This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A330532 #19 Jul 13 2025 10:59:17 %S A330532 1,6,28,120,496,8128,523776,33550336,8589869056,137438691328, %T A330532 2305843008139952128,2658455991569831744654692615953842176, %U A330532 191561942608236107294793378084303638130997321548169216,13164036458569648337239753460458722910223472318386943117783728128 %N A330532 Triangular multiply-perfect numbers. %C A330532 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 A330532 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 A330532 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 A330532 Conjecture: union of even perfect numbers from A000396 and 3-perfect numbers 120 and 523776. %C A330532 Intersection of A000217 and A007691. %H A330532 Amiram Eldar, <a href="/A330532/b330532.txt">Table of n, a(n) for n = 1..15</a> %o A330532 (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 A330532 (PARI) isok(k) = ispolygonal(k, 3) && (denominator(sigma(k)/k) == 1); \\ _Michel Marcus_, Dec 19 2019 %Y A330532 Cf. A000217, A000396, A005820, A007691. %K A330532 nonn %O A330532 1,2 %A A330532 _Jaroslav Krizek_, Dec 17 2019