A396551 Powerful numbers that are equal to the sum of their coreful divisors that are not exponential unitary divisors.

144, 576, 1568, 123008, 8258048, 9007061816311808

Offset

1,1

Comments

See A284318 and A361255 for the definitions of coreful divisors and exponential unitary divisors, respectively.

a(7) > 10^18.

If p is a Mersenne prime exponent (A000043) such that p+2 is prime power (A246655), then 2^(p+2) * (2^p-1)^2 is a term.

Of the 52 known Mersenne prime exponents, only 7 have this property: 2, 3, 5, 7, 17, 107 and 521.

a(1) and a(3)-a(6) are of this form, corresponding to p = 2, 3, 5, 7 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 A396550. Therefore, A396550 can be generated from this sequence by multiplying with coprime squarefree numbers, and the asymptotic density of A396550can be calculated from this sequence (see comment in A396550).

Links

Table of n, a(n) for n=1..6.

Mathematica

f1[p_, e_] = (p^(e+1)-1)/(p-1) - 1;
f2[p_, e_] = DivisorSum[e, p^# &, CoprimeQ[#, e/#] &];
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, if(gcd(d, f[i, 2]/d) == 1, f[i, 1]^d))) == k; }

Crossrefs

Intersection of A001694 and A396550.

Cf. A000043, A005117, A246655, A284318, A307959, A361255, A391281, A396546, A396549.

Sequence in context: A155707 A017522 A180413 * A390290 A389397 A137885

Adjacent sequences: A396548 A396549 A396550 * A396552 A396553 A396554

Keyword

nonn,more

Author

Amiram Eldar, May 29 2026

Status

approved