A378930 -id:A378930 - cp's OEIS frontend

A379146 Numbers k that are not in A378930 (i.e., are never the value of f(n) = n * d(n) / gcd(n, d(n))^2, where d = A000005).

18, 27, 45, 63, 64, 72, 99, 112, 117, 144, 153, 160, 171, 207, 225, 243, 252, 261, 279, 288, 320, 333, 336, 352, 360, 369, 387, 396, 416, 423, 441, 468, 477, 504, 531, 544, 549, 567, 576, 603, 608, 612, 616, 625, 639, 657, 684, 711, 728, 736, 747, 792, 801, 828, 873, 880, 891, 909, 927, 928, 936, 952, 963, 981, 992

Offset: 1

Viliam Furík, Dec 16 2024

nonn

Verified using the known values of f(n) up to a limit determined by the upper bound for divisor function 2*sqrt(n). The lower bound for f(n) is n/d(n), which can be combined with d(n) <= 2*sqrt(n) to yield f(n) >= sqrt(n)/2, and n <= 4*f(n)^2. E.g. for f(n) = 18, this means that checking f(n) for n <= 1296 is sufficient to verify it's never the value of f(n).

It appears that all numbers 9*p, p prime, are in this sequence.

Viliam Furík, Table of n, a(n) for n = 1..13935

Cf. A000005. Complement of A378930.

A378929 Number of steps it takes for the chain of f(n) = n * d(n) / gcd(n, d(n))^2 to reach a cycle from a starting number n.

Original entry on oeis.org

0, 1, 1, 3, 1, 0, 1, 2, 2, 0, 1, 2, 1, 0, 3, 4, 1, 2, 1, 4, 3, 0, 1, 2, 1, 0, 4, 4, 1, 3, 1, 6, 3, 0, 1, 4, 1, 0, 3, 2, 1, 3, 1, 4, 4, 0, 1, 5, 1, 0, 3, 4, 1, 4, 1, 2, 3, 0, 1, 2, 1, 0, 4, 8, 1, 3, 1, 4, 3, 0, 1, 1, 1, 0, 0, 4, 1, 3, 1, 3, 1, 0, 1, 2, 1, 0, 3, 2

Offset: 1

Viliam Furík, Dec 11 2024

nonn

It is conjectured that every number n eventually ends up in a cycle.

Viliam Furík, Table of n, a(n) for n = 1..10000

Cf. A000005, A378930.

f[n_] := Module[{d = DivisorSigma[0, n]}, n*d/GCD[n, d]^2]; a[n_] := Module[{s = NestWhileList[f, n, UnsameQ, All]}, FirstPosition[s, s[[-1]]][[1]] - 1]; Array[a, 100] (* Amiram Eldar, Dec 16 2024 *)

cp's OEIS Frontend

A379146 Numbers k that are not in A378930 (i.e., are never the value of f(n) = n * d(n) / gcd(n, d(n))^2, where d = A000005).

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