A038536 Odd values of k > 1 for which there are k-hyperperfect numbers.

3, 11, 19, 31, 35, 59, 75, 91, 111, 115, 131, 151, 179, 235, 255, 311, 335, 339, 371, 375, 399, 411, 431, 439, 495, 515, 531, 539, 551, 591, 619, 675, 739, 791, 795, 811, 839, 851, 871, 915, 951, 999, 1015, 1035, 1039, 1055, 1071, 1075, 1155, 1231, 1351

Offset: 1

Author

Jud McCranie

Keywords

• nonn

Comments

((3*k+1)/2)^2*(3*k+4) is k-hyperperfect.

References

• Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, p. 79.
• Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 177.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Judson S. McCranie, A study of hyperperfect numbers, J. Int. Seq., Vol. 3 (2000), Article 00.1.3.
• Daniel Minoli, Issues In Non-Linear Hyperperfect Numbers, Mathematics of Computation, Vol. 34, No. 150, April 1980, pp. 639-645.

Crossrefs

Cf. A007592.

Programs

Mathematica

q[k_] := Module[{m = (3*k + 1)^2*(3*k + 4)/4}, Divisible[m - 1, DivisorSigma[1, m] - m - 1]]; Select[Range[3, 1500, 2], q] (* Amiram Eldar, May 25 2025 *)

PARI

isok(k) = if(k == 1 || !(k % 2), 0, my(m = (3*k + 1)^2*(3*k + 4)/4); !((m-1) % (sigma(m)-m-1))); \\ Amiram Eldar, May 25 2025