A389298 Primitive terms in A389297: powerful numbers that are exponential deficient and having at least one exponential abundant divisor.

3528, 5292, 5400, 7200, 8712, 9000, 9800, 10584, 10800, 12168, 13068, 13500, 14112, 14400, 15876, 16200, 18000, 18252, 19600, 20808, 21168, 21600, 24300, 24336, 24500, 24696, 25992, 26136, 27000, 28224, 28800, 31212, 31752, 32400, 34300, 34848, 36000, 36504, 37044

Offset

1,1

Comments

Subsequence of A389297 and first differs from it at n = 17: A389297(17) = 17640 = 2^3 * 3^2 * 5 * 7^2 is not a term of this sequence.

Any term of A389297 is of the form k*m where k is a term of this sequence and m is a squarefree number coprime to k. Therefore, A389297 can be generated from this sequence by multiplying with coprime squarefree numbers, and the asymptotic density of A389297 can be evaluated from the terms of this sequence (see the Comments section of A389297).

The least odd term is a(13240) = 676350675.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Mathematica

pows[max_] := Union[Flatten[Table[i^2*j^3, {j, 1, Surd[max, 3]}, {i, 1, Sqrt[max/j^3]}]]];
f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n];
expAbQ[n_] := esigma[n] > 2*n; expDefQ[n_] := esigma[n] < 2*n; q[n_] := expDefQ[n] && AnyTrue[Divisors[n], expAbQ];
seq[max_] := Select[pows[max], q]; seq[30000]

Prog

(PARI) is1(k) = {my(f = factor(k)); prod(i = 1, #f~, sumdiv(f[i, 2], d, f[i, 1]^d)) < 2*k && prod(i = 1, #f~, if(f[i, 2] > 1, 1+1/f[i, 1], 1)) > 2; }
list(lim) = {my(s = List(), m); for(j = 1, sqrtnint(lim, 3), for(i = 1, sqrtint(lim\j^3), m = i^2*j^3; if(is1(m), listput(s, m)))); Set(s); }

Crossrefs

Intersection of A001694 and A389297.

Cf. A005117, A129575.

Sequence in context: A389331 A064256 A389297 * A232555 A389300 A330841

Adjacent sequences: A389295 A389296 A389297 * A389299 A389300 A389301

Keyword

nonn

Author

Amiram Eldar, Sep 29 2025

Status

approved