A334084 Integers m such that only 2 binomial coefficients C(m,k), with 0<=k<=m, are practical numbers.

1, 3, 5, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767

Offset

1,2

Comments

Integers m such that A334082(m) = m-1.

Integers of the form 2^k-1 (A000225) with k>0 are terms, but this condition is not necessary since 5 is a term.

Links

Table of n, a(n) for n=1..16.

Prog

(PARI) isok(n) = sum(k=0, n, !is_A005153(binomial(n, k))) == n-1;
(Python)
from itertools import count, islice
from math import comb
from sympy import factorint
def A334084_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue, 1)):
        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:
                    break
        else:
            yield n
A334084_list = list(islice(A334084_gen(), 10)) # Chai Wah Wu, Jul 05 2023

Crossrefs

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

Cf. A334082, A334083.

Sequence in context: A261646 A303027 A217615 * A050553 A005540 A155802

Adjacent sequences: A334081 A334082 A334083 * A334085 A334086 A334087

Keyword

nonn,more

Author

Michel Marcus, Apr 14 2020

Extensions

a(11) from Jinyuan Wang, Apr 14 2020

a(12)-a(16) from Chai Wah Wu, Jul 05 2023

Status

approved