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A177025
Number of ways to represent n as a polygonal number.
13
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, 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, 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
OFFSET
3,4
COMMENTS
Frequency of n in the array A139601 or A086270 of polygonal numbers.
Since n is always n-gonal number, a(n) >= 1.
Conjecture: Every positive integer appears in the sequence.
Records of 2, 3, 4, 5, ... are reached at n = 6, 15, 36, 225, 561, 1225, ... see A063778. [R. J. Mathar, Aug 15 2010]
REFERENCES
J. J. Tattersall, Elementary Number Theory in Nine chapters, 2nd ed (2005), Cambridge Univ. Press, page 22 Problem 26, citing Wertheim (1897)
LINKS
E. Deza and M. Deza, Figurate Numbers, World Scientific, 2012; see p. 45.
FORMULA
a(n) = A129654(n) - 1.
G.f.: x * Sum_{k>=2} x^k / (1 - x^(k*(k + 1)/2)) (conjecture). - Ilya Gutkovskiy, Apr 09 2020
MAPLE
A177025 := proc(p)
local ii, a, n, s, m ;
ii := 2*p ;
a := 0 ;
for n in numtheory[divisors](ii) do
if n > 2 then
s := ii/n ;
if (s-2) mod (n-1) = 0 then
a := a+1 ;
end if;
end if;
end do:
return a;
end proc: # R. J. Mathar, Jan 10 2013
MATHEMATICA
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 *)
Table[Length[Intersection[Divisors[2 n - 2] + 1, Divisors[2 n]]] - 1, {n, 3, 100}] (* Jonathan Sondow, May 09 2014 *)
PROG
(PARI) a(n) = sum(i=3, n, ispolygonal(n, i)); \\ Michel Marcus, Jul 08 2014
(Python)
from sympy import divisors
def a(n):
i=2*n
x=0
for d in divisors(i):
if d>2:
s=i/d
if (s - 2)%(d - 1)==0: x+=1
return x # Indranil Ghosh, Apr 28 2017, translated from Maple code by R. J. Mathar
KEYWORD
nonn
AUTHOR
Vladimir Shevelev, May 01 2010
EXTENSIONS
Extended by R. J. Mathar, Aug 15 2010
STATUS
approved