A090467 Numbers which are not regular figurative or polygonal numbers of order greater than 2. That is, numbers not of the form 1 + k*m*(m-1)/2 - (m-1)^2 where k > 2 and m > 2.

1, 2, 3, 4, 5, 7, 8, 11, 13, 14, 17, 19, 20, 23, 26, 29, 31, 32, 37, 38, 41, 43, 44, 47, 50, 53, 56, 59, 61, 62, 67, 68, 71, 73, 74, 77, 79, 80, 83, 86, 89, 97, 98, 101, 103, 104, 107, 109, 110, 113, 116, 119, 122, 127, 128, 131, 134, 137, 139, 140, 143, 146, 149, 151, 152

Offset: 1

Author

Robert G. Wilson v, Dec 01 2003

Keywords

• easy
• nonn

Comments

The m-th k-gonal number is 1 + k*m*(m-1)/2 - (m-1)^2 = A057145(k,m).

Numbers that are strictly trivially polygonal: numbers m that are only 2-gonal and m-gonal. - Daniel Mondot, Jun 13 2024

Examples

			3 is a triangular number, but is not a k-gonal number for any other k, so 3 is a term.
6 is both a triangular number and a hexagonal number, so 6 is not a term.
		

References

• Albert H. Beiler, Recreations In The Theory Of Numbers, The Queen Of Mathematics Entertains, Dover, NY, 1964, pp. 185-199.

Links

• Michel Marcus, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• Eric Weisstein's World of Mathematics, Figurate Number.
• Index to sequences related to polygonal numbers

Crossrefs

Complement is A090466.

Cf. A057145, A176774, A176775.

Programs

Mathematica

Complement[ Table[i, {i, 300}], Take[ Union[ Flatten[ Table[1 + k*n(n - 1)/2 - (n - 1)^2, {n, 3, 40}, {k, 3, 300}]]], 300]]

PARI

isok(n) = (n < 3) || (vecsum(vector(n-2, k, k+=2; ispolygonal(n, k))) == 1); \\ Michel Marcus, May 01 2016

Formula

An integer n >= 3 is in this sequence iff A176774(n) = n (or, equivalently, A176775(n) = 2). - Max Alekseyev, Apr 24 2018

Extensions

Verified by Don Reble, Mar 12 2006