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.

A342806 Integers m such that A342805(m) = m+3.

Original entry on oeis.org

15, 21, 78, 300, 528, 903, 990, 1830, 2628, 3240, 3570, 4278, 5253, 5460, 7503, 8778, 9870, 13203, 13530, 16653, 18528, 20100, 22578, 24753, 25200, 29403, 31878, 37128, 39903, 45753, 48828, 55278, 64980, 65703, 72390, 73920, 81003, 88410, 98790, 106953, 107880
Offset: 1

Views

Author

Michel Marcus, Mar 22 2021

Keywords

Examples

			For m=15, A177028 row is (15, 6, 3); 6*3 = 18 and 18-3 = 15, so 15 is a term.

Crossrefs

Cf. A177028, A342805.
Subsequence of A090466.

Programs

  • PARI
    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
f(n) = vecprod(row(n))/n; \\ A342805
isok(m) = f(m) == m+3;

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

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

Views

Author

Robert G. Wilson v, Jun 22 2024

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    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]
Showing 1-3 of 3 results.

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