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A246775
Smallest number x such that phi(x) = phi(x(n)), where x(n) is the n-th arithmetic derivatives of x and x is not equal to x(n).
1
2, 104, 15, 2565, 4947, 2827, 2834, 153, 3462, 16109, 3201, 7413, 31842, 24541, 66563, 15111, 223995, 542270, 733502, 717874, 521666, 2622290, 2111642, 16169763
OFFSET
1,1
EXAMPLE
First 5 arithmetic derivatives of x = 4947 are 1991, 192, 640, 2368, 7168 and phi(4947) = phi(7178) = 3072.
First 9 arithmetic derivatives of x = 3462 are 2891, 875, 650, 635, 132, 188, 192, 640, 2368 and phi(3462) = phi(2368) = 1152.
MAPLE
with(numtheory); P:= proc(q) local a, b, k, n, p, t;
for n from 1 to q do for k from 1 to q do t:=1; b:=k;
while t<=n do a:=b*add(op(2, p)/op(1, p), p=ifactors(b)[2]); b:=a; t:=t+1; od;
if phi(k)=phi(a) and k<>a then lprint(n, k); break; fi;
od; od; end: P(10^6);
MATHEMATICA
d[n_] := If[n < 2, 0, n*Total[#2/#1 & @@@ FactorInteger[n]]]; dd[n_, k_] := Nest[ d, n, k]; aQ[x_, n_] := Module[{xx = dd[x, n]}, xx != x && EulerPhi[x] == EulerPhi[xx]]; a[n_] := Module[{k = 1}, While[! aQ[k, n], k++]; k]; Array[a, 10] (* Amiram Eldar, Apr 06 2019 *)
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Paolo P. Lava, Sep 03 2014
EXTENSIONS
a(17)-a(24) from Amiram Eldar, Apr 06 2019
STATUS
approved