A275740 Sums of the next n consecutive nonsquare integers.

0, 2, 8, 21, 46, 83, 136, 210, 306, 426, 575, 758, 972, 1223, 1519, 1855, 2236, 2669, 3156, 3694, 4290, 4956, 5678, 6467, 7332, 8269, 9278, 10368, 11548, 12804, 14148, 15593, 17126, 18753, 20485, 22325, 24262, 26308, 28481, 30756, 33148

Offset: 0

Author

Olivier Gérard, Aug 07 2016

Keywords

• nonn
• easy

Comments

Row sums of nonsquare integers (A000037), seen as a regular triangle:

.

2 | 2,

8 | 3, 5,

21 | 6, 7, 8,

46 | 10, 11, 12, 13,

83 | 14, 15, 17, 18, 19,

136 | 20, 21, 22, 23, 24, 26,

210 | 27, 28, 29, 30, 31, 32, 33,

306 | 34, 35, 37, 38, 39, 40, 41, 42,

...

The equivalent for all integers are A006003 (starting from 1), A229183 (starting from 2) and A027480 (starting from 0).

There are several sequences close to nonsquares whose sum of groups of n terms starts like this sequence, notably A028761, A158276, A167759.

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000037, A006003, A229183, A027480.

Programs

Maple

R:= 0: s:= 1:
for n from 1 to 100 do
  if floor(sqrt(s+n)) = floor(sqrt(s)) then
    R:= R, n*s + n*(n+1)/2; s:= s+n;
  else
    R:= R, n*s + n*(n+1)/2 - floor(sqrt(s+n))^2 + s+n+1; s:= s+n+1;
  fi
od:
R; # Robert Israel, Oct 02 2022

Mathematica

Table[Sum[
  i + Floor[1/2 + Sqrt[i]], {i, n (n - 1)/2 + 1, (n + 1) (n)/2}], {n,
  0, 40}]
Join[{0},Module[{nn=1000,nsi,len},nsi=Select[Range[nn],!IntegerQ[Sqrt[#]]&];len=Floor[ (Sqrt[ 8*Length[nsi]+1]-1)/2];Total/@TakeList[nsi,Range[len]]]] (* Harvey P. Dale, Jan 04 2024 *)

Extensions

Definition clarified by Harvey P. Dale, Jan 04 2024