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A269459 Numbers that are equal to the sum of the number of divisors of their first k arithmetic derivatives, for some k. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

Table of n, a(n) for n=1..34.

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,more

AUTHOR

Paolo P. Lava, Apr 06 2016

STATUS

approved

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Last modified October 20 21:31 EDT 2018. Contains 316404 sequences. (Running on oeis4.)