A133759 Numbers that are the sum of a nonzero pentagonal number and a nonzero square in at least one way.

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

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*i-1)/2 + j^2 for i, j > 0}.

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.

Crossrefs

Cf. A000290, A000326, A134935-A134938.

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