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A354206 a(n) = A354203(sigma(A354202(n))), where A354202 is fully multiplicative with a(p) = A354200(A000720(p)), and A354203 is its left inverse.

%I A354206 #15 May 24 2022 16:31:53
%S A354206 1,1,1,23,3,1,1,5,11,3,2,23,1,1,3,469,2,11,1,69,1,2,1,5,53,1,4,23,11,
%T A354206 3,7,69,2,2,3,253,3,1,1,15,1,1,1,46,33,1,2,469,33,53,2,23,23,4,6,5,1,
%U A354206 11,13,69,29,7,11,19507,3,2,1,46,1,3,2,55,2,3,53,23,2,1,3,1407,2797,1,5,23,6,1,11,10,9,33
%N A354206 a(n) = A354203(sigma(A354202(n))), where A354202 is fully multiplicative with a(p) = A354200(A000720(p)), and A354203 is its left inverse.
%H A354206 Antti Karttunen, <a href="/A354206/b354206.txt">Table of n, a(n) for n = 1..20000</a>
%H A354206 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A354206 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A354206 Multiplicative with a(p^e) = A354203((q^(e+1)-1)/(q-1)) where q = A354200(A000720(p)).
%F A354206 a(n) = A354203(A354205(n)) = A354203(sigma(A354202(n))).
%F A354206 a(n) = n - A354207(n).
%o A354206 (PARI)
%o A354206 A354200(n) = if(1==n,5,my(p=prime(n), m=p%4); forprime(q=1+p,,if(m==(q%4),return(q))));
%o A354206 A354201(n) = if(n<=3,(n+1)\2,my(m=prime(n)%4); forstep(i=n-1,0,-1,if(m==(prime(i)%4),return(prime(i)))));
%o A354206 A354202(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354200(primepi(f[k,1]))); factorback(f); };
%o A354206 A354203(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354201(primepi(f[k,1]))); factorback(f); };
%o A354206 A354206(n) = A354203(sigma(A354202(n)));
%Y A354206 Cf. A000203, A000720, A354200, A354201, A354202, A354203, A354205, A354207.
%Y A354206 Cf. A354361 (positions of 1's).
%Y A354206 Cf. also A326042, A348750, A354088, A354096 for similar constructions.
%K A354206 nonn,mult
%O A354206 1,4
%A A354206 _Antti Karttunen_, May 23 2022