A377747 Centered triangular numbers that are pseudoperfect (semiperfect).

460, 760, 2584, 4960, 5860, 14260, 15760, 22144, 27136, 28360, 30460, 41584, 47260, 49960, 70960, 74260, 79696, 99460, 103360, 118864, 132760, 137260, 147424, 159904, 170860, 175960, 179056, 213760, 219460, 261460, 267760, 309856, 313960, 320860, 371260, 378760, 383296, 428536, 433360, 441460, 459544, 486496

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.

Centered triangular numbers are the sum of three consecutive triangular numbers (A000217).

If x == 18 or 23 (mod 40), then the centered triangular number k = 3 * x * (x-1)/2 + 1 is divisible by 20, and k is a term since k = k/2 + k/4 + k/5 + k/20. - Robert Israel, Nov 07 2024

Links

Robert Israel, Table of n, a(n) for n = 1..841

Example

460 is a term in the sequence because 460 = 3 * 18 * 17 / 2 + 1 is a centered triangular number and 460 = 23 + 92 + 115 + 230 where 23, 92, 115 and 230 are proper divisors of 460.
760 is a term in the sequence because 760 = 3 * 23 * 22 / 2 + 1 is a centered triangular number and 760 = 8 + 10 + 20 + 152 + 190 + 380 where 8, 10, 20, 152, 190 and 380 are proper divisors of 760.

Maple

filter:= proc(n) local S, d;
  S:= mul(1+x^d, d=numtheory:-divisors(n) minus {n});
  coeff(S, x, n) > 0
end proc:
select(filter, [seq(3*n*(n-1)/2+1, n=2..1000)]); # Robert Israel, Nov 07 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[Table[3*n*(n-1)/2 + 1, {n, 1, 400}], psQ] (* Amiram Eldar, Nov 06 2024 *)

Crossrefs

Intersection of A005448 and A005835.

Cf. A376997.

Sequence in context: A048923 A329292 A252535 * A264242 A358894 A107973

Adjacent sequences: A377744 A377745 A377746 * A377748 A377749 A377750

Keyword

nonn

Author

Massimo Kofler, Nov 06 2024

Status

approved