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

Author

Paolo P. Lava, Apr 06 2016

Keywords

• nonn

Examples

			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.
		

Crossrefs

Cf. A000005, A003415, A270389, A270713.

Programs

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 a0 then a:=a+tau(b); else break; fi; od;
if n=a then print(n); fi; od; end: P(10^6);

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

Extensions

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