A334082 a(n) is the number of binomial coefficients C(n,k), with 0<=k<=n, that are not practical.

0, 0, 2, 0, 4, 2, 6, 1, 2, 4, 6, 2, 8, 10, 14, 0, 6, 2, 8, 5, 10, 10, 20, 2, 8, 14, 14, 6, 16, 14, 30, 0, 2, 4, 6, 2, 6, 10, 16, 4, 10, 10, 24, 16, 14, 20, 36, 4, 8, 10, 16, 12, 30, 14, 30, 6, 14, 18, 34, 14, 34, 38, 62, 0, 2, 2, 6, 10, 12, 10, 24, 2, 8, 14, 14

Offset

1,3

Links

Table of n, a(n) for n=1..75.

Paolo Leonetti and Carlo Sanna, Practical numbers among the binomial coefficients, arXiv:1905.12023 [math.NT], 2019.

Maple

A334082 := proc(n)
    local a, k ;
    a := 0 ;
    for k from 0 to n do
        if not isA005153(binomial(n, k)) then # reuses code of A005153
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A334082(n), n=1..100) ; # R. J. Mathar, Jul 07 2023

Mathematica

PracticalQ[n_] := Module[{f, p, e, prod = 1, ok = True}, If[n < 1 || (n > 1 && OddQ[n]), False, If[n == 1, True, f = FactorInteger[n]; {p, e} = Transpose[f]; Do[If[p[[i]] > 1 + DivisorSigma[1, prod], ok = False; Break[]]; prod = prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]];
a[n_] := Select[Table[Binomial[n, k], {k, 0, n}], !PracticalQ[#]&] // Length;
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Aug 11 2023, after T. D. Noe in A005153 *)

Prog

(PARI) a(n) = sum(k=0, n, !is_A005153(binomial(n, k)));
(Python)
from math import comb
from sympy import factorint
def A334082(n):
    m = 0
    for k in range(1, n):
        c = comb(n, k)
        l = (~c & c-1).bit_length()
        if l>0:
            P = (1<<l+1)-1
            for p, e in factorint(c>>l).items():
                if p > 1+P:
                    break
                P *= (p**(e+1)-1)//(p-1)
            else:
                continue
            m += 1
        else:
            m += 1
    return m # Chai Wah Wu, Jul 05 2023

Crossrefs

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

Cf. A334083, A334084.

Sequence in context: A153733 A083218 A203908 * A346612 A352528 A307704

Adjacent sequences: A334079 A334080 A334081 * A334083 A334084 A334085

Keyword

nonn

Author

Michel Marcus, Apr 14 2020

Status

approved