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.

A071562 Numbers n such that the sum of the middle divisors of n (A071090) is not zero.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 15, 16, 18, 20, 24, 25, 28, 30, 32, 35, 36, 40, 42, 45, 48, 49, 50, 54, 56, 60, 63, 64, 66, 70, 72, 77, 80, 81, 84, 88, 90, 91, 96, 98, 99, 100, 104, 108, 110, 112, 117, 120, 121, 126, 128, 130, 132, 135, 140, 143, 144, 150, 153, 154, 156, 160
Offset: 1

Views

Author

Robert G. Wilson v, May 30 2002

Keywords

Comments

Numbers n such that A067742(n) is nonzero.
Numbers of the form m*k with m <= k <= 2m. - Vladeta Jovovic, May 07 2005
Numbers occurring in A100345 (except 0). - Franklin T. Adams-Watters, Apr 04 2010
This sequence is closed under multiplication. If n = a*b with a <= b <= 2a, and m = c*d with c <= d <= 2c, then min(a*d,b*c)*max(a*d,b*c) is a factorization of m*n with the specified property. - Franklin T. Adams-Watters, Apr 07 2010
Also numbers n with the property that the number of parts in the symmetric representation of sigma(n) is odd. - Michel Marcus and Omar E. Pol, Apr 25 2014. (For a proof see the link in A071561.) - Hartmut F. W. Hoft, Sep 09 2015
Among these numbers, those with sigma(n) also odd are 1, 2, 4, 8, 9, 16, ..., that is, probably A028982 (squares and twice squares). - Michel Marcus, Jun 21 2014
Records in A244367. - Omar E. Pol, Jul 27 2014
Starting from a(5), the sequence is a subset of a sequence formed out of the prime factorization of A129912(m), m >= 5; see associated PARI code in Prog section. - Bill McEachen, Jan 25 2018
For numbers n = 2^m * q, m >= 0, q odd, and where r(n) = floor( (sqrt(8n+1) - 1)/2 ), the symmetric representation of sigma(n) has an odd number of parts precisely when there exists an odd divisor d of n satisfying d <= r(n) and d*2^(m+1) > r(n); see the link for a proof and see the associated Mathematica code. - Hartmut F. W. Hoft, Feb 12 2018
All hexagonal numbers A000384 > 0 are in the sequence. - Omar E. Pol, Aug 28 2018

Examples

			From _Hartmut F. W. Hoft_, Feb 12 2018: (Start)
63 = 3^2*7 is in the sequence since 7*2^1 > r(63) = 10.
80 = 2^4*5 is in the sequence since 1*2^5 > r(80) = 12. (End)

Links

Crossrefs

Cf. A067742.
The complement is A071561.
Cf. A000203, A028982, A071090, A100345, A175040, A176039 (primitive elements), A237270, A237271, A237593, A244367.

Programs

  • Mathematica
    f[n_] := Plus @@ Select[ Divisors[n], Sqrt[n/2] <= # < Sqrt[n*2] &]; Select[ Range[175], f[ # ] != 0 &]
(* Related to the symmetric representation of sigma *)
(* subsequence of odd parts of number k for m <= k <= n *)
(* Function a237270[] is defined in A237270 *)
(* Using Wilson's Mathematica program (see above) I verified the equality of both for numbers k <= 10000 *)
a071562[m_,n_]:=Select[Range[m,n],OddQ[Length[a237270[#]]]&]
a071562[1,160] (* data *)
(* Hartmut F. W. Hoft, Jun 23 2014 *)
(* implementation using the odd divisor property *)
evenExp[n_] := First[NestWhile[{#[[1]]+1, #[[2]]/2}&, {0, n}, EvenQ[Last[#]]&]]
oddSRQ[n_] := Module[{e=2^evenExp[n], Floor[(Sqrt[8n+1]-1)/2]}, Select[Divisors[n/e], #<=r&&2 e #>r&]!={}]
a071562D[m_, n_] := Select[Range[m, n], oddSRQ]
a071562D[1, 160] (* data *) (* Hartmut F. W. Hoft, Feb 12 2018 *)
  • PARI
    is(n)=fordiv(n,d, if(d^2>=n/2 && d^2<2*n, return(1))); 0 \\ Charles R Greathouse IV, Aug 01 2016
  • PARI
    is(n,f=factor(n))=my(t=(n+1)\2); fordiv(f,d, if(d^2>=t, return(d^2<2*n))); 0 \\ Charles R Greathouse IV, Jan 22 2018
  • PARI
    list(lim)=my(v=List(),t); forfactored(n=1,lim\1, t=(n[1]+1)\2; fordiv(n[2],d, if(d^2>=t, if(d^2<2*n[1], listput(v,n[1])); break))); Vec(v) \\ Charles R Greathouse IV, Jan 22 2018
  • PARI
    /* functional code associated to the A129912 comment above */ for(j5=5, length(A129912), a=Mat(); a=factor(A129912[j5]); sum2=0; for(i5=1, length(a[,2]), sum2=sum2+a[i5,2]); listput(final,length(a[,1])*sum2)); v=Set(final); \\ Bill McEachen, Jan 25 2018

A281007 Number of middle divisors of the n-th number that has middle divisors.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 3, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 4, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 4, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 4, 1, 2, 4, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2
Offset: 1

Views

Author

Omar E. Pol, Feb 11 2017

Keywords

Comments

Conjecture 1: also widths of the successive terraces that we can find descending by the main diagonal of the pyramid described in A245092. Hence, bisection of A281012.
Conjecture 2: also number of central subparts in the symmetric representation of sigma of the numbers j that have the property that the number of parts in the symmetric representation of sigma(j) is odd.
Conjecture 3: Partial sums give A282131.

Crossrefs

Positive terms in A067742.
Cf. A071562, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A240542, A244050, A244367, A244580, A245092, A262626, A279387, A280919, A281012, A282131.

Programs

  • Mathematica
    DeleteCases[#, 0] &@ Table[Count[Divisors@ n, d_ /; Sqrt[n/2] <= d < Sqrt[2 n]], {n, 300}] (* Michael De Vlieger, Feb 12 2017 *)

Formula

a(n) = A067742(A071562(n)).

A259179 Number of Dyck paths described in A237593 that contain the point (n,n) in the diagram of the symmetric representation of sigma.

Original entry on oeis.org

1, 2, 2, 0, 2, 1, 3, 0, 3, 0, 1, 2, 2, 0, 4, 0, 1, 3, 0, 2, 0, 2, 3, 0, 1, 4, 0, 2, 0, 3, 0, 3, 0, 1, 1, 4, 0, 2, 0, 4, 0, 3, 0, 1, 2, 0, 4, 0, 2, 0, 0, 5, 0, 3, 0, 1, 3, 0, 4, 0, 2, 0, 1, 0, 5, 0, 2, 1, 0, 1, 4, 0, 4, 0, 2, 0, 2, 0, 5, 0, 3, 0, 0, 0, 1, 5, 0, 2, 2, 0, 2, 0, 3, 0, 5, 0, 3, 0, 1, 0, 0, 6
Offset: 1

Views

Author

Omar E. Pol, Aug 11 2015

Keywords

Comments

Since the diagram of the symmetric representation of sigma is also the top view of the stepped pyramid described in A245092, and the diagram is also the top view of the staircase described in A244580, so we have that a(n) is also the height difference (or length of the vertical line segment) at the point (n,n) in the main diagonal of the mentioned structures.
a(n) is the number of occurrences of n in A240542. - Omar E. Pol, Dec 09 2016
Nonzero terms give A280919, the first differences of A071562. - Omar E. Pol, Apr 17 2018
Also first differences of A244367. Where records occur gives A279286. - Omar E. Pol, Apr 20 2020

Examples

			Illustration of initial terms:
--------------------------------------------------------
                           Diagram with 15 Dyck paths
n   A000203(n)  a(n)         to evaluate a(1)..a(10)
--------------------------------------------------------
.                         _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
1        1        1      |_| | | | | | | | | | | | | | |
2        3        2      |_ _|_| | | | | | | | | | | | |
3        4        2      |_ _|  _|_| | | | | | | | | | |
4        7        0      |_ _ _|    _|_| | | | | | | | |
5        6        2      |_ _ _|  _|  _ _|_| | | | | | |
6       12        1      |_ _ _ _|  _| |  _ _|_| | | | |
7        8        3      |_ _ _ _| |_ _|_|    _ _|_| | |
8       15        0      |_ _ _ _ _|  _|     |  _ _ _|_|
9       13        3      |_ _ _ _ _| |      _|_| |
10      18        0      |_ _ _ _ _ _|  _ _|    _|
.                        |_ _ _ _ _ _| |  _|  _|
.                        |_ _ _ _ _ _ _| |_ _|
.                        |_ _ _ _ _ _ _| |
.                        |_ _ _ _ _ _ _ _|
.                        |_ _ _ _ _ _ _ _|
.
For n = 3 there are two Dyck paths that contain the point (3,3) so a(3) = 2.
For n = 4 there are no Dyck paths that contain the point (4,4) so a(4) = 0.

Crossrefs

Cf. A000203, A024916, A071562, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A240542, A244050, A244367, A244580, A245092, A249351, A256533, A259179, A262626, A279286, A280919.

Programs

  • Mathematica
    a240542[n_] := Sum[(-1)^(k+1)Ceiling[(n+1)/k - (k+1)/2], {k, 1, Floor[(Sqrt[8n+1]-1)/2]}]
a259179[n_] := Module[{t=Table[0, n], k=1, d=1}, While[d<=n, t[[d]]+=1; d=a240542[++k]]; t] (* a(1..n) *)
a259179[102] (* Hartmut F. W. Hoft, Aug 06 2020 *)

Extensions

More terms from Omar E. Pol, Dec 09 2016

A244580 Square array read by antidiagonals related to the symmetric representation of sigma.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 3, 3, 4, 5, 4, 4, 4, 5, 6, 5, 4, 4, 5, 6, 7, 6, 5, 6, 5, 6, 7, 8, 7, 6, 6, 6, 6, 7, 8, 9, 8, 7, 6, 6, 6, 7, 8, 9, 10, 9, 8, 7, 8, 8, 7, 8, 9, 10, 11, 10, 9, 8, 8, 8, 8, 8, 9, 10, 11, 12, 11, 10, 9, 8, 9, 9, 8, 9, 10, 11, 12
Offset: 1

Views

Author

Omar E. Pol, Jul 04 2014

Keywords

Comments

The number of parts k in the square array is equal to A000203(k) hence the sum of parts k is equal to A064987(k).
The structure has a three-dimensional representation using polycubes. T(n,k) is the height of a column. The total area in the horizontal level z gives A000203(z).
The main diagonal gives A244367.

Examples

			.                         _ _ _ _ _ _ _ _ _
1,2,3,4,5,6,7,8,9...     |_| | | | | | | | |
2,2,3,4,5,6,7,8,9...     |_ _|_| | | | | | |
3,3,4,4,5,6,7,8,9...     |_ _|  _|_| | | | |
4,4,4,6,6,6,7,8,9...     |_ _ _|    _|_| | |
5,5,5,6,6,8,8,8,9...     |_ _ _|  _|  _ _|_|
6,6,6,6,8,8,9...         |_ _ _ _|  _| |
7,7,7,7,8,9,9...         |_ _ _ _| |_ _|
8,8,8,8,8...             |_ _ _ _ _|
9,9,9,9,9...             |_ _ _ _ _|
.

Crossrefs

Cf. A000203, A064987, A143128, A185283, A196020, A236104, A235791, A237270, A237271, A237591, A237593, A071562, A244367.
Showing 1-4 of 4 results.

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