cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Robert G. Wilson v, Dec 01 2003

Keywords

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

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

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

Views

Author

Vladimir Shevelev, May 01 2010

Keywords

Comments

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]

References

  • J. J. Tattersall, Elementary Number Theory in Nine chapters, 2nd ed (2005), Cambridge Univ. Press, page 22 Problem 26, citing Wertheim (1897)

Links

Crossrefs

Cf. A129654, A139601, A090428, A176949, A176948, A176774, A176744, A176747, A176775, A175873, A176874.

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    a(n) = sum(i=3, n, ispolygonal(n, i)); \\ Michel Marcus, Jul 08 2014
  • Python
    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

Formula

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

Extensions

Extended by R. J. Mathar, Aug 15 2010

A279831 Numbers that are centered k-gonal numbers for two or more values of k.

Original entry on oeis.org

1, 7, 13, 16, 19, 22, 25, 31, 37, 43, 46, 49, 55, 61, 64, 67, 73, 76, 79, 85, 91, 97, 103, 106, 109, 111, 115, 121, 127, 133, 136, 139, 141, 145, 148, 151, 154, 157, 163, 166, 169, 172, 175, 181, 187, 190, 191, 193, 196, 199, 205, 211, 217, 221, 223, 226, 229, 232, 235, 241, 247, 253, 256
Offset: 1

Views

Author

Daniel Sterman, Dec 20 2016

Keywords

Comments

Numbers satisfying 1 + n*m*(m+1)/2 for two or more values of (n,m), where n>=0 m>1.
Numbers in this sequence appear in A101321 at least three times (because the second column contains every positive integer).

Examples

			19 is in the sequence because 19 is both a centered triangular number and a centered hexagonal number.

Links

Crossrefs

Cf. A090428 (rough equivalent for polygonal numbers).
Cf. A101321 (table of all centered polygonal numbers).
Cf. A275340 (list of nontrivial centered polygonal numbers).
Showing 1-3 of 3 results.

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