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 1-6 of 6 results.

A062858 In the triangular multiplication table of size A062859(n), the smallest number which appears n times.

Original entry on oeis.org

1, 4, 12, 36, 60, 120, 120, 360, 360, 360, 840, 840, 2520, 2520, 2520, 2520, 2520, 5040, 5040, 5040, 5040, 5040, 5040, 10080, 10080, 27720, 27720, 27720, 27720, 55440, 55440, 55440, 55440, 55440, 55440, 55440, 55440, 55440, 55440, 55440
Offset: 1

Views

Author

Ron Lalonde (ronronronlalonde(AT)hotmail.com), Jun 25 2001

Keywords

Comments

Smallest number to appear n times in any m X m multiplication table (excluding the numbers below the diagonals).

Examples

			a(4)=36 because 36 is the smallest number to appear 4 times in the triangular multiplication table of size A062859(4)=18.

Crossrefs

Cf. A027424, A062851, A062854, A062855, A062856, A062857, A062859, A000217.

Programs

  • Python
    from itertools import count
from collections import Counter
def A062858(n):
    c = Counter()
    for m in count(1):
        for i in range(1,m+1):
            ij = i*m
            c[ij] += 1
            if c[ij]>=n:
                return ij # Chai Wah Wu, Oct 16 2023

Extensions

More terms from Don Reble, Nov 08 2001

A062854 First differences of A027424.

Original entry on oeis.org

1, 2, 3, 3, 5, 4, 7, 5, 6, 6, 11, 6, 13, 8, 9, 8, 17, 9, 19, 10, 12, 12, 23, 10, 16, 14, 15, 13, 29, 12, 31, 15, 18, 18, 20, 13, 37, 20, 21, 16, 41, 17, 43, 20, 21, 24, 47, 17, 31, 22, 27, 23, 53, 22, 31, 22, 30, 30, 59, 19, 61, 32, 28, 26, 36, 26, 67, 30, 36, 26, 71, 23, 73, 38
Offset: 1

Views

Author

Ron Lalonde (ronronronlalonde(AT)hotmail.com), Jun 25 2001

Keywords

Comments

For prime p, a(p) = p. - Ralf Stephan, Jun 02 2005
a(n) is the number of times n appears in A033677. - Franklin T. Adams-Watters, Nov 18 2005
Conjecture: a(n) > n/log(n) for n > 2. - Thomas Ordowski, Jan 28 2017
a(n) is the number of integers 1<=i<=n such that all divisors of i*n are either <=i or >=n. - Chai Wah Wu, Oct 13 2023

Examples

			a(4)=3 because there are 9 unique products in the 4 X 4 multiplication table (1 2 3 4 6 8 9 12 16), which is 3 more than the 6 unique products in the 3 X 3 multiplication table (1 2 3 4 6 9).

Links

Crossrefs

Cf. A027421, A062854, A062855, A062856, A062857, A062859.

Programs

  • Maple
    A062854 := proc(n)
    A027424(n)-A027424(n-1) ;
end proc:
seq(A062854(n),n=1..40) ; # R. J. Mathar, Oct 02 2020
  • Mathematica
    Prepend[Differences@ #, First@ #] &@ Module[{ u = {}}, Table[Length[u = Union[u, n Range@ n]], {n, 100}]] (* Michael De Vlieger, Jan 30 2017 *)
  • PARI
    b(n) = #setbinop((x, y)->x*y, vector(n, i, i); );
a(n) = b(n) - b(n-1); \\ Michel Marcus, Jan 28 2017
  • Python
    from itertools import takewhile
from sympy import divisors
def A062854(n): return sum(1 for i in range(1,n+1) if all(d<=i for d in takewhile(lambda d:dChai Wah Wu, Oct 13 2023

Extensions

More terms from Ralf Stephan, Jun 02 2005

A062851 Number of k such that 1 < k < n X n and k not of the form ij for 1 <= {i, j} <= n.

Original entry on oeis.org

0, 1, 3, 7, 11, 18, 24, 34, 45, 58, 68, 85, 97, 116, 136, 159, 175, 201, 219, 248, 277, 308, 330, 367, 400, 437, 475, 517, 545, 592, 622, 670, 717, 766, 815, 873, 909, 964, 1020, 1083, 1123, 1189, 1231, 1298, 1366, 1433, 1479, 1557, 1623, 1700, 1774, 1854
Offset: 1

Views

Author

Ron A. Lalonde (ronronronlalonde(AT)hotmail.com), Jun 25 2001

Keywords

Comments

Smallest k for given n is given by A007918, largest by A005563 (except for some initial terms).

Examples

			a(4)=7 because there are 9 unique products in the 4 X 4 multiplication table (1 2 3 4 6 8 9 12 16), which excludes 7 non-product integers within the range 1 to 16 (5 7 10 11 13 14 15).

Crossrefs

a(n) = n^2 - A027424.
Cf. A027424, A062854, A062855, A062856, A062857, A062859.

Programs

  • Python
    def A062851(n): return n**2-len({i*j for i in range(1,n+1) for j in range(1,i+1)}) # Chai Wah Wu, Oct 13 2023

A062855 First differences of A062851.

Original entry on oeis.org

0, 1, 2, 4, 4, 7, 6, 10, 11, 13, 10, 17, 12, 19, 20, 23, 16, 26, 18, 29, 29, 31, 22, 37, 33, 37, 38, 42, 28, 47, 30, 48, 47, 49, 49, 58, 36, 55, 56, 63, 40, 66, 42, 67, 68, 67, 46, 78, 66, 77, 74, 80, 52, 85, 78, 89, 83
Offset: 1

Views

Author

Ron A. Lalonde (ronronronlalonde(AT)hotmail.com), Jun 25 2001

Keywords

Examples

			a(4)=4 because there are 7 integer non-products for the 4 X 4 multiplication table (5 7 10 11 13 14 15), which is 4 more than the 3 non-products for the 3 X 3 multiplication table (5 7 8).

Crossrefs

Cf. A027424, A062851, A062854, A062856, A062857, A062858, A062859.

Programs

  • Python
    from itertools import takewhile
from sympy import divisors
def A062855(n): return (n<<1)-1-sum(1 for i in range(1,n+1) if all(d<=i for d in takewhile(lambda d:dChai Wah Wu, Oct 13 2023

Formula

a(n) = 2*n-1-A062854(n). - Chai Wah Wu, Oct 13 2023

A062856 In the square multiplication table of size A062857(n), the smallest number which appears n times.

Original entry on oeis.org

1, 2, 4, 6, 12, 12, 36, 60, 60, 60, 120, 120, 120, 120, 360, 360, 360, 360, 360, 360, 840, 840, 840, 840, 2520, 2520, 2520, 2520, 2520, 2520, 2520, 2520, 2520, 2520, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 10080, 10080
Offset: 1

Views

Author

Ron Lalonde (ronronronlalonde(AT)hotmail.com), Jun 25 2001

Keywords

Comments

Smallest number to appear n times in any m X m multiplication table.

Examples

			a(7)=36 because 36 is the first product to appear in the m X m multiplication tables 7 times as m increases from 1 to infinity.

Crossrefs

Cf. A027424, A062851, A062854, A062855, A062857, A062858, A062859.

Programs

  • Mathematica
    b[1] = {1, 1}; b[n_] := b[n] = For[m = b[n-1][[1]], True, m++, T = Table[i j, {i, m}, {j, m}] // Flatten // Tally; sel = SelectFirst[T, #[[2]] >= n&]; If[sel != {}, Print[n, " ", m, " ", sel[[1]]]; Return[{m, sel[[1]]}] ]];
a[n_] := b[n][[2]];
Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Mar 25 2019 *)
  • Python
    from itertools import count
from collections import Counter
def A062856(n):
    if n == 1: return 1
    c = Counter()
    for m in count(1):
        for i in range(1,m):
            ij = i*m
            c[ij] += 2
            if c[ij]>=n:
                return ij
        c[m*m] = 1 # Chai Wah Wu, Oct 16 2023

Extensions

More terms from Don Reble, Nov 08 2001

A062857 Size of smallest square multiplication table which contains some number at least n times.

Original entry on oeis.org

1, 2, 4, 6, 12, 12, 18, 20, 30, 30, 40, 40, 60, 60, 72, 72, 90, 90, 120, 120, 140, 140, 168, 168, 180, 180, 210, 210, 252, 252, 280, 280, 315, 315, 336, 336, 360, 360, 420, 420, 504, 504, 560, 560, 630, 630, 672, 672, 720, 720, 792, 792, 840, 840, 924, 924, 990
Offset: 1

Views

Author

Ron Lalonde (ronronronlalonde(AT)hotmail.com), Jun 25 2001

Keywords

Comments

a(n) is the least number m such that there exists k with 1 <= k <= m^2 such that k has at least n divisors t with k/m <= t <= m. - Robert Israel, Jan 30 2017

Examples

			a(7)=18 because the 18 X 18 multiplication table is the smallest to contain a product of frequency 7 (namely the number A062856(7)=36).

Crossrefs

The least such number is A062856(n).
Cf. A027424, A062851, A062854, A062855, A062856, A062858, A062859.

Programs

  • MATLAB
    N = 1000; % to get all terms with a(n) <= N
M = sparse(1,N^2);
A(1) = 1;
imax = 1;
for k = 2:N
  M(k*[1:k-1]) = M(k*[1:k-1])+2;
  M(k^2) = 1;
  newimax = max(M);
  A(imax+1:newimax) = k;
  imax = newimax;
end
A  % Robert Israel, Jan 30 2017
  • Mathematica
    a[1] = 1; a[n_] := a[n] = For[m = a[n-1], True, m++, T = Table[i j, {i, m}, {j, m}] // Flatten // Tally; sel = SelectFirst[T, #[[2]] >= n&]; If[sel != {}, Print[n, " ", m, " ", sel[[1]]]; Return[m]]];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Mar 25 2019 *)
  • Python
    from itertools import count
from collections import Counter
def A062857(n):
    if n == 1: return 1
    c = Counter()
    for m in count(1):
        for i in range(1,m):
            ij = i*m
            c[ij] += 2
            if c[ij]>=n:
                return m
        c[m*m] = 1 # Chai Wah Wu, Oct 16 2023

Extensions

More terms from Don Reble, Nov 08 2001
Name clarified by Robert Israel, Jan 30 2017
Showing 1-6 of 6 results.

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