A396546 Powerful numbers that are equal to the sum of their coreful divisors that are not exponential divisors.

576, 1568, 123008, 9007061816311808

Offset

1,1

Comments

Powerful numbers k such that A396544(k) = k.

See A284318 and A322791 for the definitions of coreful divisors and exponential divisors, respectively.

a(5) > 10^18.

If p is a Mersenne prime exponent (A000043) such that p+2 is prime, then 2^(p+2) * (2^p-1)^2 is a term.

Of the 52 known Mersenne prime exponents, only 5 have this property: 3, 5, 17, 107 and 521 (A087858).

a(2)-a(4) are of this form, corresponding to p = 3, 5, and 17. The next terms of this form are 2^109 * (2^107 - 1)^2 = 1.708...*10^97 and 2^523 * (2^521 - 1)^2 = 1.294*10^471.

If m is a term and k is a squarefree number coprime to m, then k*m is a term of A396545. Therefore, A396545 can be generated from this sequence by multiplying with coprime squarefree numbers, and the asymptotic density of A396545 can be calculated from this sequence (see comment in A396545).

Links

Table of n, a(n) for n=1..4.

Index entries for sequences related to powerful numbers.

Mathematica

f1[p_, e_] = (p^(e+1)-1)/(p-1) - 1;
f2[p_, e_] = DivisorSum[e, p^# &];
q[k_] := Module[{fct = FactorInteger[k]}, Min[fct[[;; , 2]]] > 1 && Times @@ f1 @@@ fct - Times @@ f2 @@@ fct == k]; q[1] = False; Select[Range[150000], q]

Prog

(PARI) isok(k) = {my(f = factor(k)); ispowerful(f) && prod(i = 1, #f~, (f[i, 1]^(f[i, 2]+1) - 1)/(f[i, 1]-1) - 1) - prod(i = 1, #f~, sumdiv(f[i, 2], d, f[i, 1]^d)) == k; }

Crossrefs

Intersection of A001694 and A396545.

Cf. A000043, A005117, A054980, A087858, A284318, A307959, A322791, A396544.

Sequence in context: A235181 A268773 A396545 * A250789 A137484 A189990

Adjacent sequences: A396543 A396544 A396545 * A396547 A396548 A396549

Keyword

nonn,more

Author

Amiram Eldar, May 29 2026

Status

approved