A354088 Sum of divisors function conjugated by Pythagorean prime shift: a(n) = A348747(sigma(A348746(n))).

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, 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, 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

Offset: 1

Author

Antti Karttunen, May 17 2022

Keywords

• nonn
• mult

Comments

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

Links

• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

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

Cf. also A326042, A354096 for variants.

Programs

PARI

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); };
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); };
A354088(n) = A348747(sigma(A348746(n)));

Formula

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

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