A352483 -id:A352483 - cp's OEIS frontend

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

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

A352710 Least k such that A352483(k) = 2*n-1, or -1 if no such k exists.

Original entry on oeis.org

3, 5, 7, 20, 11, 13, 32, 17, 19, 44, 23, 27, 216, 29, 31, 35, 280, 37, 117, 41, 43, 92, 47, 51, 153, 53, 57, 116, 59, 61, 65, 520, 67, 207, 71, 73, 77, 616, 79, 164, 83, 87, 261, 89, 93, 95, 760, 97, 105, 101, 103, 212, 107, 109, 333, 113, 920, 119, 952, 123, 125, 1000, 127, 135, 131

Offset: 1

Michel Marcus, Mar 30 2022

nonn

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A352483.

Bisection of A351913.

f(n) = my(d=numdiv(n)); denominator(n*d/(n-d)); \\ A352483
a(n) = {my(k=3, q=2*n-1); while (f(k) != q, k++); k; }

a(n) = A351913(2*n-1).

A146566 Numbers k such that k*sigma_0(k) is divisible by (k - sigma_0(k)).

Original entry on oeis.org

3, 4, 6, 8, 12, 18, 24, 36, 40, 60, 84, 156, 180, 600

Offset: 1

Ctibor O. Zizka, Nov 01 2008

No other term < 10000000. - Michel Marcus, Jun 02 2013

No other term multiple of 12 below 10^9. - M. F. Hasler, Apr 16 2022

Cf. A000005, A049820, A352483.

nsiQ[n_]:=Module[{s=DivisorSigma[0,n]},IntegerQ[(n*s)/(n-s)]]; Select[ Range[3,600],nsiQ] (* Harvey P. Dale, Dec 05 2019 *)

isok(n) = {my(nd = numdiv(n)); type(n*nd/(n-nd)) == "t_INT"}  \\ Michel Marcus, Jun 02 2013

is_A146566(n,d=numdiv(n))={n*d%(n-d)==0} \\ M. F. Hasler, Apr 16 2022

A352483(a(n)) = 1. - Bernard Schott, Mar 23 2022

Corrected and extended by Michel Marcus, Jun 02 2013

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

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

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

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A352710 Least k such that A352483(k) = 2*n-1, or -1 if no such k exists.

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A146566 Numbers k such that k*sigma_0(k) is divisible by (k - sigma_0(k)).

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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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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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