A354088 - OEIS

%I #8 May 17 2022 17:50:05

%S 1,1,2,5,7,2,1,3,31,7,2,10,4,1,14,121,6,31,3,35,2,2,2,6,106,4,10,5,19,

%T 14,1,35,4,6,7,155,14,3,8,21,8,2,11,10,217,2,2,242,38,106,12,20,31,10,

%U 14,3,6,19,6,70,29,1,31,1069,28,4,13,30,4,7,4,93,12,14,212,15,2,8,3,847,781,8,14,10,42,11,38

%N Sum of divisors function conjugated by Pythagorean prime shift: a(n) = A348747(sigma(A348746(n))).

%C This is variant of A326042, and like that sequence, also this one is multiplicative.

%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) = A348747((q^(e+1)-1)/(q-1)), where q = A348744(A000720(p)).

%F a(n) = A348747(A354089(n)) = A348747(A000203(A348746(n))).

%o (PARI)

%o A348746(n) = { my(f=factor(n)); for(k=1,#f~, if(2==f[k,1], f[k,1]=3, if(3==f[k,1], f[k,1]=5, if(1==(f[k,1]%4), for(i=1+primepi(f[k,1]),oo,if(1==(prime(i)%4), f[k,1]=prime(i); break)))))); factorback(f); };

%o A348747(n) = { my(f=factor(n)); for(k=1,#f~, if(f[k,1]<=3, f[k,1]--, if(5==f[k,1], f[k,1]=3, if(1==(f[k,1]%4), forstep(i=primepi(f[k,1])-1,0,-1,if(1==(prime(i)%4), f[k,1]=prime(i); break)))))); factorback(f); };

%o A354088(n) = A348747(sigma(A348746(n)));

%Y Cf. A000203, A326042, A348744, A348746, A348747, A354089.

%Y Cf. also A326042, A354096 for variants.

%K nonn,mult

%O 1,3

%A _Antti Karttunen_, May 17 2022