A058084 -id:A058084 - cp's OEIS frontend

A357327 a(n) is the unique nonnegative integer k <= A058084(n)/2 such that binomial(A058084(n),k) = n.

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1

Offset: 1

Pontus von Brömssen, Sep 24 2022

nonn

			The first occurrence of 6 in Pascal's triangle is in row 4 = A058084(6) and binomial(4,2) = 6, so a(6) = 2.

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A007318, A058084.

a(n) = my(k=0); while (!vecsearch(vector((k+2)\2, i, binomial(k, i-1)), n), k++); select(x->(x==n), vector((k+2)\2, i, binomial(k, i-1)), 1)[1] - 1; \\ Michel Marcus, Sep 24 2022

A007318(A058084(n),a(n)) = n.

A006987 Binomial coefficients: C(n,k), 2 <= k <= n-2, sorted, duplicates removed.

Original entry on oeis.org

6, 10, 15, 20, 21, 28, 35, 36, 45, 55, 56, 66, 70, 78, 84, 91, 105, 120, 126, 136, 153, 165, 171, 190, 210, 220, 231, 252, 253, 276, 286, 300, 325, 330, 351, 364, 378, 406, 435, 455, 462, 465, 495, 496, 528, 560, 561, 595, 630, 666, 680, 703, 715, 741, 780, 792, 816, 820

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

nonn

Complement of A137905; a(n) > A058084(a(n)). - Reinhard Zumkeller, Mar 20 2009
Or numbers l which, for the first time, appear in m-th row of the Pascal triangle for m < l. - Vladimir Shevelev, Apr 28 2010
Appears to be the set of simplex numbers of order > 2 and dimension > 1. - Dylan Hamilton, Nov 05 2010
This is correct (assuming the notational choice of giving the first n-simplicial number index 1), as the n-th diagonal or antidiagonal of Pascal's triangle gives the n-simplicial numbers. - Thomas Anton, Dec 04 2018

			Pascal's triangle (A007318) with the outer two layers removed:
             6
          10  10
        15  20  15
      21  35  35  21
    28  56  70  56  28
  36  84 126 126  84  36
  ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
R. G. Wilson v, Letter to N. J. A. Sloane, Nov. 1988
R. G. Wilson v, Letter to N. J. A. Sloane, Jan. 1989

Cf. A002808, A007318.

Take[ Union[ Flatten[ Table[ Binomial[n, k], {n, 2, 45}, {k, 2, n - 2}]]], 58] (* Robert G. Wilson v, May 25 2004 *)

list(lim)=my(v=List(), t); for(n=4, sqrtint(2*lim)+1, for(k=2, n\2, t=binomial(n, k); if(t>lim, break, listput(v, t)))); vecsort(Vec(v), , 8) \\ Charles R Greathouse IV, Apr 03 2012

More terms from David W. Wilson

Spelling corrected by Jason G. Wurtzel, Aug 22 2010

A137905 Numbers that appear as binomial coefficients exactly twice.

Original entry on oeis.org

3, 4, 5, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 85, 86, 87, 88

Offset: 1

David Wasserman, Feb 21 2008

Complement of A006987; a(n) = A058084(a(n)). - Reinhard Zumkeller, Mar 20 2009

			7 is a member because 7 = binomial(7, 1) = binomial(7, 6) and no other binomial coefficient equals 7. [clarified by _Jonathan Sondow_, Jan 12 2018]

Cf. A006987, A058084, A062527, A098564, A098566.

isok(n) = (sum(i=0, n, sum(j=0, i, binomial(i,j)==n)) == 2) \\ Michel Marcus, Jun 16 2013

a(n) = A185024(n+1). - Elijah Beregovsky, May 14 2019

A349958 a(n) is the index of the first row in Pascal's triangle that contains a multiple of n.

Original entry on oeis.org

0, 2, 3, 4, 5, 4, 7, 8, 9, 5, 11, 9, 13, 8, 6, 16, 17, 9, 19, 6, 7, 11, 23, 10, 25, 13, 27, 8, 29, 10, 31, 32, 11, 17, 7, 9, 37, 19, 13, 10, 41, 9, 43, 12, 10, 23, 47, 16, 49, 25, 18, 13, 53, 27, 11, 8, 19, 29, 59, 10, 61, 32, 9, 64, 13, 11, 67, 17, 23, 8, 71, 12, 73, 37, 25

Offset: 1

Nathan M Epstein, Dec 06 2021

nonn

a(n) is the minimum j such that binomial(j,k) is divisible by n for some k in 0..j.

a(n) is at most equal to A058084(n), the least m such that binomial(m,k) = n for some k.

			In the table below, the k value shown is the minimum k such that n divides binomial(a(n), k).
.
   n  a(n)  k  C(a(n), k)
  --  ----  -  ----------
   1    0   0       1
   2    2   1       2
   3    3   1       3
   4    4   1       4
   5    5   1       5
   6    4   2       6
   7    7   1       7
   8    8   1       8
   9    9   1       9
  10    5   2      10
  11   11   1      11
  12    9   2      36
.
The table below shows the left half (and middle column) of rows j = 0..12 of Pascal's triangle; each number in parentheses there is the first term encountered in Pascal's triangle (read by rows from left to right) that is a multiple of some number n in 1..12, and the corresponding term of {a(n)} whose value is j appears in the column at the right.
E.g., the first multiple of 12 encountered in Pascal's triangle is binomial(9,2) = 36; it appears in row 9, so a(12) = 9, and the column at the right includes a(12) in row 9.
                                               | terms in a(1)..a(12)
   j | left half of row j of Pascal's triangle | that are equal to j
  ---+-----------------------------------------+---------------------
   0 |                                    (1)  |        a(1)  =  0
   1 |                                  1      |
   2 |                               1    (2)  |        a(2)  =  2
   3 |                            1    (3)     |        a(3)  =  3
   4 |                         1    (4)   (6)  |  a(4), a(6)  =  4
   5 |                      1    (5)  (10)     |  a(5), a(10) =  5
   6 |                   1     6    15    20   |
   7 |                1    (7)   21    35      |        a(7)  =  7
   8 |             1    (8)   28    56    70   |        a(8)  =  8
   9 |          1    (9)  (36)   84   126      |  a(9), a(12) =  9
  10 |       1    10    45   120   210   252   |
  11 |    1   (11)   55   165   330   462      |        a(11) = 11
  12 | 1    12    66   210   496   792   924   |

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A007318, A058084, A374959.

a[n_] := Module[{k = 0}, While[!AnyTrue[Binomial[k, Range[0, Floor[k/2]]], Divisible[#, n] &], k++]; k]; Array[a, 75] (* Amiram Eldar, Dec 07 2021 *)

a(n) = my(k=0); while (!#select(x->(x==1), apply(denominator, vector((k+2)\2, i, binomial(k, i-1))/n)), k++); k; \\ Michel Marcus, Dec 07 2021

a(n) = { my (r = [1 % n]); for (i = 0, oo, if (vecmin(r)==0, return (i), r = (concat(0, r) + concat(r, 0)) % n;);); }

import numpy as np
def pascals(n):
  a = np.ones(1)
  f = np.ones(2)
  triangle = [a]
  for i in range(n):
    a = np.convolve(a,f)
    triangle.append(a)
  return triangle
def test(n,tri):
  for i, element in enumerate(tri):
    for sub_e in element:
      if sub_e % n == 0:
        return i
tri = pascals(500)
for i in range(1,50):
  print(test(i,tri),end=',')

from math import comb
def A349958(n):
    for j in range(n+1):
        for k in range(j+1):
            if comb(j,k) % n == 0: return j # Chai Wah Wu, Dec 10 2021

More terms from Michel Marcus, Dec 07 2021

A199424 Index of first row in triangle A199333 containing n-th prime.

Original entry on oeis.org

2, 3, 4, 4, 6, 5, 8, 9, 6, 6, 12, 7, 14, 15, 16, 7, 18, 19, 20, 9, 22, 23, 24, 25, 8, 27, 28, 8, 30, 31, 11, 33, 34, 35, 36, 9, 12, 39, 40, 41, 42, 43, 13, 45, 46, 47, 9, 10, 50, 14, 52, 53, 54, 55, 56, 57, 58, 15, 60, 61, 62, 63, 64, 65, 66, 16, 11, 69, 70

Offset: 1

Reinhard Zumkeller, Nov 09 2011

nonn

a(n) <= n + 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..150

Cf. A199425, A058084.

import Data.List (findIndex)
import Data.Maybe (fromJust)
a199424 n = fromJust $ findIndex (elem $ a000040 n) a199333_tabl

A375570 Smallest m such that A008949(m,k) = n for some k.

Original entry on oeis.org

0, 1, 2, 2, 4, 5, 3, 3, 8, 9, 4, 11, 12, 13, 4, 4, 16, 17, 18, 19, 20, 6, 22, 23, 24, 5, 26, 27, 7, 29, 5, 5, 32, 33, 34, 35, 8, 37, 38, 39, 40, 6, 42, 43, 44, 9, 46, 47, 48, 49, 50, 51, 52, 53, 54, 10, 6, 57, 58, 59, 60, 61, 6, 6, 64, 65, 11, 67, 68, 69, 70

Offset: 1

Pontus von Brömssen, Aug 19 2024

nonn

Cf. A008949, A058084, A375571, A375572, A375573.

A008949(a(n),A375571(n)) = n.
a(n) <= n-1.
a(2^n) = n.
a(2^n-1) = n for n >= 2.

A257646 Index of first row of triangle A103284 containing n.

Original entry on oeis.org

0, 2, 3, 4, 4, 5, 5, 9, 5, 11, 6, 13, 6, 7, 16, 6, 18, 7, 20, 7, 22, 23, 24, 7, 26, 27, 9, 29, 7, 8, 32, 33, 34, 8, 10, 37, 38, 8, 40, 41, 42, 43, 44, 8, 46, 47, 48, 49, 9, 51, 52, 53, 8, 12, 56, 57, 58, 59, 9, 61, 62, 63, 64, 9, 13, 67, 68, 69, 70, 71, 72

Offset: 1

Reinhard Zumkeller, Nov 19 2015

nonn

a(n) <= n + 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A103284, A058084.

import Data.List (findIndex); import Data.Maybe (fromJust)
a257646 n = fromJust $ findIndex (elem n) a103284_tabl

A264856 Index of first row of triangle A125605 containing n.

Original entry on oeis.org

0, 2, 3, 4, 5, 6, 7, 6, 9, 10, 11, 8, 11, 10, 15, 16, 17, 18, 14, 12, 15, 22, 12, 18, 12, 22, 27, 18, 13, 20, 15, 32, 33, 34, 13, 36, 37, 14, 15, 14, 38, 42, 43, 44, 45, 16, 24, 13, 25, 50, 51, 52, 53, 54, 31, 56, 45, 58, 59, 60, 19, 62, 63, 64, 65, 66, 67

Offset: 1

Reinhard Zumkeller, Nov 26 2015

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A125605, A058084.

import Data.List (findIndex); import Data.Maybe (fromJust)
a264856 n = fromJust $ findIndex (elem n) a125605_tabl

cp's OEIS Frontend

A357327 a(n) is the unique nonnegative integer k <= A058084(n)/2 such that binomial(A058084(n),k) = n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A006987 Binomial coefficients: C(n,k), 2 <= k <= n-2, sorted, duplicates removed.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A137905 Numbers that appear as binomial coefficients exactly twice.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A349958 a(n) is the index of the first row in Pascal's triangle that contains a multiple of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Python

Extensions

A199424 Index of first row in triangle A199333 containing n-th prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

A375570 Smallest m such that A008949(m,k) = n for some k.

Views

Author

Keywords

Crossrefs

Formula

A257646 Index of first row of triangle A103284 containing n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

A264856 Index of first row of triangle A125605 containing n.

Views

Author

Keywords

Links

Crossrefs

Programs