A051377 a(1)=1; for n > 1, a(n) = sum of exponential divisors (or e-divisors) of n.

1, 2, 3, 6, 5, 6, 7, 10, 12, 10, 11, 18, 13, 14, 15, 22, 17, 24, 19, 30, 21, 22, 23, 30, 30, 26, 30, 42, 29, 30, 31, 34, 33, 34, 35, 72, 37, 38, 39, 50, 41, 42, 43, 66, 60, 46, 47, 66, 56, 60, 51, 78, 53, 60, 55, 70, 57, 58, 59, 90, 61, 62, 84, 78, 65, 66, 67, 102, 69, 70, 71

Offset

1,2

Comments

The e-divisors (or exponential divisors) of x=Product p(i)^r(i) are all numbers of the form Product p(i)^s(i) where s(i) divides r(i) for all i.

a(n) = n if and only if n is squarefree. - Jon Perry, Nov 13 2012

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

J. Fabrykowski and M. V. Subbarao, The maximal order and the average order of multiplicative function sigma^(e)(n), in Théorie des nombres/Number theory (Quebec, PQ, 1987), 201-206, de Gruyter, Berlin, 1989.

Nicussor Minculete, Concerning some arithmetic functions which use exponential divisors, Acta Universitatis Apulensis, No. 28/2011, pp. 125-133.

Y.-F. S. Pétermann and J. Wu, On the sum of exponential divisors of an integer, Acta Math. Hungar. 77 (1997), 159-175.

Eric Weisstein's World of Mathematics, e-Divisor

Formula

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

a(n) = A126164(n)+n. - R. J. Mathar, Oct 05 2017

The average order of a(n) is Dn + O(n^e) for any e > 0, due to Fabrykowski & Subbarao, where D is about 0.568. (D >= 0.5 since a(n) >= n.) - Charles R Greathouse IV, Sep 22 2023

Example

a(8)=10 because 2 and 2^3 are e-divisors of 8 and 2+2^3=10.

Maple

A051377 := proc(n)
    local a, pe, p, e;
    a := 1;
    for pe in ifactors(n)[2] do
        p := pe[1] ;
        e := pe[2] ;
        add(p^d, d=numtheory[divisors](e)) ;
        a := a*% ;
    end do:
    a ;
end proc:
seq(A051377(n), n=1..100) ; # R. J. Mathar, Oct 05 2017

Mathematica

a[n_] := Times @@ (Sum[ First[#]^d, {d, Divisors[Last[#]]}] & ) /@ FactorInteger[n]; Table[a[n], {n, 1, 71}] (* Jean-François Alcover, Apr 06 2012 *)

Prog

(PARI) a(n)=my(f=factor(n)); prod(i=1, #f[, 1], sumdiv(f[i, 2], d, f[i, 1]^d)) \\ Charles R Greathouse IV, Nov 22 2011
(PARI) ediv(n, f=factor(n))=my(v=List(), D=apply(divisors, f[, 2]~), t=#f~); forvec(u=vector(t, i, [1, #D[i]]), listput(v, prod(j=1, t, f[j, 1]^D[j][u[j]]))); Set(v)
a(n)=vecsum(ediv(n)) \\ Charles R Greathouse IV, Oct 29 2018
(Haskell)
a051377 n = product $ zipWith sum_e (a027748_row n) (a124010_row n) where
   sum_e p e = sum [p ^ d | d <- a027750_row e]
-- Reinhard Zumkeller, Mar 13 2012
(GAP) A051377:=n->Product(List(Collected(Factors(n)), p -> Sum(DivisorsInt(p[2]), d->p[1]^d))); List([1..10^4], n -> A051377(n)); # Muniru A Asiru, Oct 29 2017

Crossrefs

Cf. A051378, A049419 (number of e-divisors).

Cf. A027748, A124010, A027750.

Row sums of A322791.

See A307042 and A275480 where the formula and constant appear.

Sequence in context: A323309 A322857 A361175 * A369319 A336465 A340774

Adjacent sequences: A051374 A051375 A051376 * A051378 A051379 A051380

Keyword

nonn,easy,nice,mult

Author

Yasutoshi Kohmoto

Extensions

More terms from Jud McCranie, May 29 2000

Definition corrected by Jaroslav Krizek, Feb 27 2009

Status

approved