A058084 Smallest m such that binomial(m,k) = n for some k.

0, 2, 3, 4, 5, 4, 7, 8, 9, 5, 11, 12, 13, 14, 6, 16, 17, 18, 19, 6, 7, 22, 23, 24, 25, 26, 27, 8, 29, 30, 31, 32, 33, 34, 7, 9, 37, 38, 39, 40, 41, 42, 43, 44, 10, 46, 47, 48, 49, 50, 51, 52, 53, 54, 11, 8, 57, 58, 59, 60, 61, 62, 63, 64, 65, 12, 67, 68, 69, 8, 71, 72, 73, 74, 75, 76

Offset: 1

Author

Fabian Rothelius, Nov 25 2000

Keywords

• nice
• nonn
• easy
• changed

Comments

Index of first row of Pascal's triangle (A007318) containing n.

Examples

			a(28)=8 because 28 is first found in row 8 of Pascal's triangle (where the first row is counted as 0).
		

Links

• Pontus von Brömssen, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe, with 3 corrections).

Crossrefs

Cf. A007318, A006987, A137905, A357327.

Programs

Haskell

import Data.List (findIndex); import Data.Maybe (fromJust)
a058084 n = fromJust $ findIndex (elem n) a007318_tabl
-- Reinhard Zumkeller, Nov 09 2011

Maple

with(combinat): for n from 2 to 150 do flag := 0: for m from 1 to 150 do for k from 1 to m do if binomial(m,k) = n then printf(`%d,`,m); flag := 1; break fi: od: if flag=1 then break fi; od: od:

Mathematica

nmax = 76; t = Table[Binomial[m, k], {m, 0, nmax}, {k, 0, m}]; a[n_] := Position[t, n, 2, 1][[1, 1]]-1; Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Nov 30 2011 *)

PARI

a(n) = my(k=0); while (!vecsearch(vector((k+2)\2, i, binomial(k, i-1)), n), k++); k; \\ Michel Marcus, Dec 07 2021

Python

def A058084(n):
    if n == 1: return 0
    c = [1]
    for m in range(n):
        d = [1]
        for i in range(m):
            a = c[i]+c[i+1]
            if a == n:
                return m+1
            d.append(a)
        c = d+[1] # Chai Wah Wu, Aug 23 2025

Formula

a(A006987(n)) < A006987(n); a(A137905(n)) = A137905(n). - Reinhard Zumkeller, Mar 20 2009

A007318(a(n), A357327(n)) = n. - Pontus von Brömssen, Sep 24 2022

Extensions

More terms from James Sellers, Nov 27 2000