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A376997
Triangular numbers that are pseudoperfect (semiperfect).
2
6, 28, 36, 66, 78, 120, 210, 276, 300, 378, 496, 528, 630, 666, 780, 820, 990, 1128, 1176, 1326, 1540, 1596, 1770, 1830, 2016, 2080, 2346, 2556, 2628, 2850, 3160, 3240, 3486, 3570, 3828, 4095, 4278, 4560, 4656, 4950, 5460, 5778, 5886, 6216, 6328, 6786, 7140, 7260, 7626, 7875, 8128, 8256, 8646
OFFSET
1,1
COMMENTS
A number is called pseudoperfect (or semiperfect) if it is equal to the sum of a subset of its proper divisors.
If x == 0, 1, 4 or 9 (mod 12), the triangular number k = x * (x-1)/2 is divisible by 6, and k is a term since k = k/2 + k/3 + k/6. - Robert Israel, Nov 07 2024
LINKS
EXAMPLE
6 is a term because has three proper divisors (1, 2, 3) and the sum of subset (1,2,3) is 6.
36 is a term because has five proper divisors (1, 2, 3, 4, 6, 9, 12, 18, 36) and the sum of the subset (6,12,18) is 36.
MAPLE
ispp:= proc(n) uses Optimization;
local D, d, R;
D:= numtheory:-divisors(n) minus {n};
R:= traperror(Minimize(add(x[d], d=D), {add(d*x[d], d=D)=n}, assume=binary));
R <> lasterror and R[1] > 0
end proc;
select(ispp, [seq(i*(i+1)/2, i=1..200)]); # Robert Israel, Oct 16 2024
MATHEMATICA
psQ[n_] := Module[{d = Most[Divisors[n]], x}, Total[d] >= n && SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, 1, Length[d]}], {x, 0, n}], n] > 0]; Select[Accumulate[Range[132]], psQ] (* Amiram Eldar, Oct 12 2024 *)
CROSSREFS
Intersection of A000217 and A005835.
Sequence in context: A117948 A252234 A334405 * A242344 A344588 A247111
KEYWORD
nonn
AUTHOR
Massimo Kofler, Oct 12 2024
STATUS
approved