A241405 - OEIS

%I #71 Sep 03 2023 08:44:40

%S 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 42,31,40,30,72,32,39,48,54,48,50,38,60,56,66,42,96,44,60,60,72,48,68,

%U 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 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 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 Number of modified exponential divisors coincides with number of exponential divisors A049419.

%C 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 Antti Karttunen, <a href="/A241405/b241405.txt">Table of n, a(n) for n = 1..16384</a>

%H David Moews, <a href="http://djm.cc/amicable.html">Perfect, amicable and sociable numbers</a>.

%H <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>.

%F a(n / A007947(n)) = A051377(n).

%F Multiplicative with a(p^a) = sum p^b such that b+1 divides a+1.

%t 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 (PARI) A241405(n) = {my(f=factor(n)); prod(i=1, #f[, 1], sumdiv(f[i, 2]+1, d, f[i, 1]^(d-1)))}

%Y Cf. A007947, A049419, A051377, A054980, A051378.

%K nonn,easy,mult

%O 1,2

%A _Andrew Lelechenko_, May 06 2014

%E More terms from _Antti Karttunen_, Nov 23 2017