A176747 -id:A176747 - 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

A176948 a(n) is the smallest solution x to A176774(x)=n; a(n)=0 if this equation has no solution.

Original entry on oeis.org

3, 4, 5, 0, 7, 8, 24, 27, 11, 33, 13, 14, 42, 88, 17, 165, 19, 20, 60, 63, 23, 69, 72, 26, 255, 160, 29, 87, 31, 32, 315, 99, 102, 208, 37, 38, 114, 805, 41, 123, 43, 44, 132, 268, 47, 696, 475, 50, 150, 304, 53, 159, 162, 56, 168, 340, 59, 177, 61, 62, 615, 1309, 192, 388

Offset: 3

Vladimir Shevelev, Apr 29 2010

nonn

A greedy inverse function to A176774.
Conjecture: For every n >= 4, except for n=6, there exists an n-gonal number N which is not k-gonal for 3 <= k < n.
This means that the sequence contains only one 0: a(6)=0. For n=6 it is easy to prove that every hexagonal number (A000384) is also triangular (A000217), i.e., N does not exist. - Vladimir Shevelev, Apr 30 2010

			For n=9, 24 is a nonagonal number, but not an octagonal number, heptagonal number, hexagonal number, etc. The smaller nonagonal number 9 is also a square number. Thus, a(9) = 24. - _Michael B. Porter_, Jul 16 2016

Chai Wah Wu, Table of n, a(n) for n = 3..10000

Cf. A176744, A176747, A176775, A175873, A176874.

A139601 := proc(k,n) option remember ; n/2*( (k-2)*n-k+4) ; end proc:
A176774 := proc(n) option remember ; local k,m,pol ; for k from 3 do for m from 0 do pol := A139601(k,m) ; if pol = n then return k ; elif pol > n then break; end if; end do: end do: end proc:
A176948 := proc(n) if n = 6 then 0; else for x from 3 do if A176774(x)= n then return x ; end if; end do: end if; end proc:
seq(A176948(n),n=3..80) ; # R. J. Mathar, May 03 2010

A176774[n_] := A176774[n] = (m = 3; While[Reduce[k >= 1 && n == k (k (m - 2) - m + 4)/2, k, Integers] == False, m++]; m); a[6] = 0; a[p_?PrimeQ] := p; a[n_] := (x = 3; While[A176774[x] != n, x++]; x); Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 3, 100}] (* Jean-François Alcover, Sep 04 2016 *)

a(p) = p if p is any odd prime.

More terms from R. J. Mathar, May 03 2010

A176949 Composite numbers n for which A176948(n) = n.

Original entry on oeis.org

4, 8, 14, 20, 26, 32, 38, 44, 50, 56, 62, 68, 74, 77, 80, 86, 98, 104, 110, 116, 119, 122, 128, 134, 140, 143, 146, 152, 158, 161, 164, 170, 182, 187, 188, 194, 200, 203, 206, 209, 212, 218, 221, 224, 230, 236, 242, 248, 254, 266, 272, 278, 284, 290, 296, 299, 302

Offset: 1

Vladimir Shevelev, Apr 29 2010, Apr 30 2010

nonn

If p >= 3 is prime, then A176948(p) = p. The sequence lists composite numbers with this property.
It is interesting that there is a large overlap with terms in A140164 (but there are exceptions, e.g., 77).
From Daniel Forgues, Jul 15 2016: (Start)
Composite numbers n which are not of form (k/2)*[(m-2)*k - (m-4)] for any m >= 3 and k >= 3, thus not m-gonal numbers for any order k >= 3.
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 k >= 3.
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 order 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.
Odd composite numbers n which are divisible by 5 are m-gonal numbers of order 5, with m = (n + 15)/10. Thus all odd a(n) are coprime to 5.
a(1) = 4 is the only square number: 4-gonal with order k = 2. (End)
An integer n which is congruent to k (mod t_{k-1}) with 3 <= t_{k-1} < n, i.e. n = j * t_{k-1} + k with k >= 3 and j >= 1, is an m-gonal number of order k, with m = j + 2, where t_{k-1} is a triangular number. If all the congruence tests fail, a composite n belongs to this sequence. - Daniel Forgues, Aug 02 2016
From Jonathan Dushoff, Apr 05 2022: (Start)
All numbers n>2 are trivially n-gonal numbers, and will thus have A176948(n)=n unless they have a nontrivial polygonal decomposition. Thus this is just the sequence of non-polygonal composite numbers.
Note that the 2nd through 13th terms are in arithmetic progression.
Some reasons: many of the smaller odd numbers are prime (and thus don't appear); numbers of the form 6x (or 6x+3) are always order-3 numbers; numbers of the form 6x+4 are always order-4 numbers; small odd composites not divisible by 3 are usually divisible by 5, and are thus order-5 numbers.
In fact, the first number to break the arithmetic progression is the first product of distinct primes > 5.
Conversely, 6x+2 numbers cannot be order-3 or -6 numbers (those are divisible by 3); order-4 numbers (all == 4 (mod 6)); order-5 numbers (all odd); or order-7 numbers (all == 1 (mod 3)).
The first 6x+2 composite not in the list is order-8 pentagonal number 92. (End)

			8 is in the sequence since it is composite and is an octagonal number, but not a heptagonal number, hexagonal number, pentagonal number, etc. 10 is not in the sequence because even though it is composite and a decagonal number, it is also a triangular number: 10 = 1 + 2 + 3 + 4. - _Michael B. Porter_, Jul 16 2016

Giovanni Resta, Table of n, a(n) for n = 1..10000
OEIS Wiki, Polygonal numbers

Cf. A175873, A176744, A176747, A176774, A176775, A176874, A176948, A274967, A274968.

Select[Range[302], CompositeQ@ # && FindInstance[n*(4 + n*(s-2) - s)/2 == # && s >= 3 && n >= 3, {s, n}, Integers] == {} &] (* Giovanni Resta, Jul 13 2016 *)

listc(nn) = {forcomposite(c=1, nn, sp = c; forstep(k=c, 3, -1, if (ispolygonal(c, k), sp=k);); if (sp == c, print1(c, ", ")););} \\ Michel Marcus, Sep 06 2016

def is_a(n):
    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..302) if is_a(n)]) # Peter Luschny, Jul 28 2016

Offset corrected and sequence extended by R. J. Mathar, May 03 2010

A355774 An extension of the generalized pentagonal numbers such that every positive integer can be represented as the sum of at most two terms of the sequence.

Original entry on oeis.org

0, 1, 2, 5, 7, 11, 12, 15, 21, 22, 25, 26, 35, 39, 40, 49, 51, 57, 67, 70, 77, 87, 92, 100, 117, 120, 123, 126, 145, 153, 155, 173, 176, 182, 186, 187, 205, 210, 214, 222, 228, 241, 247, 251, 260, 283, 287, 301, 319, 330, 345, 376, 382, 392, 425, 435, 442, 448

Offset: 0

Peter Luschny, Jul 17 2022

nonn

The sequence is defined inductively. Starting from the empty sequence, the terms are added one after the other. A term is added if it is a generalized pentagonal number or if it cannot be represented as the sum of two preceding terms. Note that these exceptions form a proper subsequence of A093519.

Thus any positive number can be expressed as the sum of at most two positive terms by Euler's Pentagonal Number Theorem. Every pentagonal number and every generalized pentagonal number is in this sequence.

			32 = 7 + 25; 195 = 22 + 173.

Andreas Enge, William Hart and Fredrik Johansson, Short addition sequences for theta functions, arXiv:1608.06810 [math.NT], 2016.
Burkard Polster (Mathloger), The hardest 'What comes next?' (Euler's pentagonal formula), YouTube video, 2020.

Cf. A000326, A001318, A093519, A100878, A355717, A176747 (same construction with triangular numbers).

A355774_list := proc(upto) local P, k, issum, isgpn; P := [];
isgpn := k -> ormap(n -> 0 = 8*k-(n+irem(n,2))*(3*n+2-irem(n,2)), [$0..k]);
issum := k -> ormap(p -> member(k - p, P), P);
for k from 0 to upto do
    if isgpn(k) or not issum(k) then P := [op(P), k] fi od;
P end: print(A355774_list(448));

isgpn[k_] := AnyTrue[Range[0, k], 0 == 8*k-(#+Mod[#,2])*(3*#+2-Mod[#,2])&];
issum[k_] := AnyTrue[P, MemberQ[P, k-#]&];
P = {};
For[k = 0, k <= 448, k++, If[isgpn[k] || !issum[k], AppendTo[P, k]]];
P (* Jean-François Alcover, Mar 07 2024, after Peter Luschny *)

def A355774_list(upto: int) -> list[int]:
    P: list[int] = []
    for k in range(upto + 1):
        if any(
            k == ((n + n % 2) * (3 * n + 2 - n % 2)) >> 3
            for n in range(k + 1)
        ) or not any([(k - p) in P for p in P]):
            P.append(k)
    return P
print(A355774_list(448))

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

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

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A176948 a(n) is the smallest solution x to A176774(x)=n; a(n)=0 if this equation has no solution.

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A176949 Composite numbers n for which A176948(n) = n.

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A355774 An extension of the generalized pentagonal numbers such that every positive integer can be represented as the sum of at most two terms of the sequence.

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