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A301939 Integers whose arithmetic derivative is equal to their Dedekind function.

%I A301939 #25 Feb 23 2022 22:53:47
%S A301939 8,81,108,2500,2700,3375,5292,13068,15625,18252,31212,38988,57132,
%T A301939 67228,90828,94500,103788,147852,181548,199692,231525,238572,303372,
%U A301939 375948,401868,484812,544428,575532,674028,713097,744012,855468,1016172,1058841,1101708,1145772
%N A301939 Integers whose arithmetic derivative is equal to their Dedekind function.
%C A301939 If n = Product_{k=1..j} p_k ^ i_k with each p_k prime, then psi(n) = n * Product_{k=1..j} (p_k + 1)/p_k and n' = n*Sum_{k=1..j} i_k/p_k.
%C A301939 Thus every number of the form p^(p+1), where p is prime, is in the sequence.
%C A301939 The sequence also contains every number of the form 108*p^2 where p is a prime > 3, or 108*p^3*(p+2) where p > 3 is in A001359. - _Robert Israel_, Mar 29 2018
%H A301939 Paolo P. Lava, <a href="/A301939/b301939.txt">Table of n, a(n) for n = 1..100</a>
%F A301939 Solutions of the equation n' = psi(n).
%e A301939 5292 = 2^2 * 3^3 * 7^2.
%e A301939 n' = 5292*(2/2 + 3/3 + 2/7) = 12096,
%e A301939 psi(n) = 5292*(1 + 1/2)*(1 + 1/3)*(1 + 1/7) = 12096.
%p A301939 with(numtheory): P:=proc(n) local a,p; a:=ifactors(n)[2];
%p A301939 if add(op(2,p)/op(1,p),p=a)=mul(1+1/op(1,p),p=a) then n; fi; end:
%p A301939 seq(P(i),i=1..10^6);
%t A301939 selQ[n_] := Module[{f = FactorInteger[n], p, e}, Product[{p, e} = pe; p^e + p^(e-1), {pe, f}] == Sum[{p, e} = pe; (n/p)e, {pe, f}]];
%t A301939 Select[Range[10^6], selQ] (* _Jean-François Alcover_, Oct 16 2020 *)
%o A301939 (PARI) dpsi(f) = prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1));
%o A301939 ader(n, f) = sum(i=1, #f~, n/f[i, 1]*f[i, 2]);
%o A301939 isok(n) = my(f=factor(n)); dpsi(f) == ader(n, f); \\ _Michel Marcus_, Mar 29 2018
%Y A301939 Cf. A001359, A001615, A003415, A166374, A342458. A345005 (gives the odd terms).
%Y A301939 Subsequence of A345003.
%K A301939 nonn,easy
%O A301939 1,1
%A A301939 _Paolo P. Lava_, Mar 29 2018