A269459 Numbers that are equal to the sum of the number of divisors of their first k arithmetic derivatives, for some k.

15, 16, 45, 60, 69, 75, 112, 116, 236, 296, 319, 452, 576, 586, 843, 1047, 1184, 1704, 1902, 2852, 2966, 3068, 3122, 4708, 4805, 5684, 6150, 6712, 7126, 10920, 10950, 13107, 16700, 18698, 27828, 29309, 31142, 31448, 31764, 43152, 48584, 51609, 53822, 62472, 63008

Offset

1,1

Links

Table of n, a(n) for n=1..45.

Example

The first eight arithmetic derivatives of 75 are 55, 16, 32, 80, 176, 368, 752, 1520 and d(55) + d(16) + d(32) + d(80) + d(176) + d(368) + d(752) + d(1520) = 4 + 5 + 6 + 10 + 10 + 10 + 10 + 20 = 75.

Maple

with(numtheory): P:=proc(q) local a, b, k, n, p; for n from 1 to q do a:=0; k:=1; b:=n;
while a<n do b:=b*add(op(2, p)/op(1, p), p=ifactors(b)[2]); if b>0 then a:=a+tau(b); else break; fi; od;
if n=a then print(n); fi; od; end: P(10^6);

Prog

(PARI) ad(n) = if (n<1, 0, my(f = factor(n)); n*sum(k=1, #f~, f[k, 2]/f[k, 1]));
isok(n) = {ss = 0; kn = n; while (ss < n, der = ad(kn); if (der == 0, break); ss += numdiv(der); kn = der); ss == n; } \\ Michel Marcus, Apr 08 2016

Crossrefs

Cf. A000005, A003415, A270389, A270713.

Sequence in context: A041450 A042867 A296863 * A041462 A041464 A041460

Adjacent sequences: A269456 A269457 A269458 * A269460 A269461 A269462

Keyword

nonn

Author

Paolo P. Lava, Apr 06 2016

Extensions

a(35)-a(45) from Amiram Eldar, Jun 23 2023

Status

approved