

A133759


Numbers that are the sum of a nonzero pentagonal number and a nonzero square in at least one way.


0



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
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OFFSET

1,1


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?


LINKS

Table of n, a(n) for n=1..65.


FORMULA

{A000326(i) + A000290(j) for i, j > 0}. {i(3*i1)/2 + j^2 for i, j > 0}.


EXAMPLE

Let P(n) = nth 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.


CROSSREFS

Cf. A000290, A000326, A134935A134938.
Sequence in context: A031461 A085183 A340289 * A188258 A308395 A227149
Adjacent sequences: A133756 A133757 A133758 * A133760 A133761 A133762


KEYWORD

easy,nonn


AUTHOR

Jonathan Vos Post, Jan 21 2008


EXTENSIONS

Corrected and extended by R. J. Mathar, Jan 21 2008


STATUS

approved



