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

1,1

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

Formula

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

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 *)

Prog

(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

Crossrefs

Cf. A057145, A001248. Complement is A090467.

Sequence A090428 (excluding 1) is a subsequence of this sequence. - T. D. Noe, Jun 14 2012

Sequence in context: A053869 A085275 A177201 * A090428 A039725 A262362

Adjacent sequences: A090463 A090464 A090465 * A090467 A090468 A090469

Keyword

easy,nonn

Author

Robert G. Wilson v, Dec 01 2003

Extensions

Verified by Don Reble, Mar 12 2006

Status

approved