A241405 - OEIS

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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).

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

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OFFSET

1,2

COMMENTS

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.

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

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.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

David Moews, Perfect, amicable and sociable numbers.

Index entries for sequences related to sums of divisors.

FORMULA

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

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

MATHEMATICA

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 *)

PROG

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

CROSSREFS