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

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

Author

Amiram Eldar, Jun 06 2018

Keywords

• nonn

Comments

Includes all the powers of 3 (A000244).

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

Examples

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..74
• Bhabesh Das and Helen K. Saikia, Identities for Near and Deficient Hyperperfect Numbers, Indian Journal in Number Theory, Vol. 3 (2016), pp. 124-134.

Crossrefs

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

Programs

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]

PARI

isok(n) = (n % 2) && (k = (3*n+1)/2 - sigma(n)) && (k > 0) && !(n % k) && (k != n); \\ Michel Marcus, Jun 07 2018, corrected by Amiram Eldar, Dec 23 2024