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A348283 Numbers that are multiples of their arithmetic derivative, A003415.

%I A348283 #38 May 22 2025 09:38:57
%S A348283 0,2,3,4,5,7,11,13,17,19,23,27,29,31,37,41,43,47,53,59,61,67,71,73,79,
%T A348283 83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,
%U A348283 179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293
%N A348283 Numbers that are multiples of their arithmetic derivative, A003415.
%C A348283 Here, m' denotes the arithmetic derivative of m (A003415).
%C A348283 Not the same as A211781 since this sequence does not contain 225, 252, etc.
%C A348283 All prime numbers p are in the sequence since p' = 1 | p.
%C A348283 Numbers k such that k' | k. - The original definition of the sequence.
%C A348283 Sequence consists of 0, primes, and the prime powers of the form p^p (A051674, that together with 0 give the only fixed points of A003415). This can be seen from theorems 4-6 given in the Ufnarovski & Åhlander paper. - _Antti Karttunen_, May 17 2025
%H A348283 Antti Karttunen & R. J. Mathar, <a href="/A348283/b348283.txt">Table of n, a(n) for n = 1..20001</a>
%H A348283 Victor Ufnarovski and Bo Åhlander, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarovski.html">How to Differentiate a Number</a>, J. Integer Seqs., Vol. 6, 2003, #03.3.4.
%e A348283 0 is in the sequence as A003415(0) = 0 and 0 is a multiple 0.
%e A348283 27 is in the sequence as A003415(27) = 27' = 27, and 27 is a multiple of 27.
%e A348283 127 (like any prime) is in the sequence since 127' = 1 | 127.
%p A348283 q:= n-> is(irem(n, n*add(i[2]/i[1], i=ifactors(n)[2]))=0):
%p A348283 select(q, [$2..300])[];  # _Alois P. Heinz_, Oct 11 2021
%o A348283 (PARI) ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
%o A348283 isok(k) = (0==k) || ((k>1) && !(k % ad(k))); \\ _Michel Marcus_, Oct 10 2021
%Y A348283 Cf. A003415, A211781.
%Y A348283 After the initial zero, gives the indices of 0's in A369049.
%Y A348283 Disjoint union of {0}, A000040 and A051674.
%Y A348283 Apart from term 2, a subsequence of A383300.
%K A348283 nonn
%O A348283 1,2
%A A348283 _Wesley Ivan Hurt_, Oct 09 2021
%E A348283 a(1) = 0 inserted because of a new, more inclusive definition. - _Antti Karttunen_, May 17 2025