A051378 Sum of (1+e)-divisors of n. Let n = Product_i p(i)^r(i) then (1+e)-sigma(n) = Product_i (1 + Sum_{s|r(i)} p(i)^s).

1, 3, 4, 7, 6, 12, 8, 11, 13, 18, 12, 28, 14, 24, 24, 23, 18, 39, 20, 42, 32, 36, 24, 44, 31, 42, 31, 56, 30, 72, 32, 35, 48, 54, 48, 91, 38, 60, 56, 66, 42, 96, 44, 84, 78, 72, 48, 92, 57, 93, 72, 98, 54, 93, 72, 88, 80, 90, 60, 168, 62, 96, 104, 79, 84, 144, 68, 126, 96

Offset

1,2

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Formula

Multiplicative with a(p^e) = 1 + Sum_{d|e} p^d. - Vladeta Jovovic, Apr 23 2002

a(n) = Sum_{d|n, gcd(d, n/d) = 1} A051377(d). - Daniel Suteu, Nov 01 2022

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 + (1-1/p)*Sum_{k>=2} p^k/(p^(2*k)-1)) = 0.76636964336546210751... . - Amiram Eldar, Oct 31 2023

Maple

A051378 := proc(n)
    local a, d, p, e, sp;
    a := 1;
    for d in ifactors(n)[2] do
        p := op(1, d) ;
        e := op(2, d) ;
        sp := 1;
        for s in numtheory[divisors](e) do
            sp := sp+p^s ;
        end do:
        a := a*sp ;
    end do:
    a;
end proc: # R. J. Mathar, Oct 26 2015

Mathematica

a[1] = 1; a[p_?PrimeQ] = p+1; a[n_] := Times @@ (1 + Sum[First[#]^d, {d, Divisors[Last[#]]}] & ) /@ FactorInteger[n]; Table[a[n], {n, 1, 69}] (* Jean-François Alcover, May 04 2012 *)

Prog

(PARI) a(n)=my(f=factor(n)); prod(i=1, #f[, 1], sumdiv(f[i, 2], d, f[i, 1]^d)+1) \\ Charles R Greathouse IV, Nov 22 2011
(Haskell)
a051378 n = product $ zipWith sum_1e (a027748_row n) (a124010_row n)
   where sum_1e p e = 1 + sum [p ^ d | d <- a027750_row e]
-- Reinhard Zumkeller, Mar 13 2012

Crossrefs

Cf. A051377, A049599.

Cf. A027748, A124010, A027750, A069915, A107758, A107759.

Sequence in context: A366903 A049418 A333926 * A366440 A344575 A254981

Adjacent sequences: A051375 A051376 A051377 * A051379 A051380 A051381

Keyword

nonn,easy,nice,mult

Author

Yasutoshi Kohmoto

Extensions

Corrected and extended by Naohiro Nomoto, Apr 12 2001

Status

approved