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

1,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to sigma(n)

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).

Mathematica

f[n_]:=Module[{fac, p, i}, fac=FactorInteger[n]; Do[p=fac[[k, 1]]; If[Mod[p, 3]==2, i=PrimePi[p]+1; While[Mod[Prime[i], 3]!=2, i++; ]; fac[[k, 1]]=Prime[i]; ]; , {k, Length[fac]}]; DivisorSigma[1, Times@@(Power@@@fac)]]; Table[f[n], {n, 1, 70}]  (* Vincenzo Librandi, Nov 05 2025 *)

Prog

(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)); };

Crossrefs

Inverse Möbius transform of A354091.

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

Cf. A003973, A354089 for variants.

Sequence in context: A191567 A274707 A372524 * A377018 A163934 A163939

Adjacent sequences: A354090 A354091 A354092 * A354094 A354095 A354096

Keyword

nonn,mult

Author

Antti Karttunen, May 17 2022

Status

approved