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A241405 Sum of modified exponential divisors: if n = Product p_i^r_i then me-sigma(x) = Product (sum p_i^s_i such that s_i+1 divides r_i+1).

%I A241405 #78 Dec 14 2024 09:16:34
%S A241405 1,3,4,5,6,12,8,11,10,18,12,20,14,24,24,17,18,30,20,30,32,36,24,44,26,
%T A241405 42,31,40,30,72,32,39,48,54,48,50,38,60,56,66,42,96,44,60,60,72,48,68,
%U A241405 50,78,72,70,54,93,72,88,80,90,60,120,62,96,80,65,84,144,68,90,96,144,72,110,74,114,104,100,96,168,80
%N A241405 Sum of modified exponential divisors: if n = Product p_i^r_i then me-sigma(x) = Product (sum p_i^s_i such that s_i+1 divides r_i+1).
%C A241405 The modified exponential divisors of a number n = product p_i^r_i are all numbers of the form product p_i^s_i such that s_i+1 divides r_i+1 for each i.
%C A241405 The concept of modified exponential divisors simplifies combinatorial problems on the sum of exponential divisors A051377 such as a search of e-perfect numbers. Each primitive e-perfect number A054980 corresponds to a unique me-perfect number of smaller magnitude.
%H A241405 Antti Karttunen, <a href="/A241405/b241405.txt">Table of n, a(n) for n = 1..16384</a>
%H A241405 David Moews, <a href="http://djm.cc/amicable.html">Perfect, amicable and sociable numbers</a>.
%H A241405 <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>.
%F A241405 a(n / A007947(n)) = A051377(n).
%F A241405 Multiplicative with a(p^a) = sum p^b such that b+1 divides a+1.
%t A241405 f[p_, e_] := DivisorSum[e+1, p^(#-1)&]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 03 2023 *)
%o A241405 (PARI) A241405(n) = {my(f=factor(n)); prod(i=1, #f[, 1], sumdiv(f[i, 2]+1, d, f[i, 1]^(d-1)))}
%Y A241405 Cf. A007947, A049419, A051377, A054980, A051378, A379027, A379028.
%K A241405 nonn,easy,mult
%O A241405 1,2
%A A241405 _Andrew Lelechenko_, May 06 2014
%E A241405 More terms from _Antti Karttunen_, Nov 23 2017
%E A241405 Incorrect comment removed by _Amiram Eldar_, Dec 14 2024