A305617 Deficient 2-hyperperfect numbers: numbers n such that 3n/2 + 1/2 - sigma(n) is a proper divisor of n.

3, 9, 27, 35, 39, 55, 81, 243, 279, 387, 715, 729, 1443, 2187, 2619, 3655, 5635, 6561, 10855, 12635, 19683, 59049, 77283, 177147, 178119, 294759, 443135, 531441, 817167, 1170723, 1594323, 1605987, 1632231, 1710963, 1947159, 2410239, 2624375, 2655747, 3944255

Offset

1,1

Comments

Includes all the powers of 3 (A000244).

A combination of the notions 2-hyperperfect numbers (A007593) and deficient-perfect numbers (A271816).

Links

Table of n, a(n) for n=1..39.

Bhabesh Das and Helen K. Saikia, Identities for Near and Deficient Hyperperfect Numbers, Indian Journal in Number Theory, Vol. 3 (2016), pp. 124-134, alternative link.

Example

35 is in the sequence since sigma(35) = 48 and 3*35/2 + 1/2 - 48 = 5 is a proper divisor of 35.

Mathematica

aQ[n_]:=Module[{d = 3n/2+1/2-DivisorSigma[1, n]}, d>0 && d!=n && IntegerQ[d] && Divisible[n, d]]; Select[Range[2, 1000000], aQ]

Prog

(PARI) isok(n) = (n % 2) && (k = (3*n+1)/2 - sigma(n)) && !(n % k) && (k != n); \\ Michel Marcus, Jun 07 2018

Crossrefs

Cf. A000203, A000244, A007593, A271816, A305616.

Sequence in context: A057238 A070364 A070363 * A292390 A211219 A248959

Adjacent sequences: A305614 A305615 A305616 * A305618 A305619 A305620

Keyword

nonn

Author

Amiram Eldar, Jun 06 2018

Status

approved