A376997 Triangular numbers that are pseudoperfect (semiperfect).

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

Robert Israel, Table of n, a(n) for n = 1..10000

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

Adjacent sequences: A376994 A376995 A376996 * A376998 A376999 A377000

Keyword

nonn,changed

Author

Massimo Kofler, Oct 12 2024

Status

approved