A342772 -id:A342772 - cp's OEIS frontend

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

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

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;

A373921 The last entry in the difference table for {the n-th row of A177028 arranged in increasing order}.

Original entry on oeis.org

3, 4, 5, 3, 7, 8, 5, 7, 11, 7, 13, 14, 6, 12, 17, 11, 19, 20, 8, 17, 23, 15, 21, 26, 17, 19, 29, 19, 31, 32, 21, 27, 30, 6, 37, 38, 25, 32, 41, 27, 43, 44, 12, 37, 47, 31, 45, 50, 20, 42, 53, 35, 44, 56, 37, 47, 59, 39, 61, 62, 41, 44, 57, 12, 67, 68, 45, 49, 71, 47, 73, 74, 32

Offset: 3

Robert G. Wilson v, Jun 22 2024

Inspired by A342772 and A187202.
The n-th row of A177028 are all integers k for which n is a k-gonal number.
As an example: row 10 of A177028 contain 3 and 10, because 10 is a 10-gonal number but also a triangular number.
-3n/2 < a(n) <= n.
a(n) = n if n is an odd prime (A065091), an odd composite number in A274967, or even numbers in A274968.
a(n) = 0: 231, tested up to 150000.
a(n) < 0: 441, 540, 561, 1089, 1128, 1296, 1521, 1701, 1716, 1881, 2016, 2211, 2541, 2556, 2601, ..., .
a(n) is negative less than 1% of the time.

			a(15) = 6, because the 15th row of A177028 is {3,6,15} -> {3,9} -> {6};
a(36) = 6, because the 36th row of A177028 is {3,4,13,36} -{1,9,23} - {8,14} -> {6};
a(225) = 37, because the 225th row of A177028 is {4,8,24,76,225} -> {4,16,52,149} -> {12,36,97} -> {24,61} -> {37};
a(561) = -82, because the 561st row of A177028 is {3,6,12,39,188,561} -> {3,6,27,149,373} -> {3,21,122,224} -> {18,101,102}, {83,1} -> {-82}; etc.

Robert G. Wilson v, Table of n, a(n) for n = 3..150000

Cf. A063778, A177025, A177028, A176774, A176948, A274967, A342550, A342805.

planeFigurateQ[n_, r_] := IntegerQ[((r -4) + Sqrt[(r -4)^2 + 8n (r -2)])/(2 (r -2))]; a[n_] := Block[{pg = Select[ Range[3, n], planeFigurateQ[n, #] &]}, Differences[pg, Length@ pg - 1][[1]]]; Array[a, 73, 3]

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

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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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A373921 The last entry in the difference table for {the n-th row of A177028 arranged in increasing order}.

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