A051377 - OEIS

%I #65 Feb 16 2025 08:32:41

%S 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,

%T 26,30,42,29,30,31,34,33,34,35,72,37,38,39,50,41,42,43,66,60,46,47,66,

%U 56,60,51,78,53,60,55,70,57,58,59,90,61,62,84,78,65,66,67,102,69,70,71

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

%C 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.

%C a(n) = n if and only if n is squarefree. - _Jon Perry_, Nov 13 2012

%H Charles R Greathouse IV, <a href="/A051377/b051377.txt">Table of n, a(n) for n = 1..10000</a>

%H J. Fabrykowski and M. V. Subbarao, <a href="https://doi.org/10.1515/9783110852790.201">The maximal order and the average order of multiplicative function sigma^(e)(n)</a>, in Théorie des nombres/Number theory (Quebec, PQ, 1987), 201-206, de Gruyter, Berlin, 1989.

%H Nicussor Minculete, <a href="http://www.emis.de/journals/AUA/acta28/Paper10-Acta28-2011.pdf">Concerning some arithmetic functions which use exponential divisors</a>, Acta Universitatis Apulensis, No. 28/2011, pp. 125-133.

%H Y.-F. S. Pétermann and J. Wu, <a href="http://dx.doi.org/10.1023/A:1006596009566">On the sum of exponential divisors of an integer</a>, Acta Math. Hungar. 77 (1997), 159-175.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/e-Divisor.html">e-Divisor</a>

%F Multiplicative with a(p^e) = Sum_{d|e} p^d. - _Vladeta Jovovic_, Apr 23 2002

%F a(n) = A126164(n)+n. - _R. J. Mathar_, Oct 05 2017

%F 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

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

%p A051377 := proc(n)

%p     local a,pe,p,e;

%p     a := 1;

%p     for pe in ifactors(n)[2] do

%p         p := pe[1] ;

%p         e := pe[2] ;

%p         add(p^d,d=numtheory[divisors](e)) ;

%p         a := a*% ;

%p     end do:

%p     a ;

%p end proc:

%p seq(A051377(n),n=1..100) ; # _R. J. Mathar_, Oct 05 2017

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

%o (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

%o (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)

%o a(n)=vecsum(ediv(n)) \\ _Charles R Greathouse IV_, Oct 29 2018

%o (Haskell)

%o a051377 n = product $ zipWith sum_e (a027748_row n) (a124010_row n) where

%o    sum_e p e = sum [p ^ d | d <- a027750_row e]

%o -- _Reinhard Zumkeller_, Mar 13 2012

%o (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

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

%Y Cf. A027748, A124010, A027750.

%Y Row sums of A322791.

%Y See A307042 and A275480 where the formula and constant appear.

%K nonn,easy,nice,mult,changed

%O 1,2

%A _Yasutoshi Kohmoto_

%E More terms from _Jud McCranie_, May 29 2000

%E Definition corrected by _Jaroslav Krizek_, Feb 27 2009