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%I #25 Sep 04 2016 12:10:08
%S 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,
%T 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,
%U 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
%N Smallest polygonality of n = smallest integer m>=3 such that n is m-gonal number.
%C A176775(n) gives the index of n as a(n)-gonal number.
%C Since n is the second n-gonal number, a(n) <= n.
%C Furthermore, a(n)=n iff A176775(n)=2.
%H Michel Marcus, <a href="/A176774/b176774.txt">Table of n, a(n) for n = 3..10000</a>
%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/PolygonalNumber.html">Polygonal Number</a>. MathWorld.
%e a(12) = 5 since 12 is a pentagonal number, but not a square or triangular number. - _Michael B. Porter_, Jul 16 2016
%t 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 *)
%o (PARI) a(n) = {k=3; while (! ispolygonal(n, k), k++); k;} \\ _Michel Marcus_, Mar 25 2015
%o (Python)
%o from __future__ import division
%o from gmpy2 import isqrt
%o def A176774(n):
%o k = (isqrt(8*n+1)-1)//2
%o while k >= 2:
%o a, b = divmod(2*(k*(k-2)+n),k*(k-1))
%o if not b:
%o return a
%o k -= 1 # _Chai Wah Wu_, Jul 28 2016
%Y Cf. A090466, A090467.
%K nonn
%O 3,1
%A _Max Alekseyev_, Apr 25 2010
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