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Triangular multiply-perfect numbers.
2

%I #16 May 12 2024 00:46:00

%S 1,6,28,120,496,8128,523776,33550336,8589869056,137438691328,

%T 2305843008139952128,2658455991569831744654692615953842176,

%U 191561942608236107294793378084303638130997321548169216,13164036458569648337239753460458722910223472318386943117783728128

%N Triangular multiply-perfect numbers.

%C 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.

%C 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).

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

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

%C Intersection of A000217 and A007691.

%H Amiram Eldar, <a href="/A330532/b330532.txt">Table of n, a(n) for n = 1..15</a>

%o (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))))]

%o (PARI) isok(k) = ispolygonal(k, 3) && (denominator(sigma(k)/k) == 1); \\ _Michel Marcus_, Dec 19 2019

%Y Cf. A000217, A000396, A005820, A007691.

%K nonn

%O 1,2

%A _Jaroslav Krizek_, Dec 17 2019