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Search: id:A109470
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| A109470 |
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Sum of first n non-cubes. |
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+0
1
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| 2, 5, 9, 14, 20, 27, 36, 46, 57, 69, 82, 96, 111, 127, 144, 162, 181, 201, 222, 244, 267, 291, 316, 342, 370, 399, 429, 460, 492, 525, 559, 594, 630, 667, 705, 744, 784, 825, 867, 910, 954, 999, 1045, 1092, 1140, 1189, 1239, 1290, 1342, 1395, 1449, 1504, 1560
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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1^3 + 2^3 + 3^3 +...+ n^3=(1+2+3+...+n)^2. Note that the sum of noncubes can be cube: a(6) = 3^3. Note that the sum of noncubes can be square: a(4) = 3^2, a(7) = 6^2, a(15) = 12^2, a(37) = 28^2, a(69) = 51^2. Primes in this sequence include: a(1) = 2, a(2) = 5, a(14) = 127, a(17) = 181, a(62) = 2111, a(73) = 2903, a(77) = 3221.
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FORMULA
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a(n) = SUM{from i = 1 to n} A007412(i). a(n) = SUM{from i = 1 to n} (i +[(i+[i^{1/3}])^{1/3}]) where [x] = floor(x). a(n) = A000217(A007412(n)) - SUM{from i = 1 to [(A007412(n)^(1/3))]} i^3. a(n) = A000217(A007412(n)) - (A000217([(A007412(n))^(1/3)])^2).
Set R=a007412(n), S=FLOOR(R^(1/3)), then a(n)=(R*(R+1))/2-((S*(S+1))/2)^2 [From Gerald Hillier (adr.rabbicat(AT)gmail.com), Dec 21 2008]
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EXAMPLE
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a(6) = 2 + 3 + 4 + 5 + 6 + 7 = 27.
a(7) = 2 + 3 + 4 + 5 + 6 + 7 + 9 = 36.
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CROSSREFS
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Cf. A000537, A007412, A048766, A064524, A086849.
Sequence in context: A080956 A132337 A134189 this_sequence A112873 A048093 A024669
Adjacent sequences: A109467 A109468 A109469 this_sequence A109471 A109472 A109473
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KEYWORD
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easy,nonn
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AUTHOR
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Jonathan Vos Post (jvospost3(AT)gmail.com), Aug 28 2005
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