A295236 Hemi-imperfect numbers: numbers such that the denominator of k/A206369(k) is equal to 2.

3, 10, 42, 60, 63, 840, 1260, 12642, 18480, 18900, 18963, 154350, 228480, 252840, 379260, 3458700, 5562480, 5688900, 68772480, 1041068700, 15032156160, 53621568000, 4524679004160, 9812746944000

Offset: 1

Author

Michel Marcus, Nov 19 2017

Keywords

• nonn
• more

Comments

This is to rho (A206369) what hemiperfect numbers are to sigma (A000203).

After 3, 10 and 42, whose quotients are resp. 3/2, 5/2 and 7/2, 373316437260251755241798182764378479569038727298776522806597255168000000 is an instance of a term with quotient 9/2. - Michel Marcus, Dec 17 2017

a(25) > 10^13. - Giovanni Resta, Feb 17 2020

Examples

			3 is a term since rho(3) = 2, so 3/rho(3) is 3/2.
10 is a term since rho(10) = 4, so 10/rho(10) is 5/2.
42 is a term since rho(42) = 12, so 42/rho(42) is 7/2.
		

Links

• Douglas E. Iannucci, On a variation of perfect numbers, INTEGERS: Electronic Journal of Combinatorial Number Theory, 6 (2006), #A41.
• László Tóth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842 [math.NT], 2011-2014.
• Wikipedia, Hemiperfect number

Crossrefs

Cf. A127724 (k-imperfect), A206369 (rho).

Cf. A159907 (hemiperfect).

Programs

Maple

rho:= proc(n) local f;
  mul((f[1]^(f[2]+1)+(-1)^f[2])/(f[1]+1), f = ifactors(n)[2]);
end proc:
select(t -> denom(t/rho(t)) = 2, [$1..10^6]); # Robert Israel, Nov 20 2017

Mathematica

(* b = A209369 *) b[n_] := n*DivisorSum[n, LiouvilleLambda[#]/# &];
Select[Range[10^6], If[Denominator[#/b[#]] == 2, Print[#]; True, False]&] (* Jean-François Alcover, Dec 04 2017 *)

PARI

rho(n) = {my(f = factor(n), res = q = 1); for(i=1, #f~, q = 1; for(j = 1, f[i, 2], q = -q + f[i, 1]^j); res *= q); res;}
isok(n) = denominator(n/rho(n))==2;

Extensions

a(20) from Jinyuan Wang, Feb 15 2020

a(21)-a(24) from Giovanni Resta, Feb 17 2020