A090467 -id:A090467 - cp's OEIS frontend

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

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

A090466 Regular figurative or polygonal numbers of order greater than 2.

Original entry on oeis.org

6, 9, 10, 12, 15, 16, 18, 21, 22, 24, 25, 27, 28, 30, 33, 34, 35, 36, 39, 40, 42, 45, 46, 48, 49, 51, 52, 54, 55, 57, 58, 60, 63, 64, 65, 66, 69, 70, 72, 75, 76, 78, 81, 82, 84, 85, 87, 88, 90, 91, 92, 93, 94, 95, 96, 99, 100, 102, 105, 106, 108, 111, 112, 114, 115, 117, 118

Offset: 1

Robert G. Wilson v, Dec 01 2003

The sorted k-gonal numbers of order greater than 2. If one were to include either the rank 2 or the 2-gonal numbers, then every number would appear.
Number of terms less than or equal to 10^k for k = 1,2,3,...: 3, 57, 622, 6357, 63889, 639946, 6402325, 64032121, 640349979, 6403587409, 64036148166, 640362343980, ..., . - Robert G. Wilson v, May 29 2014
The n-th k-gonal number is 1 + k*n(n-1)/2 - (n-1)^2 = A057145(k,n).
For all squares (A001248) of primes p >= 5 at least one a(n) exists with p^2 = a(n) + 1. Thus the subset P_s(3) of rank 3 only is sufficient. Proof: For p >= 5, p^2 == 1 (mod {3,4,6,8,12,24}) and also P_s(3) + 1 = 3*s - 2 == 1 (mod 3). Thus the set {p^2} is a subset of {P_s(3) + 1}; Q.E.D. - Ralf Steiner, Jul 15 2018
For all primes p > 5, at least one polygonal number exists with P_s(k) + 1 = p when k = 3 or 4, dependent on p mod 6. - Ralf Steiner, Jul 16 2018
Numbers m such that r = (2*m/d - 2)/(d - 1) is an integer for some d, where 2 < d < m is a divisor of 2*m. If r is an integer, then m is the d-th (r+2)-gonal number. - Jianing Song, Mar 14 2021

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

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 1000 terms are from T. D. Noe)
Felix Ponomarenko and Vadim Ponomarenko, Polygonal Simplex Numbers, San Diego State Univ., 2025.
Eric Weisstein's World of Mathematics, Figurate Number
Index to sequences related to polygonal numbers

Cf. A057145, A001248, A177028 (A342772, A342805), A177201, A316676, A364693 (characteristic function).
Complement is A090467.
Sequence A090428 (excluding 1) is a subsequence of this sequence. - T. D. Noe, Jun 14 2012
Other subsequences: A324972 (squarefree terms), A324973, A342806, A364694.
Cf. also A275340.

isA090466 := proc(n)
    local nsearch,ksearch;
    for nsearch from 3 do
        if A057145(nsearch,3) > n then
            return false;
        end if;
        for ksearch from 3 do
            if A057145(nsearch,ksearch) = n then
                return true;
            elif A057145(nsearch,ksearch) > n then
                break;
            end if;
        end do:
    end do:
end proc:
for n from 1 to 1000 do
    if isA090466(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jul 28 2016

Take[Union[Flatten[Table[1+k*n (n-1)/2-(n-1)^2,{n,3,100},{k,3,40}]]],67] (* corrected by Ant King, Sep 19 2011 *)
mx = 150; n = k = 3; lst = {}; While[n < Floor[mx/3]+2, a = PolygonalNumber[n, k]; If[a < mx+1, AppendTo[ lst, a], (n++; k = 2)]; k++]; lst = Union@ lst (* Robert G. Wilson v, May 29 2014 and updated Jul 23 2018; PolygonalNumber requires version 10.4 or higher *)

list(lim)=my(v=List()); lim\=1; for(n=3,sqrtint(8*lim+1)\2, for(k=3,2*(lim-2*n+n^2)\n\(n-1), listput(v, 1+k*n*(n-1)/2-(n-1)^2))); Set(v); \\ Charles R Greathouse IV, Jan 19 2017

is(n)=for(s=3,n\3+1,ispolygonal(n,s)&&return(s)); \\ M. F. Hasler, Jan 19 2017

isA090466(m) = my(v=divisors(2*m)); for(i=3, #v, my(d=v[i]); if(d==m, return(0)); if((2*m/d - 2)%(d - 1)==0, return(1))); 0 \\ Jianing Song, Mar 14 2021

Integer k is in this sequence iff A176774(k) < k. - Max Alekseyev, Apr 24 2018

Verified by Don Reble, Mar 12 2006

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

A342772 Row sums of triangle A177028.

Original entry on oeis.org

3, 4, 5, 9, 7, 8, 13, 13, 11, 17, 13, 14, 24, 20, 17, 25, 19, 20, 32, 27, 23, 33, 29, 26, 37, 37, 29, 41, 31, 32, 45, 41, 40, 56, 37, 38, 53, 48, 41, 57, 43, 44, 70, 55, 47, 65, 53, 50, 74, 62, 53, 73, 65, 56, 77, 69, 59, 81, 61, 62, 85, 80, 73, 98, 67, 68, 93, 88, 71, 97

Offset: 3

Michel Marcus, Mar 21 2021

nonn

			6 is a triangular and hexagonal number, so a(6) = 3 + 6 = 9.

Michel Marcus, Table of n, a(n) for n = 3..10000

Cf. A090466, A090467, A177028.

a(n) = sum(k=3, n, if (ispolygonal(n, k), k));

a(n) >= n.
For m in A090466, a(m) > m.
For m in A090467, a(m) = m.

A342805 a(n) = (1/n)*(product of the terms of the n-th row of A177028).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 4, 3, 1, 5, 1, 1, 18, 4, 1, 7, 1, 1, 24, 5, 1, 9, 4, 1, 10, 18, 1, 11, 1, 1, 12, 7, 5, 156, 1, 1, 14, 8, 1, 15, 1, 1, 288, 9, 1, 17, 4, 1, 90, 10, 1, 19, 21, 1, 20, 11, 1, 21, 1, 1, 22, 48, 8, 414, 1, 1, 24, 65, 1, 25, 1, 1, 234, 14, 1, 81, 1, 1, 784, 15

Offset: 3

Michel Marcus, Mar 22 2021

nonn

			For n=3, A177028 row is (3), so a(3) = 3/3 = 1.
For n=6, A177028 row is (6, 3), so a(6) = 6*3/6 = 3.
For n=15, A177028 row is (15, 6, 3), so a(15) = 15*6*3/15 = 18.

Michel Marcus, Table of n, a(n) for n = 3..10000

Cf. A090466, A090467, A177028, A342772.

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); \\ A177028
a(n) = vecprod(row(n))/n;

For m in A090466, a(m) > 1.

For m in A090467, a(m) = 1.

A342550 For n>=3, a(n) is the sum of the indices of n seen as an m-gonal number.

Original entry on oeis.org

2, 2, 2, 5, 2, 2, 5, 6, 2, 5, 2, 2, 10, 6, 2, 5, 2, 2, 11, 6, 2, 5, 7, 2, 5, 13, 2, 5, 2, 2, 5, 6, 7, 19, 2, 2, 5, 6, 2, 5, 2, 2, 19, 6, 2, 5, 9, 2, 11, 6, 2, 5, 17, 2, 5, 6, 2, 5, 2, 2, 5, 14, 7, 22, 2, 2, 5, 13, 2, 5, 2, 2, 10, 6, 2, 17, 2, 2, 20, 6, 2, 5, 7, 2, 5

Offset: 3

Michel Marcus, Mar 27 2021

nonn

This sum is constituted of A177025(n) terms related to the n-row of A177028 triangle.
For m in A090467, a(m) = 2.
By definition, a(n) can never be equal to 3 or 4.
Up to 10^7, no n has been found with a(n) = 8, 12 or 18.

			15 is the 5th triangular, the 3rd hexagonal and the 2nd 15-gonal, so a(15) = 5+3+2 = 10.

Michel Marcus, Table of n, a(n) for n = 3..10000

Cf. A090467, A177025, A177028, A342772.

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))); Vecrev(v); \\ A177028
a(n) = my(v=row(n), s=0); for (k=1, #v, if ((v[k]>2) && ispolygonal(n, v[k], &i), s += i)); s;

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.

Original entry on oeis.org

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);}

cp's OEIS Frontend

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

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A090466 Regular figurative or polygonal numbers of order greater than 2.

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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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A342772 Row sums of triangle A177028.

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A342805 a(n) = (1/n)*(product of the terms of the n-th row of A177028).

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A342550 For n>=3, a(n) is the sum of the indices of n seen as an m-gonal number.

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