A176774 Smallest polygonality of n = smallest integer m>=3 such that n is m-gonal number.

3, 4, 5, 3, 7, 8, 4, 3, 11, 5, 13, 14, 3, 4, 17, 7, 19, 20, 3, 5, 23, 9, 4, 26, 10, 3, 29, 11, 31, 32, 12, 7, 5, 3, 37, 38, 14, 8, 41, 15, 43, 44, 3, 9, 47, 17, 4, 50, 5, 10, 53, 19, 3, 56, 20, 11, 59, 21, 61, 62, 22, 4, 8, 3, 67, 68, 24, 5, 71, 25, 73, 74, 9, 14, 77, 3, 79, 80, 4, 15, 83

Offset: 3

Author

Max Alekseyev, Apr 25 2010

Keywords

• nonn

Comments

A176775(n) gives the index of n as a(n)-gonal number.

Since n is the second n-gonal number, a(n) <= n.

Furthermore, a(n)=n iff A176775(n)=2.

Examples

			a(12) = 5 since 12 is a pentagonal number, but not a square or triangular number. - _Michael B. Porter_, Jul 16 2016
		

Links

• Michel Marcus, Table of n, a(n) for n = 3..10000
• Eric W. Weisstein, Polygonal Number. MathWorld.

Crossrefs

Cf. A090466, A090467.

Programs

Mathematica

a[n_] := (m = 3; While[Reduce[k >= 1 && n == k (k (m - 2) - m + 4)/2, k, Integers] == False, m++]; m); Table[a[n], {n, 3, 100}] (* Jean-François Alcover, Sep 04 2016 *)

PARI

a(n) = {k=3; while (! ispolygonal(n, k), k++); k;} \\ Michel Marcus, Mar 25 2015

Python

from _future_ import division
from gmpy2 import isqrt
def A176774(n):
    k = (isqrt(8*n+1)-1)//2
    while k >= 2:
        a, b = divmod(2*(k*(k-2)+n),k*(k-1))
        if not b:
            return a
        k -= 1 # Chai Wah Wu, Jul 28 2016