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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 (list; graph; refs; listen; history; text; internal format)
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: A213713 A031461 A085183 * A188258 A227149 A042963

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

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Last modified April 25 04:12 EDT 2019. Contains 322451 sequences. (Running on oeis4.)