login
Integers m such that all binomial coefficients C(m,k), with 0<=k<=m, are practical numbers.
2

%I #22 Jul 06 2023 01:52:38

%S 1,2,4,16,32,64,128,256,1024,2048,4096,8192,16384

%N Integers m such that all binomial coefficients C(m,k), with 0<=k<=m, are practical numbers.

%C Integers m such that A334082(m) = 0.

%C All terms are powers of 2, but this is not a sufficient condition since A334082(8) = 1.

%o (PARI) isok(n) = sum(k=0, n, !is_A005153(binomial(n,k))) == 0;

%o (Python)

%o from itertools import count, islice

%o from math import comb

%o from sympy import factorint

%o def A334083_gen(): # generator of terms

%o for n in count(0):

%o m, flag = 1<<n, True

%o for k in range(1,m):

%o c = comb(m,k)

%o if c > 1:

%o l = (~c & c-1).bit_length()

%o if l>0:

%o P = (1<<l+1)-1

%o for p, e in factorint(c>>l).items():

%o if p > 1+P:

%o flag = False

%o break

%o P *= (p**(e+1)-1)//(p-1)

%o else:

%o flag = False

%o break

%o if flag:

%o yield m

%o A334083_list = list(islice(A334083_gen(),10)) # _Chai Wah Wu_, Jul 05 2023

%Y Cf. A005153 (practical numbers), A007318 (binomial coefficients).

%Y Cf. A334082, A334084.

%K nonn,more

%O 1,2

%A _Michel Marcus_, Apr 14 2020

%E a(9) from _Jinyuan Wang_, Apr 14 2020

%E a(10)-a(13) from _Chai Wah Wu_, Jul 05 2023