A354093 a(n) = sigma(A354091(n)), where A354091 is fully multiplicative prime shift which replaces the primes of the form 3k+2 by the next larger such prime, while other primes stay as they are, and sigma is the sum of divisors function.

1, 6, 4, 31, 12, 24, 8, 156, 13, 72, 18, 124, 14, 48, 48, 781, 24, 78, 20, 372, 32, 108, 30, 624, 133, 84, 40, 248, 42, 288, 32, 3906, 72, 144, 96, 403, 38, 120, 56, 1872, 48, 192, 44, 558, 156, 180, 54, 3124, 57, 798, 96, 434, 60, 240, 216, 1248, 80, 252, 72, 1488, 62, 192, 104, 19531, 168, 432, 68, 744, 120, 576

Offset: 1

Author

Antti Karttunen, May 17 2022

Keywords

• nonn
• mult

Links

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

Crossrefs

Inverse Möbius transform of A354091.

Cf. A000203, A003627, A010872, A074941, A354095.

Cf. A003973, A354089 for variants.

Programs

PARI

A354093(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)))); sigma(factorback(f)); };

Formula

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

a(n) = Sum_{d|n} A354091(d).

For all n >= 1, A010872(a(n)) = A010872(A000203(n)) = A074941(n).