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A282515 Numbers m such that phi(sum of the divisors of m) = phi(sum of the distinct prime divisors of m) where phi is the Euler totient function.

%I A282515 #28 Dec 12 2023 18:16:41
%S A282515 3,6,10,22,34,142,178,214,382,862,1402,2302,5182,9098,15398,17398,
%T A282515 21178,23602,279934,289558,296734,368062,900754,944782,1079374,
%U A282515 1563442,1572862,1990654,2116342,2505886,2584882,2691574,2858698,2883058,3351214,3909046
%N A282515 Numbers m such that phi(sum of the divisors of m) = phi(sum of the distinct prime divisors of m) where phi is the Euler totient function.
%C A282515 Or numbers m such that A000010(A000203(m)) = A000010(A008472(m)).
%C A282515 For n > 1, we observe that a(n) is semiprime of the form a(n) = 2p with p = 3, 5, 11, 17, 71, 89, 107, 191, 431, 701, 1151, 2591, 4549, 7699, 8699, 10589, 11801, ... Except for the primes 3, 4549 and 7699 in the first 35 terms (from 6 until 3909046), the primes p are of the form 6k - 1.
%H A282515 Amiram Eldar, <a href="/A282515/b282515.txt">Table of n, a(n) for n = 1..289</a> (terms below 10^10)
%e A282515 34 is in the sequence because A000010(A000203(34)) = A000010(54) = 18 and A000010(A008472(34)) = A000010(19) = 18.
%p A282515 with(numtheory):
%p A282515 for n from 2 to 200000 do:
%p A282515 x:=divisors(n):n0:=nops(x):y:=factorset(n):n1:=nops(y):
%p A282515    s0:=sum(‘x[i]’, ‘i’=1..n0):s1:=sum(‘y[i]’, ‘i’=1..n1):
%p A282515     if phi(s1)=phi(s0)
%p A282515      then
%p A282515        print(n):
%p A282515        else
%p A282515      fi:
%p A282515 od:
%t A282515 Select[Range[10^6], EulerPhi@ DivisorSigma[1, #] == EulerPhi[Total@ FactorInteger[#][[All, 1]]] &] (* _Michael De Vlieger_, Feb 17 2017 *)
%o A282515 (PARI) isok(n) = my(f=factor(n)); eulerphi(sigma(n)) == eulerphi(vecsum(f[,1])); \\ _Michel Marcus_, Feb 25 2017
%Y A282515 Cf. A000010, A000203, A008472, A062401.
%K A282515 nonn
%O A282515 1,1
%A A282515 _Michel Lagneau_, Feb 17 2017