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

Author

Michel Marcus, Mar 18 2022

Keywords

• nonn

Comments

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

Links

• 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).

Crossrefs

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

Programs

Mathematica

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 *)

PARI

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

Python

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

Formula

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