A329189 3-admirable numbers: 3-abundant numbers (A068403) k such that exists a proper divisor d of k such that sigma(k) - 2*d = 3*k, where sigma(k) is the sum of divisors of k (A000203).

180, 240, 360, 420, 504, 540, 600, 780, 1080, 1344, 1872, 1890, 2016, 2184, 2352, 2376, 2688, 3192, 3276, 3744, 4284, 4320, 4680, 5292, 5376, 5796, 6048, 6552, 7128, 7344, 7440, 8190, 9504, 10296, 10416, 13776, 14850, 18600, 19824, 19872, 20496, 21528, 22932

Offset

1,1

Comments

Analogous to admirable numbers (A111592) as 3-perfect numbers (A005820) are analogous to perfect numbers (A000396).

The proper divisors of each term k can be added to a sum of 2*k with one divisor taken with a minus sign.

Are there odd terms in this sequence? If there are, they are squares that are larger than 10^20. - Amiram Eldar, Jul 06 2026

Links

Amiram Eldar, Table of n, a(n) for n = 1..810 (terms below 10^11)

Example

180 is a term since its proper divisors can be added to 1 + 2 - 3 + 4 + 5 + 6 + 9 + 10 + 12 + 15 + 18 + 20 + 30 + 36 + 45 + 60 + 90 = 360 = 2 * 180, with one divisor, 3, taken with a minus sign.

Mathematica

aQ[n_] := (ab = DivisorSigma[1, n] - 3 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2]; Select[Range[23000], aQ]

Prog

(PARI) isok(k) = {my((ab = sigma(k) - 3*k)); ab > 0 && !(ab % 2) && ab/2 < k && !(k % (ab/2)); } \\ Amiram Eldar, Jul 06 2026

Crossrefs

Cf. A000203, A000396, A068403, A111592.

Sequence in context: A068545 A068403 A291457 * A364976 A289056 A309380

Adjacent sequences: A329186 A329187 A329188 * A329190 A329191 A329192

Keyword

nonn,changed

Author

Amiram Eldar, Nov 07 2019

Status

approved