A380446 Numbers k for which A339792(k), the third coefficient of the lindep transform of sigma, is 0.

1, 6, 24, 28, 40, 84, 90, 120, 140, 168, 216, 224, 234, 252, 270, 280, 308, 360, 364, 420, 440, 496, 528, 540, 546, 585, 588, 600, 672, 728, 819, 840, 864, 918, 924, 936, 1080, 1120, 1140, 1170, 1260, 1320, 1428, 1456, 1488, 1550, 1560, 1638, 1656, 1710, 1782, 1890, 1920, 1932, 1989, 2016, 2040, 2160, 2184, 2208, 2280, 2295

Offset

1,2

Comments

Numbers k for which A000203(k)*A339790(k) = -k*A339791(k).

See A339790 for a definition of lindep-transform by Benoit Cloitre.

Question: Is it certain that all multiply perfect numbers (A007691) are included in this sequence?

Squares in this sequence are: 1, 480249, 1920996, 2140369, 3538161, 5659641, 8561476, etc, see A380448. The odd squares are in A380449.

Yes, all multiperfect numbers (A007691) are in this sequence: For these numbers, with k = sigma(n)/n, the minimum N = |x| + |y| + |z| = x + |y| + |k x + y|*n is N = 1 + k, for (x, y, z) = (1, -k, 0). (If |z| is any nonzero multiple of n, it will be much larger than 1+k, since the least k-perfect n is larger than exp(exp(0.56 k)), cf. A007539.) - M. F. Hasler, Jan 27 2025

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Index entries for sequences related to sigma(n).

Example

24 is included as sigma(24)*A339790(24) = -24*A339791(24), i.e., 60*2 = -24 * -5 = 120.
28 is included as sigma(28)*A339790(28) = -28*A339791(28), i.e., 56*1 = -28 * -2 = 56.
120 is included as sigma(120)*A339790(120) = -120*A339791(120), i.e., 360*1 = -120 * -3 = 360.
585 is included as sigma(585)*A339790(585) = -585*A339791(585), i.e., 1092*15 = -585 * -28 = 16380.

Prog

(PARI) is_A380446 = A380445;

Crossrefs

Cf. A000203, A339790, A339791, A339792, A380445 (characteristic function).

Subsequences: A000396, A007691, A380447 (odd terms), A380448 (square terms), A380449 (odd squares).

Sequence in context: A349686 A069235 A175200 * A293453 A344754 A364977

Adjacent sequences: A380443 A380444 A380445 * A380447 A380448 A380449

Keyword

nonn,new

Author

Antti Karttunen, Jan 25 2025

Status

approved