A330532 Triangular multiply-perfect numbers.

1, 6, 28, 120, 496, 8128, 523776, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216, 13164036458569648337239753460458722910223472318386943117783728128

Offset: 1

Author

Jaroslav Krizek, Dec 17 2019

Keywords

• nonn

Comments

Multiply-perfect numbers of the form 2^(k - 1) * (2^k - 1) that can be written as sum of the first h natural numbers for some h.

Corresponding values of numbers k and h: (1, 2, 3, 4, 5, 7, 10, 13, 17, 19, 31, 61, 89, 107, 127, ...), (1, 3, 7, 15, 31, 127, 1023, 8191, 131071, 524287, 2147483647, ...), where h = 2^k - 1. Conjecture: numbers h are numbers from A066175 (sigma(phi(sigma(h))) = h).

Corresponding values of abundancies sigma(a(n)) / a(n): 1, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, ...

Conjecture: union of even perfect numbers from A000396 and 3-perfect numbers 120 and 523776.

Intersection of A000217 and A007691.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..15

Crossrefs

Cf. A000217, A000396, A005820, A007691.

Programs

Magma

[(2^k - 1) * (2^(k - 1)): k in [1..200] | IsIntegral(SumOfDivisors((2^k - 1) * (2^(k - 1)))/( (2^k - 1) * (2^(k - 1))))];

PARI

isok(k) = ispolygonal(k, 3) && (denominator(sigma(k)/k) == 1); \\ Michel Marcus, Dec 19 2019