cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Viliam Furík

Viliam Furík's wiki page.

Viliam Furík has authored 3 sequences.

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).

Original entry on oeis.org

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

Views

Author

Viliam Furík, Dec 16 2024

Keywords

Comments

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.

Links

Crossrefs

Cf. A000005. Complement of A378930.

A378930 a(n) = n * d(n) / gcd(n, d(n))^2, where d = A000005.

Original entry on oeis.org

1, 1, 6, 12, 10, 6, 14, 2, 3, 10, 22, 2, 26, 14, 60, 80, 34, 3, 38, 30, 84, 22, 46, 3, 75, 26, 108, 42, 58, 60, 62, 48, 132, 34, 140, 4, 74, 38, 156, 5, 82, 84, 86, 66, 30, 46, 94, 120, 147, 75, 204, 78, 106, 108, 220, 7, 228, 58, 118, 5, 122, 62, 42, 448, 260
Offset: 1

Views

Author

Viliam Furík, Dec 11 2024

Keywords

Links

Crossrefs

Cf. A000005, A009191, A009230, A038040.

Programs

  • Maple
    a:= n-> (d-> n*d/igcd(n,d)^2)(numtheory[tau](n)):
seq(a(n), n=1..65);  # Alois P. Heinz, Dec 11 2024
  • Mathematica
    a[n_]:= n * DivisorSigma[0,n]/GCD[n, DivisorSigma[0,n]]^2; Array[a,65] (* Stefano Spezia, Dec 11 2024 *)
  • PARI
    a(n) = my(d = numdiv(n)); n * d / gcd(n, d)^2; \\ Amiram Eldar, Dec 11 2024

Formula

a(n) = A038040(n)/A009191(n)^2.
a(n) = A009230(n)/A009191(n).

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

Views

Author

Viliam Furík, Dec 11 2024

Keywords

Comments

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

Links

Crossrefs

Cf. A000005, A378930.

Programs

  • Mathematica
    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 *)

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