login
Smallest m such that binomial(m,k) = n for some k.
9

%I #39 Sep 24 2022 15:31:29

%S 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,

%T 8,29,30,31,32,33,34,7,9,37,38,39,40,41,42,43,44,10,46,47,48,49,50,51,

%U 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

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

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

%H Pontus von Brömssen, <a href="/A058084/b058084.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe, with 3 corrections).

%F a(A006987(n)) < A006987(n); a(A137905(n)) = A137905(n). - _Reinhard Zumkeller_, Mar 20 2009

%F A007318(a(n), A357327(n)) = n. - _Pontus von Brömssen_, Sep 24 2022

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

%p 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:

%t 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 *)

%o (Haskell)

%o import Data.List (findIndex); import Data.Maybe (fromJust)

%o a058084 n = fromJust $ findIndex (elem n) a007318_tabl

%o -- _Reinhard Zumkeller_, Nov 09 2011

%o (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

%Y Cf. A007318, A006987, A137905, A357327.

%K nice,nonn,easy

%O 1,2

%A _Fabian Rothelius_, Nov 25 2000

%E More terms from _James A. Sellers_, Nov 27 2000