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

1,1

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

Links

Table of n, a(n) for n=1..24.

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

Example

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.

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 *)

Prog

(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;

Crossrefs

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

Cf. A159907 (hemiperfect).

Sequence in context: A009329 A009364 A308951 * A370537 A214835 A149059

Adjacent sequences: A295233 A295234 A295235 * A295237 A295238 A295239

Keyword

nonn,more

Author

Michel Marcus, Nov 19 2017

Extensions

a(20) from Jinyuan Wang, Feb 15 2020

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

Status

approved