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A133759
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Numbers that are the sum of a nonzero pentagonal number and a nonzero square in at least one way.
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0
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2, 5, 6, 9, 10, 13, 14, 16, 17, 21, 23, 26, 28, 30, 31, 36, 37, 38, 39, 41, 44, 47, 48, 50, 51, 52, 54, 55, 58, 60, 61, 65, 67, 69, 71, 74, 76, 79, 82, 84, 86, 87, 93, 95, 96, 99, 100, 101, 103, 105, 106, 108, 112, 115, 116, 117, 118, 119, 121, 122, 126, 128, 132, 133, 134
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| These pentagonal numbers P(k) that can be represented as the sum of P(i)+j^2, i,j>0, are at k= 2, 6, 9, 10, 13, 17, 21, 22, 24, 26, 29, 34, 35, 38, 41, 45, 46, 53. Are almost all positive integers in this sequence and, if so, what is the largest value in the complement? The largest square in the complement? The largest pentagonal number in the complement?
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FORMULA
| {A000326(i) + A000290(j) for i, j > 0}. {i(3*i-1)/2 + j^2 for i, j > 0}.
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EXAMPLE
| Let P(n) = n-th pentagonal number:
a(1) = P(1) + 1^2 = 1 + 1 = 2.
a(2) = P(1) + 2^2 = 1 + 4 = 5 = P(2).
a(3) = P(2) + 1^2 = 5 + 1 = 6.
a(4) = P(2) + 2^2 = 5 + 4 = 9 = 3^2.
a(5) = P(1) + 3^2 = 1 + 9 = 10 = a(P(2)).
a(8) = P(3) + 2^2 = 12 + 4 = 16 = 4^2.
a(10) = P(2) + 4^2 = 5 + 16 = P(3) + 3^2 = 12 + 9 = 21.
a(12) = P(1) + 5^2 = 1 + 25 = P(4) + 2^2 = 22 + 4 = 26 = a(P(3)).
a(16) = P(5) + 1^2 = 35 + 1 = 36 = 6^2.
a(17) = P(1) + 6^2 = 1 + 36 = P(3) + 5^2 = 12 + 25 = 37.
a(25) = P(5) + 4^2 = 35 + 16 = 51 = P(6).
a(30) = P(6) + 3^2 = 51 + 9 = P(5) + 5^2 = 35 + 25 = 60.
a(35) = P(7) + 1^2 = 70 + 1 = P(5) + 6^2 = 35 + 36 = P(4) + 7^2 = 22 + 49 = 71 = a(P(5)).
a(37) = P(6) + 5^2 = 51 + 25 = P(3) + 8^2 = 12 + 64 = 76.
a(41) = P(7) + 4^2 = 70 + 16 = P(4) + 8^2 = 22 + 64 = 86.
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CROSSREFS
| Cf. A000290, A000326, A134935-A134938.
Sequence in context: A028946 A031461 A085183 * A188258 A042963 A166097
Adjacent sequences: A133756 A133757 A133758 * A133760 A133761 A133762
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KEYWORD
| easy,nonn
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AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 21 2008
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EXTENSIONS
| Corrected and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 21 2008
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