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 1-4 of 4 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

A364691 Pentagonal numbers which are the sum of the first k primes, for some k >= 0.

Original entry on oeis.org

0, 5, 13490, 3299391550, 22042432252064127, 2387505511919644051, 680588297594638712735
Offset: 1

Views

Author

Paolo Xausa, Aug 03 2023

Keywords

Examples

			5 is a term because it's both a pentagonal number and the sum of the first two primes (2 + 3).

Links

Crossrefs

Intersection of A000326 with A007504.
Cf. A066527, A061890, A364694, A364696, A366269.

Programs

  • Mathematica
    A364691list[kmax_]:=Module[{p=0},Join[{0},Table[If[IntegerQ[(Sqrt[24(p+=Prime[k])+1]+1)/6],p,Nothing],{k,kmax}]]];A364691list[25000] (* Paolo Xausa, Oct 06 2023 *)
  • PARI
    ispenta(n) = my(s); issquare(24*n+1,&s)&&s%6==5;
my(S=0); forprime (p=2, oo, S+=p; if (ispenta(S), print1(S,", "))) \\ Hugo Pfoertner, Aug 03 2023

Extensions

a(5)-a(7) from Hugo Pfoertner, Aug 04 2023

A364695 Positive integers k such that the sum of the first k primes is a polygonal number of order greater than 2 (A090466).

Original entry on oeis.org

3, 5, 7, 9, 10, 11, 13, 15, 16, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 39, 40, 42, 44, 46, 47, 49, 51, 52, 53, 54, 56, 57, 62, 68, 70, 72, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92, 97, 99, 103, 105, 106
Offset: 1

Views

Author

Paolo Xausa, Aug 03 2023

Keywords

Examples

			5 is a term because the sum of the first 5 primes (2 + 3 + 5 + 7 = 28) is a triangular number.
7 is a term because the sum of the first 7 primes (2 + 3 + 5 + 7 + 11 + 13 = 58) is an octagonal number.

Links

Crossrefs

Cf. A007504, A033997, A090466, A175133, A364693, A364694, A364696.

Programs

  • Mathematica
    A364693Q[n_]:=With[{d=Divisors[2n]},Catch[For[i=3,iJianing Song in A090466 *)
A364695list[kmax_]:=Flatten[Position[Map[A364693Q,Accumulate[Prime[Range[kmax]]]],True]];A364695list[100]
  • PARI
    isok(k) = my(s = sum(i=1, k, prime(i))); for (j=3, s-1, if (ispolygonal(s, j), return(1))); \\ Michel Marcus, Aug 03 2023

A366269 Hexagonal numbers which are the sum of the first k primes, for some k >= 0.

Original entry on oeis.org

0, 28, 54047322253, 14756071005948636, 600605016143706003, 41181981873797476176, 240580227206205322973571
Offset: 1

Views

Author

Paolo Xausa, Oct 06 2023

Keywords

Examples

			28 is a term because it's both a hexagonal number and the sum of the first five primes (2 + 3 + 5 + 7 + 11).

Links

Crossrefs

Intersection of A000384 with A007504.
Subsequence of A066527.
Cf. A061890, A364691, A364694, A366270 (corresponding k values).

Programs

  • Mathematica
    A366269list[kmax_]:=Module[{p=0},Join[{0},Table[If[IntegerQ[(Sqrt[8(p+=Prime[k])+1]+1)/4],p,Nothing],{k,kmax}]]];A366269list[10^5]

Formula

a(n) = A007504(A366270(n)).
Showing 1-4 of 4 results.

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