A177025 Number of ways to represent n as a polygonal number.

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

T. D. Noe, Table of n, a(n) for n = 3..10000

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

Crossrefs

Cf. A129654, A139601, A090428, A176949, A176948, A176774, A176744, A176747, A176775, A175873, A176874.

Sequence in context: A154402 A210682 A293433 * A265210 A023396 A091221

Adjacent sequences: A177022 A177023 A177024 * A177026 A177027 A177028

Keyword

nonn

Author

Vladimir Shevelev, May 01 2010

Extensions

Extended by R. J. Mathar, Aug 15 2010

Status

approved