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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).
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%I #35 Oct 31 2023 04:45:26

%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,

%T 42,31,56,30,72,32,35,48,54,48,91,38,60,56,66,42,96,44,84,78,72,48,92,

%U 57,93,72,98,54,93,72,88,80,90,60,168,62,96,104,79,84,144,68,126,96

%N 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).

%H Reinhard Zumkeller, <a href="/A051378/b051378.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = 1 + Sum_{d|e} p^d. - _Vladeta Jovovic_, Apr 23 2002

%F a(n) = Sum_{d|n, gcd(d, n/d) = 1} A051377(d). - _Daniel Suteu_, Nov 01 2022

%F 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

%p A051378 := proc(n)

%p local a,d,p,e,sp;

%p a := 1;

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

%p p := op(1,d) ;

%p e := op(2,d) ;

%p sp := 1;

%p for s in numtheory[divisors](e) do

%p sp := sp+p^s ;

%p end do:

%p a := a*sp ;

%p end do:

%p a;

%p end proc: # _R. J. Mathar_, Oct 26 2015

%t 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 *)

%o (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

%o (Haskell)

%o a051378 n = product $ zipWith sum_1e (a027748_row n) (a124010_row n)

%o where sum_1e p e = 1 + sum [p ^ d | d <- a027750_row e]

%o -- _Reinhard Zumkeller_, Mar 13 2012

%Y Cf. A051377, A049599.

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

%K nonn,easy,nice,mult

%O 1,2

%A _Yasutoshi Kohmoto_

%E Corrected and extended by _Naohiro Nomoto_, Apr 12 2001