A176775 -id:A176775 - cp's OEIS frontend

A344083 a(n) = f(x)+f(y)+f(z), where (x,y,h) is the n-th Pythagorean triple listed in (A046083, A046084, A009000), and f(m)=A176775(m) is the index of m as k-gonal number for the smallest possible k.

6, 9, 7, 11, 9, 9, 12, 10, 9, 10, 9, 11, 18, 10, 16, 9, 9, 20, 9, 7, 18, 9, 18, 15, 11, 14, 7, 12, 10, 13, 12, 7, 12, 15, 12, 17, 14, 18, 13, 9, 13, 14, 15, 10, 9, 7, 9, 21, 12, 10, 15, 23, 7, 9, 12, 20, 9, 18, 17, 28, 14, 16, 7, 21, 18, 24, 21, 21, 20, 16, 25

Offset: 1

Michel Marcus, May 09 2021

nonn

6 is the minimum possible value, and A176775(3,4,5) gives this minimum.

Conjecture: there are no other Pythagorean triples that give this minimum. In other words, it is the only triple with 3 A090467 terms.

Michel Marcus, Table of n, a(n) for n = 1..12471 (hypotenuses <= 10000).

Cf. A176774, A176775, A342491.

Cf. A090466, A090467.

tp(n) = my(k=3); while( !ispolygonal(n,k), k++); k; \\ A176774
itp(n) = my(m=tp(n)); (m-4+sqrtint((m-4)^2+8*(m-2)*n)) / (2*m-4); \\ A176775
f(v) = vecsum(apply(itp, v));
list(lim) = {my(v=List(), m2, s2, h2, h); for(middle=4, lim-1, m2=middle^2; for(small=1, middle, s2=small^2; if(issquare(h2=m2+s2, &h), if(h>lim, break); listput(v, [h, middle, small]);););); v = vecsort(Vec(v)); apply(f, v);}

A176774 Smallest polygonality of n = smallest integer m>=3 such that n is m-gonal number.

Original entry on oeis.org

3, 4, 5, 3, 7, 8, 4, 3, 11, 5, 13, 14, 3, 4, 17, 7, 19, 20, 3, 5, 23, 9, 4, 26, 10, 3, 29, 11, 31, 32, 12, 7, 5, 3, 37, 38, 14, 8, 41, 15, 43, 44, 3, 9, 47, 17, 4, 50, 5, 10, 53, 19, 3, 56, 20, 11, 59, 21, 61, 62, 22, 4, 8, 3, 67, 68, 24, 5, 71, 25, 73, 74, 9, 14, 77, 3, 79, 80, 4, 15, 83

Offset: 3

Max Alekseyev, Apr 25 2010

nonn

A176775(n) gives the index of n as a(n)-gonal number.
Since n is the second n-gonal number, a(n) <= n.
Furthermore, a(n)=n iff A176775(n)=2.

			a(12) = 5 since 12 is a pentagonal number, but not a square or triangular number. - _Michael B. Porter_, Jul 16 2016

Michel Marcus, Table of n, a(n) for n = 3..10000
Eric W. Weisstein, Polygonal Number. MathWorld.

Cf. A090466, A090467.

a[n_] := (m = 3; While[Reduce[k >= 1 && n == k (k (m - 2) - m + 4)/2, k, Integers] == False, m++]; m); Table[a[n], {n, 3, 100}] (* Jean-François Alcover, Sep 04 2016 *)

a(n) = {k=3; while (! ispolygonal(n, k), k++); k;} \\ Michel Marcus, Mar 25 2015

from _future_ import division
from gmpy2 import isqrt
def A176774(n):
    k = (isqrt(8*n+1)-1)//2
    while k >= 2:
        a, b = divmod(2*(k*(k-2)+n),k*(k-1))
        if not b:
            return a
        k -= 1 # Chai Wah Wu, Jul 28 2016

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

Original entry on oeis.org

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

A177028 Irregular table: row n contains values k (in descending order) for which n is a k-gonal number.

Original entry on oeis.org

3, 4, 5, 6, 3, 7, 8, 9, 4, 10, 3, 11, 12, 5, 13, 14, 15, 6, 3, 16, 4, 17, 18, 7, 19, 20, 21, 8, 3, 22, 5, 23, 24, 9, 25, 4, 26, 27, 10, 28, 6, 3, 29, 30, 11, 31, 32, 33, 12, 34, 7, 35, 5, 36, 13, 4, 3, 37, 38, 39, 14, 40, 8, 41, 42, 15

Offset: 3

Vladimir Shevelev, May 01 2010

Every row begins with n and contains all values of k for which n is a k-gonal number.

The cardinality of row n is A177025(n). In particular, if n is prime, then row n contains only n.

			The table starts with row n=3 as:
3;
4;
5;
6, 3;
7;
8;
9, 4;
10, 3;
11;
12, 5;
13;
14;
15, 6, 3;
16, 4;
17;
18, 7;
19;
20;
Before n=37, we have row n=36: {36, 13, 4, 3}. Thus 36 is k-gonal for k=3, 4, 13 and 36.

T. D. Noe, Rows n = 3..1000, flattened

Cf. A063778, A090467, A139600, A177025, A176948, A176774, A176775.

P := proc(n,k) n/2*((k-2)*n-k+4) ;end proc:
A177028 := proc(n) local k ,j,r,kg ; r := {} ; for k from n to 3 by -1 do for j from 1 do kg := P(j,k) ; if kg = n then r := r union {k} ;elif kg > n then break ; end if; end do; end do: sort(convert(r,list),`>`) ; end proc:
for n from 3 to 20 do print(A177028(n)) ; end do: # R. J. Mathar, Apr 17 2011

nn = 100; t = Table[{}, {nn}]; Do[n = 2; While[p = n*(4 - 2*n - r + n*r)/2; p <= nn, AppendTo[t[[p]], r]; n++], {r, 3, nn}]; Flatten[Reverse /@ t] (* T. D. Noe, Apr 18 2011 *)

row(n) = my(list = List()); for (k=3, n, if (ispolygonal(n, k), listput(list, k))); Vecrev(list); \\ Michel Marcus, Mar 19 2021

row(n)=my(v=List());fordiv(2*n,k, if(k<2,next); if(k==n, break); my(s=(2*n/k-4+2*k)/(k-1)); if(denominator(s)==1, listput(v,s))); Vec(v) \\ Charles R Greathouse IV, Mar 19 2021

A090467 Numbers which are not regular figurative or polygonal numbers of order greater than 2. That is, numbers not of the form 1 + km(m-1)/2 - (m-1)^2 where k > 2 and m > 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 11, 13, 14, 17, 19, 20, 23, 26, 29, 31, 32, 37, 38, 41, 43, 44, 47, 50, 53, 56, 59, 61, 62, 67, 68, 71, 73, 74, 77, 79, 80, 83, 86, 89, 97, 98, 101, 103, 104, 107, 109, 110, 113, 116, 119, 122, 127, 128, 131, 134, 137, 139, 140, 143, 146, 149, 151, 152

Offset: 1

Robert G. Wilson v, Dec 01 2003

The m-th k-gonal number is 1 + k*m*(m-1)/2 - (m-1)^2 = A057145(k,m).

Numbers that are strictly trivially polygonal: numbers m that are only 2-gonal and m-gonal. - Daniel Mondot, Jun 13 2024

			3 is a triangular number, but is not a k-gonal number for any other k, so 3 is a term.
6 is both a triangular number and a hexagonal number, so 6 is not a term.

Albert H. Beiler, Recreations In The Theory Of Numbers, The Queen Of Mathematics Entertains, Dover, NY, 1964, pp. 185-199.

Michel Marcus, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Eric Weisstein's World of Mathematics, Figurate Number.
Index to sequences related to polygonal numbers

Complement is A090466.

Cf. A057145, A176774, A176775.

Complement[ Table[i, {i, 300}], Take[ Union[ Flatten[ Table[1 + k*n(n - 1)/2 - (n - 1)^2, {n, 3, 40}, {k, 3, 300}]]], 300]]

isok(n) = (n < 3) || (vecsum(vector(n-2, k, k+=2; ispolygonal(n, k))) == 1); \\ Michel Marcus, May 01 2016

An integer n >= 3 is in this sequence iff A176774(n) = n (or, equivalently, A176775(n) = 2). - Max Alekseyev, Apr 24 2018

Verified by Don Reble, Mar 12 2006

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

cp's OEIS Frontend

A344083 a(n) = f(x)+f(y)+f(z), where (x,y,h) is the n-th Pythagorean triple listed in (A046083, A046084, A009000), and f(m)=A176775(m) is the index of m as k-gonal number for the smallest possible k.

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A176774 Smallest polygonality of n = smallest integer m>=3 such that n is m-gonal number.

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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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A177028 Irregular table: row n contains values k (in descending order) for which n is a k-gonal number.

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A090467 Numbers which are not regular figurative or polygonal numbers of order greater than 2. That is, numbers not of the form 1 + k*m*(m-1)/2 - (m-1)^2 where k > 2 and m > 2.

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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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A090467 Numbers which are not regular figurative or polygonal numbers of order greater than 2. That is, numbers not of the form 1 + km(m-1)/2 - (m-1)^2 where k > 2 and m > 2.