A375238 Least k with exactly n partitions k = x + y + z satisfying k' = x' + y' + z', where k' is the arithmetic derivative of k.

5, 9, 22, 35, 65, 63, 70, 62, 82, 110, 75, 143, 130, 169, 142, 186, 170, 194, 230, 284, 234, 195, 147, 345, 238, 245, 323, 290, 286, 294, 285, 334, 430, 534, 458, 255, 385, 434, 390, 418, 374, 399, 441, 526, 518, 382, 748, 598, 578, 454, 455, 585, 507, 435, 582

Offset

1,1

Links

Robert Israel, Table of n, a(n) for n = 1..300

Example

a(7) = 70 and 70 has 7 partitions of three numbers, x, y and z, for which 70' = x' + y' + z' = 59. In fact:
  5' + 21' + 44' = 1 + 10 + 48 = 59;
  6' + 14' + 50' = 5 + 9 + 45 = 59;
  6' + 22' + 42' = 5 + 13 + 41 = 59;
  10' + 10' + 50' = 7 + 7 + 45 = 59;
  13' + 24' + 33' = 1 + 44 + 14 = 59;
  13' + 27' + 30' = 1 + 27 + 31 = 59;
  14' + 14' + 42' = 9 + 9 + 41 = 59.
Furthermore 70 is the minimum number to have this property.

Maple

ader:= proc(n) local F, t;
  F:= ifactors(n)[2];
  add(t[2]*n/t[1], t= F)
end proc:
g:= proc(n) local count, np, x, y, z, xp, yp, zp;
  count:= 0; np:= ader(n);
  for x from 1 to n/3 do
    xp:= ader(x);
    if xp > np then next fi;
    for y from x to (n-x)/2 do
      yp:= ader(y);
      z:= n-x-y;
      zp:= ader(z);
      if xp + yp + zp = np then count:= count+1;  fi;
  od od;
  count
end proc:
N:= 60: # for a(1) .. a(N)
count:= 0: V:= Vector(N):
for n from 1 while count < N do
  v:= g(n);
  if v > 0 and v <= N and V[v] = 0 then
     count:= count+1; V[v]:= n;
  fi
od:
convert(V, list); # Robert Israel, Jan 26 2026

Crossrefs

Cf. A003415, A212662, A373047, A375154.

Sequence in context: A110200 A063404 A102177 * A299266 A219521 A215178

Adjacent sequences: A375235 A375236 A375237 * A375239 A375240 A375241

Keyword

nonn,base

Author

Paolo P. Lava, Aug 06 2024

Status

approved