A305617 - OEIS

%I #6 Jun 18 2018 18:14:23

%S 3,9,27,35,39,55,81,243,279,387,715,729,1443,2187,2619,3655,5635,6561,

%T 10855,12635,19683,59049,77283,177147,178119,294759,443135,531441,

%U 817167,1170723,1594323,1605987,1632231,1710963,1947159,2410239,2624375,2655747,3944255

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

%C Includes all the powers of 3 (A000244).

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

%H Bhabesh Das and Helen K. Saikia, <a href="https://ijntindia.puzl.com/files/1638778/download/1_das_1-12.pdf">Identities for Near and Deficient Hyperperfect Numbers</a>, Indian Journal in Number Theory, Vol. 3 (2016), pp. 124-134, <a href="https://www.researchgate.net/profile/Bhabesh_Das/publication/303856760_Identities_for_Near_and_Deficient_Hyper-perfect_Numbers/links/57585d3f08ae414b8e3f5a50.pdf">alternative link</a>.

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

%t 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]

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

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

%K nonn

%O 1,1

%A _Amiram Eldar_, Jun 06 2018