A354096 - OEIS

%I #9 May 17 2022 17:47:49

%S 1,3,1,31,3,3,1,39,13,9,9,31,7,3,3,295,3,39,2,93,1,27,6,39,133,21,2,

%T 31,21,9,1,1953,9,9,3,403,19,6,7,117,3,3,5,279,39,18,27,295,57,399,3,

%U 217,6,6,27,39,2,63,9,93,31,3,13,19531,21,27,11,93,6,9,21,507,37,57,133,62,9,21,2,885,25,9,18,31,9

%N a(n) = A354092(sigma(A354091(n))).

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%F Multiplicative with a(p^e) = A354092((q^(e+1)-1)/(q-1)), where q = A003627(1+n) if p = A003627(n), otherwise q = p.

%F a(n) = A354092(A354093(n)) = A354095(A354091(n)) = A354092(A000203(A354091(n))).

%F For all n >= 1, A010872(a(n)) = A010872(A354095(n)).

%F For all k in A329963, A007949(a(k)) = A007949(sigma(k)) = A354100(k) = 0.

%o (PARI)

%o A354091(n) = { my(f=factor(n)); for(k=1,#f~, if(2==(f[k,1]%3), for(i=1+primepi(f[k,1]),oo,if(2==(prime(i)%3), f[k,1]=prime(i); break)))); factorback(f); };

%o A354092(n) = { my(f=factor(n)); for(k=1,#f~, if(2==(f[k,1]%3), if(2==f[k,1], f[k,1]--, forstep(i=primepi(f[k,1])-1,0,-1,if(2==(prime(i)%3), f[k,1]=prime(i); break))))); factorback(f); };

%o A354096(n) = A354092(sigma(A354091(n)));

%Y Cf. A000203, A003627, A007949, A010872, A329963, A354091, A354092, A354093, A354095.

%Y Cf. also A326042, A354088 for variants.

%K nonn,mult

%O 1,2

%A _Antti Karttunen_, May 17 2022