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
Examples
a(8)=10 because 2 and 2^3 are e-divisors of 8 and 2+2^3=10.
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
Crossrefs
Programs
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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
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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
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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
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Mathematica
a[n_] := Times @@ (Sum[ First[#]^d, {d, Divisors[Last[#]]}] & ) /@ FactorInteger[n]; Table[a[n], {n, 1, 71}] (* Jean-François Alcover, Apr 06 2012 *) -
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
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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
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
Extensions
More terms from Jud McCranie, May 29 2000
Definition corrected by Jaroslav Krizek, Feb 27 2009
Comments