A381321 Numbers k such that sigma(k)/k - 1 equals (sigma(m)/m - 1)^2 for some m <= k.

1, 6, 28, 216, 360, 496, 2016, 8128, 16758, 1571328, 1935360, 2678400, 33550336, 54758400, 101382400, 1685013120

Offset: 1

Author

Leo Hennig, Feb 21 2025

Keywords

• nonn
• more

Comments

For any perfect number sigma(k)/k - 1 = 1, so all perfect numbers are terms.

2016 = 2^(6-1)*(2^6-1) is of the form 2^(k-1)*(2^k - 1) like the perfect numbers.

4428914688 from A383482 and 155086041146982400 from A218404 are terms. - Michel Marcus, May 22 2025

Examples

			216 is a term since sigma(216)/216 - 1 = (4/3)^2 and sigma(12)/12 - 1 = 4/3.
		

Links

• David C. Luo, Generalizing the Abundancy of an Integer, arXiv:1803.10816 [math.NT], 2018-2019.

Crossrefs

Cf. A000396, A006516, A243473/A017666.

Subsequence of A383482.

Programs

PARI

isok(k)=my(t=(sigma(k)-k)*k); if(issquare(t), my(r=sqrtint(t)/k+1, s=denominator(r)); forstep(m=s, k, s, if(sigma(m)/m==r, return(1)) )); 0 \\ Andrew Howroyd, Mar 03 2025

Extensions

a(15)-a(16) from Michel Marcus, Mar 05 2025