n*d where d = A000005(n) is the number of divisors of n.

Original entry on oeis.org

1, 1, 6, 12, 10, 12, 14, 8, 9, 20, 22, 12, 26, 28, 60, 80, 34, 9, 38, 60, 84, 44, 46, 12, 75, 52, 108, 84, 58, 120, 62, 96, 132, 68, 140, 12, 74, 76, 156, 10, 82, 168, 86, 132, 90, 92, 94, 240, 147, 75, 204, 156, 106, 216, 220, 28, 228, 116, 118, 15, 122, 124, 126, 448, 260, 264, 134

Offset: 1

Michel Marcus, Mar 18 2022

The terms are of course the denominators of the fraction "in smallest terms", otherwise said: a(n) = n*d/gcd(n*d, n - d), which is unambiguous also for n = 1 and n = 2 where n - d = 0.

			The number n = 1 has d = 1 divisors, so (n-d)/(n*d) = 0/1 has denominator a(1) = 1.
The number n = 2 has d = 2 divisors, so (n-d)/(n*d) = 0/4 = 0/1 has denominator a(2) =  1 when written in smallest terms.
The number n = 3 has d = 2 divisors, so (n-d)/(n*d) = 1/6 has denominator a(3) =  6.
The number n = 4 has d = 3 divisors, so (n-d)/(n*d) = 1/12 has denominator a(4) = 12.
The number n = 6 has d = 4 divisors, so (n-d)/(n*d) = 2/24 = 1/12 has denominator a(6) = 12.

M. F. Hasler, Table of n, a(n) for n = 1..10000 (terms a(3..10^4) from Michel Marcus), Apr 17 2022

Cf. A000005, A049820, A146566, A352483 (numerator).

a[n_] := Numerator[n*(d = DivisorSigma[0, n])/(n - d)]; Array[a, 100, 3] (* Amiram Eldar, Mar 18 2022 *)

A352482(n,d=numdiv(n))=denominator((n-d)/(n*d))

Edited and extended to offset 1 by M. F. Hasler, Apr 17 2022

A152492 a(n) = number of integers of the form (n*k)^2/(k^2 - n^2).

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 0, 1, 1, 1, 0, 8, 0, 0, 4, 1, 0, 2, 0, 4, 3, 0, 0, 9, 0, 0, 1, 2, 0, 7, 0, 1, 2, 0, 1, 8, 0, 0, 1, 4, 0, 5, 0, 1, 5, 0, 0, 9, 0, 1, 1, 1, 0, 2, 1, 4, 1, 0, 0, 23, 0, 0, 3, 1, 1, 4, 0, 1, 1, 2, 0, 10, 0, 0, 4, 1, 0, 4, 0, 4, 1, 0, 0, 17, 0, 0, 1, 1, 0, 8

Offset: 1

Ctibor O. Zizka, Dec 06 2008

k needs to be checked only up through n^2+1 since beyond this n^2 < (n*k)^2/(k^2 - n^2) < n^2 + 1 and thus can't be an integer. - Micah Manary, Aug 27 2022

Micah Manary, Table of n, a(n) for n = 1..1000

Cf. A071086, A146567, A146566, A146564.

a(n) = sum(k=1, n^2+1, if (k!=n, denominator((n*k)^2/(k^2 - n^2))==1)); \\ Michel Marcus, Oct 28 2022

More terms from Micah Manary, Aug 07 2022

A351913 Least k such that A352483(k) = n, or -1 if no such k exists.

Original entry on oeis.org

3, 9, 5, 204, 7, 876, 20, 140, 11, 492, 13, 776, 32, 904, 17, 441, 19, 23364, 44, 2178, 23, 25, 27, 1544, 216, 3756, 29, 460, 31, 1928, 35, 2056, 280, 1644, 37, 5196, 117, 162, 41, 1089, 43, 2696, 92, 2824, 47, 49, 51, 6924, 153, 812, 53, 7524, 57, 3464, 116, 1521, 59, 940, 61

Offset: 1

Michel Marcus, Mar 18 2022

nonn

What is the value of a(102)?

Conjecture: a(102) and all "Unknown" values in the a-file equal -1. - Paolo Xausa, Aug 16 2022

Michel Marcus, Table of n, a(n) for n = 1..101
Michel Marcus and Paolo Xausa, Table of n, a(n) for n = 1..10000 (with a(n) noted as Unknown when the value is not known; search up to 125*10^9).
Paolo Xausa, Log-log scatterplot of a(n), n = 1..10000, with unknown values (in orange) placed at the search limit (125*10^9).

Cf. A000005, A040976, A049820, A146566, A352483.

a[n_] := Module[{k = 3}, While[Denominator[k*(d = DivisorSigma[0, k])/(k - d)] != n, k++]; k]; Array[a, 60] (* Amiram Eldar, Mar 18 2022 *)

f(n) = my(d=numdiv(n)); denominator(n*d/(n-d)); \\ A352483
a(n) = {my(k=3); while (f(k) != n, k++); k;}

from math import gcd
from sympy import divisor_count
from itertools import count, islice
def f(n): d = divisor_count(n); g = gcd(n-d, n*d); return (n-d)//g
def agen():
    n, adict = 1, dict()
    for k in count(1):
        fk = f(k)
        if fk not in adict: adict[fk] = k
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 60))) # Michael S. Branicky, Jul 23 2022

a(n) = n+2 iff n > 0 is a term of A040976. - Bernard Schott, Mar 24 2022

A153193 a(n) is the number of integers of the form n(n+1)k / (k - n*(n+1)) where k is an integer >= 1.

Original entry on oeis.org

4, 13, 22, 22, 40, 40, 31, 52, 67, 40, 67, 67, 40, 121, 121, 40, 67, 67, 67, 202, 121, 40, 94, 157, 67, 94, 157, 67, 121, 121, 49, 148, 121, 121, 337, 112, 40, 121, 283, 94, 121, 121, 67, 337, 202, 40, 121, 202, 112, 202, 202, 67, 94, 283, 283, 283, 121, 40

Offset: 1

Ctibor O. Zizka, Dec 20 2008

nonn

1/n - 1/(n+1) - 1/k = 1/c where c is an integer, k >= 1.

			The a(1)=4 integers of the form n*(n+1)*k/(k - n*(n+1)) = 1*(1+1)*k/(k - 1*(1+1)) = 2*k/(k-2) occur at
  k=1: 2*1/(1-2) = -2,
  k=3: 2*3/(3-2) =  6,
  k=4: 2*4/(4-2) =  4, and
  k=6: 2*6/(6-2) =  3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A146564, A152492, A063647, A000005, A146566.

f:= proc(n) local D;
   D:= numtheory:-divisors((n*(n+1))^2);
   nops(D) + nops(select(`<=`,D,n*(n+1)-1))
end proc:
map(f, [$1..100]); # Robert Israel, Oct 21 2024

a(13)-a(58) from Jon E. Schoenfield, Mar 15 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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A352482 Denominator of (n-d)/n*d where d = A000005(n) is the number of divisors of n.

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A152492 a(n) = number of integers of the form (n*k)^2/(k^2 - n^2).

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A351913 Least k such that A352483(k) = n, or -1 if no such k exists.

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A153193 a(n) is the number of integers of the form n*(n+1)*k / (k - n*(n+1)) where k is an integer >= 1.

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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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A153193 a(n) is the number of integers of the form n(n+1)k / (k - n*(n+1)) where k is an integer >= 1.