A383390 Numbers k such that k^2 and (k+1)^2 are both abundant numbers.

104, 495, 584, 735, 944, 1155, 1364, 1484, 2144, 2204, 2415, 2624, 2924, 2925, 3135, 3255, 3794, 3795, 4304, 4484, 4784, 4844, 5264, 5355, 5445, 5564, 5565, 5655, 5775, 5984, 6104, 6764, 7424, 7455, 7664, 7755, 7875, 8084, 8294, 8295, 8414, 8415, 8924, 9009, 9344, 9944, 9975

Offset

1,1

Comments

The numbers of terms that do not exceed 10^k, for k = 3, 4, ..., are 5, 47, 459, 4655, 46733, 460693, 4612685, 46177602, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00461... .

Links

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

Mathematica

Select[Range[10000], DivisorSigma[-1, #^2] > 2 && DivisorSigma[-1, (#+1)^2] > 2 &]

Prog

(PARI) is1(k) = {my(f = factor(k)); prod(i = 1, #f~, f[i, 2] *= 2); sigma(f, -1) > 2; }
list(lim) = {my(q1 = is1(1), q2); for(k = 2, lim, q2 = is1(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2); }

Crossrefs

Subsequence of A381738.

A383391 and A096399 are subsequences.

Cf. A005101, A063734.

Sequence in context: A234208 A372294 A234201 * A250669 A250691 A250692

Adjacent sequences: A383387 A383388 A383389 * A383391 A383392 A383393

Keyword

nonn

Author

Amiram Eldar, Apr 25 2025

Status

approved