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.

Showing 1-2 of 2 results.

A347931 Nonprime numbers that divide their second arithmetic derivative.

Original entry on oeis.org

1, 4, 8, 16, 24, 27, 48, 54, 64, 108, 128, 162, 168, 256, 320, 324, 432, 512, 704, 729, 864, 1024, 1134, 1728, 2048, 2160, 2187, 2192, 2916, 3125, 3456, 4096, 4752, 5120, 5832, 6250, 6912, 11664, 12500, 13824, 14580, 14796, 16384, 18750, 19683, 23328, 27648
Offset: 1

Views

Author

Alois P. Heinz, Oct 15 2021

Keywords

Links

Crossrefs

Cf. A000040, A003415, A018252, A068346, A348284.

Programs

  • Maple
    d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
q:= n-> not isprime(n) and irem(d(d(n)), n)=0:
select(q, [$1..30000])[];  # Alois P. Heinz, Oct 15 2021

Formula

{ A348284 } minus { A000040 }.
{ A018252 } intersect { A348284 }.

A361078 Numbers k for which k = gcd(k', k"), where k' is the arithmetic derivative of k (A003415) and k" is the second derivative of k (A068346).

Original entry on oeis.org

4, 16, 27, 64, 108, 432, 729, 1024, 2916, 3125, 4096, 6912, 12500, 16384, 19683, 27648, 46656, 50000, 84375, 110592, 186624, 314928, 337500, 746496, 800000, 823543, 1048576, 1259712, 2125764, 2278125, 3200000, 3294172, 4194304, 5038848, 5400000, 7077888, 8503056
Offset: 1

Views

Author

Marius A. Burtea, Mar 01 2023

Keywords

Comments

The sequence is infinite because for p prime, m = p^p (A051674) is a term.
For the prime number p, the number m = 4^p is a term. Indeed: (4^p)' = p*4^p, (4^p)" = (1 + p^2)*4^p and gcd((4^p)', 4^p) = gcd(p*4^p, (1 + p^2)*4^p) = 4^p*gcd(p, 1 + p^2) = 4^p.
Numbers of the form a*b with a, b in A051674 are terms. Indeed, if m = a*b then m' = a'*b + a*b' = a*a + b*b = 2*a*b = 2*m, m" = a*b + 2*a'b + 2*a*b' = a*b + 2*a*b + 2*a*b = 5*a*b = 5*m and gcd(m', m") = (2*m, 5*m) = m.

Examples

			4' = 4, 4" = 4 and gcd(4', 4") = gcd(4, 4) = 4, so 4 is a term.
16' = 32, 16" = 32' = 80 and gcd(16', 16") = gcd(32, 80) = 16, so 16 is a term.

Crossrefs

Cf. A003415 (n'), A068346 (n"), A051674, A072873, A348284.

Programs

  • Magma
    f:=func; [n:n in [2..100000]|not IsPrime(n) and  Gcd(Floor(f(n)),Floor(f(Floor(f(n))))) eq n];
  • Mathematica
    d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[10^6], GCD[d[#], d[d[#]]] == # &] (* Amiram Eldar, Mar 03 2023 *)
  • PARI
    ader(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(k) = gcd(ader(k), ader(ader(k))) == k; \\ Michel Marcus, Mar 03 2023
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.