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

Author

Robert G. Wilson v, Dec 01 2003

Keywords

• easy
• nonn

Comments

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

References

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

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 1000 terms are from T. D. Noe)
• Eric Weisstein's World of Mathematics, Figurate Number
• Index to sequences related to polygonal numbers

Crossrefs

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.

Programs

Maple

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

Mathematica

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 *)

PARI

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

PARI

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

PARI

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

Formula

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

Extensions

Verified by Don Reble, Mar 12 2006