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Numbers k such that d(k) is not an isolated number.
+10
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1, 4, 9, 16, 24, 25, 30, 36, 40, 42, 48, 49, 54, 56, 64, 66, 70, 78, 80, 81, 88, 100, 102, 104, 105, 110, 112, 114, 120, 121, 128, 130, 135, 136, 138, 144, 152, 154, 162, 165, 168, 169, 170, 174, 176, 182, 184, 186, 189, 190, 192, 195, 196, 208, 210, 216, 222
COMMENTS
A000005 = A167759 U A168086. Where 0,1,3,5,7,8,9,10,11,13,14,15,16,17,19,20,21,22,.. are nonisolated numbers A167707. The nonisolated numbers of divisors of n. The positions of isolated numbers in A000005.
FORMULA
A000005(a(n)) = nonisolated number.
EXTENSIONS
Corrected (132, 140, 148 removed, 152 inserted etc.) by R. J. Mathar, Jun 04 2010
Sums of the next n consecutive nonsquare integers.
+10
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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
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.
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:
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 *)
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