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.

A278103 Irregular triangle T(n,k) := A278101(n,k) for k = 1..A278102(n), read by rows.

Original entry on oeis.org

1, 4, 2, 9, 8, 3, 16, 8, 25, 18, 12, 36, 32, 27, 20, 49, 32, 64, 50, 48, 45, 81, 72, 100, 98, 75, 121, 98, 144, 128, 108, 169, 162, 147, 125, 196, 162, 225, 200, 192, 180, 256, 242, 289, 288, 243, 324, 288, 361, 338, 300, 400, 392, 363, 320, 441, 392, 484, 450, 432
Offset: 1

Views

Author

Jason Kimberley, Nov 15 2016

Keywords

Comments

Each row is the longest strictly decreasing prefix of the corresponding row of A278101.

Examples

			The first 23 rows are:
1;
4, 2;
9, 8, 3;
16, 8;
25, 18, 12;
36, 32, 27, 20;
49, 32;
64, 50, 48, 45;
81, 72;
100, 98, 75;
121, 98;
144, 128, 108;
169, 162, 147, 125;
196, 162;
225, 200, 192, 180;
256, 242;
289, 288, 243;
324, 288;
361, 338, 300;
400, 392, 363, 320;
441, 392;
484, 450, 432, 405, 384;
529, 512, 507, 500, 486, 448;

References

  • R. B. Eggleton, J. S. Kimberley and J. A. MacDougall, Square-free rank of integers, submitted.

Links

Crossrefs

Cf. A005117, A278101, A278102, A278104.

Programs

  • Magma
    A277647:=func;
A278101_row:=funcA277647(n,k):k in[1..n^2]|IsSquarefree(k)]>;
A278103_row:=funcA278101_row(n) >;
&cat[A278103_row(n):n in[1..23]];
  • Mathematica
    Map[TakeWhile[FoldList[Function[s, Boole[s < 0] #2][#2 - #1] &, #], # > 0 &] &, #] &@ Map[DeleteCases[#, 0] &, Table[Boole[SquareFreeQ@ k] k Floor[n/Sqrt@ k]^2, {n, 23}, {k, n^2}] ] // Flatten (* Michael De Vlieger, Nov 24 2016 *)

Formula

T(n,k) = A278104(n,k) * A005117(k) where this triangle and A278104 both have row length sequence A278102.

A278104 Irregular triangle T(n,k) := A277648(n,k) for k = 1...A278102(n), read by rows.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 2, 5, 3, 2, 6, 4, 3, 2, 7, 4, 8, 5, 4, 3, 9, 6, 10, 7, 5, 11, 7, 12, 8, 6, 13, 9, 7, 5, 14, 9, 15, 10, 8, 6, 16, 11, 17, 12, 9, 18, 12, 19, 13, 10, 20, 14, 11, 8, 21, 14, 22, 15, 12, 9, 8, 23, 16, 13, 10, 9, 8, 24, 16, 13, 10, 9, 25, 17, 26, 18, 27, 19, 15, 28, 19
Offset: 1

Views

Author

Jason Kimberley, Nov 15 2016

Keywords

Comments

This triangle lists the "descending sequences across ranks" of Eggleton et al.

Examples

			The first 23 rows are:
1;
2,  1;
3,  2,  1;
4,  2;
5,  3,  2;
6,  4,  3,  2;
7,  4;
8,  5,  4,  3;
9,  6;
10,  7,  5;
11,  7;
12,  8,  6;
13,  9,  7,  5;
14,  9;
15, 10,  8,  6;
16, 11;
17, 12,  9;
18, 12;
19, 13, 10;
20, 14, 11,  8;
21, 14;
22, 15, 12,  9,  8;
23, 16, 13, 10,  9,  8;

References

  • R. B. Eggleton, J. S. Kimberley and J. A. MacDougall, Square-free rank of integers, submitted.

Links

Programs

  • Magma
    A277647:=func;
A277648_row:=funcA277647(n,k):k in[1..n^2]|IsSquarefree(k)]>;
A278101_row:=funcA277647(n,k)^2*k:k in[1..n^2]|IsSquarefree(k)]>;
A278104_row:=funcA277648_row(n)[1..j]:j in[1..#row-1]|row[j]le row[j+1]}select dec else[1]) where row is A278101_row(n) >;
&cat[A278104_row(n):n in[1..23]];
  • Mathematica
    Map[Last, #, {2}] &@ Map[TakeWhile[FoldList[Function[s, Boole[s < 0] {First@ #2, Last@ #2}][First@ #2 - First@ #1] &, #], Total@ # > 0 &] &, #] &@ Map[DeleteCases[#, {0, 0}] &, Table[Boole[SquareFreeQ@ k] {k #^2, #} &@ Floor[n/Sqrt@ k], {n, 32}, {k, n^2}] ] // Flatten (* Michael De Vlieger, Nov 24 2016 *)

A278106 a(n) is the index of the first occurrence of n in A278102.

Original entry on oeis.org

1, 2, 3, 6, 22, 23, 177, 129, 4954, 58976, 288436, 238773
Offset: 1

Views

Author

Jason Kimberley, Nov 26 2016

Keywords

References

  • R. B. Eggleton, J. S. Kimberley and J. A. MacDougall, Square-free rank of integers, submitted.

A278116 a(n) is the largest j such that A278115(n,k) strictly decreases for k=1..j.

Original entry on oeis.org

1, 2, 3, 3, 4, 3, 2, 2, 5, 4, 4, 2, 2, 3, 3, 5, 3, 2, 2, 4, 3, 3, 2, 2, 3, 4, 6, 6, 2, 3, 4, 3, 3, 2, 2, 3, 5, 4, 4, 2, 4, 3, 4, 3, 2, 2, 3, 4, 3, 2, 2, 4, 3, 4, 3, 2, 2, 3, 4, 3, 2, 2, 3, 3, 5, 3, 2, 2, 4, 5, 4, 2, 2, 3, 3, 4, 3, 2, 3, 4, 7, 5, 2, 2, 3, 4, 2, 2, 2, 3, 5, 5, 5, 2, 2, 3, 4, 3, 2, 2, 4, 5, 3, 3, 2
Offset: 1

Views

Author

Jason Kimberley, Feb 12 2017

Keywords

Links

Crossrefs

Cf. A278102.
This is the row length sequence for triangles A278117 and A278118.
A278119 lists first occurrences in this sequence.

Programs

  • Magma
    A:=func;
A278116:=funcA278115(n,P[j+1])}
  select j else #P) where P is PrimesUpTo(2*n^2)>;
[A278116(n):n in[1..103]];
  • Mathematica
    Map[1 + Length@ TakeWhile[Differences@ #, # < 0 &] &, #] &@ Table[# Floor[n Sqrt[2/#]]^2 &@ Prime@ k, {n, 105}, {k, PrimePi[2 n^2]}] (* Michael De Vlieger, Feb 17 2017 *)
  • Python
    def isqrt(n):
    if n < 0:
        raise ValueError('imaginary')
    if n == 0:
        return 0
    a, b = divmod(n.bit_length(),2)
    x = 2**(a+b)
    while True:
        y = (x + n//x)//2
        if y >= x:
            return x
        x = y;
def next_prime(n):
    for p in range(n+1,2*n+1):
        for i in range(2,isqrt(n)+1):
            if p % i == 0:
                break
        else:
            return p
    return None
def A278116(n):
    k = 0
    p = 2
    s2= (n**2)*p
    s = s2
    while True:
        s_= s
        k+= 1
        p = next_prime(p)
        s = (isqrt(s2//p)**2)*p
        if s > s_:
            break
    return k
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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