A176949 -id:A176949 - cp's OEIS frontend

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

Vladimir Shevelev, May 01 2010

nonn

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]

J. J. Tattersall, Elementary Number Theory in Nine chapters, 2nd ed (2005), Cambridge Univ. Press, page 22 Problem 26, citing Wertheim (1897)

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.

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

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

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 *)

a(n) = sum(i=3, n, ispolygonal(n, i)); \\ Michel Marcus, Jul 08 2014

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

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

Extended by R. J. Mathar, Aug 15 2010

A177029 Numbers that have exactly two different representations as polygonal numbers.

Original entry on oeis.org

6, 9, 10, 12, 16, 18, 22, 24, 25, 27, 30, 33, 34, 35, 39, 40, 42, 46, 48, 49, 52, 54, 57, 58, 60, 63, 65, 69, 72, 76, 82, 84, 85, 87, 88, 90, 92, 93, 94, 95, 99, 102, 106, 108, 114, 115, 118, 121, 123, 124, 125, 129, 130, 132, 133, 138, 142, 147, 150, 155, 159, 160, 162, 166, 168

Offset: 1

Vladimir Shevelev, May 01 2010

nonn

Numbers that have only A177025(.)=1 representation are listed by A090467.

			6 is a triangular and a hexagonal number, but is not any other k-gonal number.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A177025, A090467, A177028, A000217, A000326, A000384, A000566, A000567, A176949, A176948, A176774, A176744, A176747, A176775, A175873, A176874.

lista(nn) = {for (n=1, nn, if (sum(k=3, n, ispolygonal(n, k)) == 2, print1(n, ", ")););} \\ Michel Marcus, Mar 25 2015

A177029_list = []
for m in range(1,10**4):
    n, c = 3, 0
    while n*(n+1) <= 2*m:
        if not 2*(n*(n-2) + m) % (n*(n - 1)):
            c += 1
            if c > 1:
                break
        n += 1
    if c == 1:
        A177029_list.append(m) # Chai Wah Wu, Jul 28 2016

{m: A177025(m)=2}.

Extended by R. J. Mathar, Aug 15 2010

A274967 Odd composite numbers n which are not m-gonal number for 3 <= m < n.

Original entry on oeis.org

77, 119, 143, 161, 187, 203, 209, 221, 299, 319, 323, 329, 371, 377, 391, 407, 413, 437, 473, 493, 497, 517, 527, 533, 539, 551, 581, 583, 589, 611, 623, 629, 649, 667, 689, 707, 713, 731, 737, 749, 767, 779, 791, 799, 803, 817, 851, 869, 893, 899, 901, 913

Offset: 1

Daniel Forgues, Jul 12 2016

An m-gonal number, m >= 3, i.e. of form n = (k/2)*[(m-2)*k - (m-4)], yields a nontrivial factorization of n if and only if of order k >= 3.
Odd composite numbers n for which A176948(n) = n.
All odd composite n are coprime to 30 (see next comment) and have smallest prime factor >= 7, e.g.
   77 =   7*11, 119 =  7*17, 143 = 11*13, 161 =  7*23,
  187 =  11*17, 203 =  7*29, 209 = 11*19, 221 = 13*17,
  299 =  13*23, 319 = 11*29, 323 = 17*19, 329 =  7*47,
  371 =   7*53, 377 = 13*29, 391 = 17*23, 407 = 11*37,
  413 =   7*59, 437 = 19*23, 473 = 11*43, 493 = 17*29,
  497 =   7*71, 517 = 11*47, 527 = 17*31, 533 = 13*41,
  539 = 7*7*11, 551 = 19*29, 581 =  7*83, 583 = 11*53,
  589 =  19*31, 611 = 13*47, 623 =  7*89, 629 = 17*37,
  649 =  11*59, 667 = 23*29, 689 = 13*53, 707 = 7*101,
  713 =  23*31, 731 = 17*43, 737 = 11*67, 749 = 7*107,
  767 =  13*59, 779 = 19*41, 791 = 7*113, 799 = 17*47,
  803 =  11*73, 817 = 19*43, 851 = 23*37, 869 = 11*79,
  893 =  19*47, 899 = 29*31, 901 = 17*53, 913 = 11*83.
Composite numbers n which are divisible by 3 are m-gonal numbers of order 3, with m = (n + 3)/3. Thus all a(n) are coprime to 3.
Odd composite numbers n which are divisible by 5 are m-gonal numbers of order 5, with m = (n + 15)/10. Thus all a(n) are coprime to 5.
Since we are looking for solutions of (m-2)*k^2 - (m-4)*k - 2*n = 0, with m >= 3 and k >= 3, the largest k we need to consider is
    k = {(m-4) + sqrt[(m-4)^2 + 8*(m-2)*n]}/[2*(m-2)] with m = 3, thus
    k <= (1/2)*{-1 + sqrt[1 + 8*n]}.
Or, since we are looking for solutions of 2n = m*k*(k-1) - 2*k*(k-2), with m >= 3 and k >= 3, the largest m we need to consider is
    m = [2n + 2*k*(k-2)]/[k*(k-1)] with k = 3, thus m <= (n+3)/3.

			77 is in this sequence because 77 is trivially a 77-gonal number of order k = 2, but not an m-gonal number for 3 <= k <= (1/2)*{-1 + sqrt[1 + 8*77]}.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
OEIS Wiki, Polygonal numbers

Cf. A176774, A176948, A176949, A274968.

Select[Range[500]2+1, ! PrimeQ[#] && FindInstance[n*(4 + n*(s-2)-s)/2 == # && s >= 3 && n >= 3, {s, n}, Integers] == {} &] (* Giovanni Resta, Jul 13 2016 *)

from sympy import isprime
A274967_list = []
for n in range(3,10**6,2):
    if not isprime(n):
        k = 3
        while k*(k+1) <= 2*n:
            if not (2*(k*(k-2)+n)) % (k*(k - 1)):
                break
            k += 1
        else:
            A274967_list.append(n) # Chai Wah Wu, Jul 28 2016

def is_a(n):
    if is_even(n): return False
    if is_prime(n): return False
    for m in (3..(n+3)//3):
        if pari('ispolygonal')(n, m):
            return False
    return True
print([n for n in (3..913) if is_a(n)]) # Peter Luschny, Jul 28 2016

a(10)-a(52) from Giovanni Resta, Jul 13 2016

A274968 Even numbers n >= 4 which are not m-gonal number for 3 <= m < n.

Original entry on oeis.org

4, 8, 14, 20, 26, 32, 38, 44, 50, 56, 62, 68, 74, 80, 86, 98, 104, 110, 116, 122, 128, 134, 140, 146, 152, 158, 164, 170, 182, 188, 194, 200, 206, 212, 218, 224, 230, 236, 242, 248, 254, 266, 272, 278, 284, 290, 296, 302

Offset: 1

Daniel Forgues, Jul 12 2016

nonn

An m-gonal number, m >= 3, i.e., of the form n = (k/2)*[(m-2)*k - (m-4)], yields a nontrivial factorization of n if and only if of order k >= 3.
Except for a(1) = 4, all a(n) are congruent to 2 (mod 6), although from 8 to 302, the numbers
     92:  5-gonal of order 8,
    176:  5-gonal of order 11, 8-gonal of order 8,
    260: 11-gonal of order 8,
  are not in this sequence.
Even numbers n for which A176948(n) = n.
Since we are looking for solutions of (m-2)*k^2 - (m-4)*k - 2*n = 0, with m >= 3 and k >= 3, the largest order k we need to consider is
    k = {(m-4) + sqrt[(m-4)^2 + 8*(m-2)*n]}/[2*(m-2)] with m = 3, thus
    k <= (1/2)*{-1 + sqrt[1 + 8*n]}.
Or, since we are looking for solutions of 2n = m*k*(k-1) - 2*k*(k-2), with m >= 3 and k >= 3, the largest m we need to consider is
    m = [2n + 2*k*(k-2)]/[k*(k-1)] with k = 3, thus m <= (n+3)/3.
Composite numbers n which are divisible by 3 are m-gonal numbers of order 3, with m = (n + 3)/3. Thus all a(n) are coprime to 3.
a(1) = 4 is the only square number: 4-gonal with order k = 2.
All integers of the form n = 6j + 4, with j >= 1, are m-gonal numbers of order k = 4, with m = j + 2, which means that none are in this sequence. - Daniel Forgues, Aug 01 2016

			20 is in this sequence because 20 is trivially a 20-gonal number of order k = 2 (element of A051872) but not an m-gonal number for 3 <= k <= (1/2)*{-1 + sqrt[1 + 8*20]}.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
OEIS Wiki, Polygonal numbers

Cf. A051872, A176948, A176949, A274967.

lista(nn) = {forstep(n=4, nn, 2, sp = n; forstep(k=n, 3, -1, if (ispolygonal(n, k), sp=k);); if (sp == n, print1(n, ", ")););} \\ Michel Marcus, Sep 06 2016

A274968_list = []
for n in range(4,10**6,2):
    k = 3
    while k*(k+1) <= 2*n:
        if not (2*(k*(k-2)+n)) % (k*(k - 1)):
            break
        k += 1
    else:
        A274968_list.append(n) # Chai Wah Wu, Jul 28 2016

def is_A274968(n):
    if is_odd(n): return False
    for m in (3..(n+3)//3):
        if pari('ispolygonal')(n, m):
            return False
    return True
print([n for n in (3..302) if is_A274968(n)]) # Peter Luschny, Jul 28 2016

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A177029 Numbers that have exactly two different representations as polygonal numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

Formula

Extensions

A274967 Odd composite numbers n which are not m-gonal number for 3 <= m < n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Sage

Extensions

A274968 Even numbers n >= 4 which are not m-gonal number for 3 <= m < n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

Sage