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%I #35 Apr 09 2020 23:49:40
%S 1,1,1,2,1,1,2,2,1,2,1,1,3,2,1,2,1,1,3,2,1,2,2,1,2,3,1,2,1,1,2,2,2,4,
%T 1,1,2,2,1,2,1,1,4,2,1,2,2,1,3,2,1,2,3,1,2,2,1,2,1,1,2,3,2,4,1,1,2,3,
%U 1,2,1,1,3,2,1,3,1,1,4,2,1,2,2,1,2,2,1,2,3,2,2,2,2,3,1,1,2,3
%N Number of ways to represent n as a polygonal number.
%C Frequency of n in the array A139601 or A086270 of polygonal numbers.
%C Since n is always n-gonal number, a(n) >= 1.
%C Conjecture: Every positive integer appears in the sequence.
%C Records of 2, 3, 4, 5, ... are reached at n = 6, 15, 36, 225, 561, 1225, ... see A063778. [_R. J. Mathar_, Aug 15 2010]
%D J. J. Tattersall, Elementary Number Theory in Nine chapters, 2nd ed (2005), Cambridge Univ. Press, page 22 Problem 26, citing Wertheim (1897)
%H T. D. Noe, <a href="/A177025/b177025.txt">Table of n, a(n) for n = 3..10000</a>
%H E. Deza and M. Deza, <a href="http://www.worldscientific.com/doi/suppl/10.1142/8188/suppl_file/8188_chap01.pdf">Figurate Numbers</a>, World Scientific, 2012; see p. 45.
%F a(n) = A129654(n) - 1.
%F G.f.: x * Sum_{k>=2} x^k / (1 - x^(k*(k + 1)/2)) (conjecture). - _Ilya Gutkovskiy_, Apr 09 2020
%p A177025 := proc(p)
%p local ii,a,n,s,m ;
%p ii := 2*p ;
%p a := 0 ;
%p for n in numtheory[divisors](ii) do
%p if n > 2 then
%p s := ii/n ;
%p if (s-2) mod (n-1) = 0 then
%p a := a+1 ;
%p end if;
%p end if;
%p end do:
%p return a;
%p end proc: # _R. J. Mathar_, Jan 10 2013
%t nn = 100; t = Table[0, {nn}]; Do[k = 2; While[p = k*((n - 2) k - (n - 4))/2; p <= nn, t[[p]]++; k++], {n, 3, nn}]; t (* _T. D. Noe_, Apr 13 2011 *)
%t Table[Length[Intersection[Divisors[2 n - 2] + 1, Divisors[2 n]]] - 1, {n, 3, 100}] (* _Jonathan Sondow_, May 09 2014 *)
%o (PARI) a(n) = sum(i=3, n, ispolygonal(n, i)); \\ _Michel Marcus_, Jul 08 2014
%o (Python)
%o from sympy import divisors
%o def a(n):
%o i=2*n
%o x=0
%o for d in divisors(i):
%o if d>2:
%o s=i/d
%o if (s - 2)%(d - 1)==0: x+=1
%o return x # _Indranil Ghosh_, Apr 28 2017, translated from Maple code by _R. J. Mathar_
%Y Cf. A129654, A139601, A090428, A176949, A176948, A176774, A176744, A176747, A176775, A175873, A176874.
%K nonn
%O 3,4
%A _Vladimir Shevelev_, May 01 2010
%E Extended by _R. J. Mathar_, Aug 15 2010
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