A064394 Numbers k such that the exponent of highest power of 2 dividing k! equals the largest prime < k.

4, 5, 8, 9, 22, 23, 26, 27, 32, 33, 50, 51, 56, 57, 70, 71, 76, 77, 82, 83, 94, 95, 100, 101, 112, 113, 118, 119, 128, 129, 134, 135, 176, 177, 186, 187, 196, 197, 266, 267, 274, 275, 280, 281, 296, 297, 342, 343, 352, 353, 358, 359, 364, 365, 372, 373, 386, 387

Offset

1,1

Comments

[k/2]+[k/4]+[k/8]+[k/16]+... = prevprime(k).

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1001 terms from Harvey P. Dale)

Formula

A011371(a(n)) = A151799(a(n)).

Example

8! = 2^7*3^2*5*7, 23! = 2^19*3^9*5^4*7^3*11^2*13*17*19*23.

Maple

for n from 3 to 10^3 do if sum(floor(n/(2^i)), i=1..15) = prevprime(n) then printf(`%d, `, n) fi; od:
# second Maple program:
b:= proc(n) option remember;
      `if`(n<1, 0, b(n-1)+padic[ordp](n, 2))
    end:
a:= proc(n) option remember; local k; for k from 1+
      `if`(n=1, 2, a(n-1)) while b(k)<>prevprime(k) do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 10 2022

Mathematica

Select[Range[400], IntegerExponent[#!, 2]==NextPrime[#, -1]&] (* Harvey P. Dale, Sep 24 2013 *)

Prog

(Python 3.10+)
from itertools import count, islice
from sympy import prevprime
def A064394_gen(startvalue=3): # generator of terms
    return filter(lambda n:n-n.bit_count()==prevprime(n), count(max(startvalue, 3)))
A064394_list = list(islice(A064394_gen(), 30)) # Chai Wah Wu, Jul 10 2022
(PARI) isok(k) = (k>1) && (valuation(k!, 2) == precprime(k-1)); \\ Michel Marcus, Jul 10 2022

Crossrefs

Cf. A000040, A011371, A064393, A151799.

Sequence in context: A228012 A067271 A268128 * A346457 A141220 A340762

Adjacent sequences: A064391 A064392 A064393 * A064395 A064396 A064397

Keyword

nonn

Author

Vladeta Jovovic, Sep 29 2001

Extensions

More terms from James A. Sellers, Oct 01 2001

Name clarified and offset changed to 1 by Chai Wah Wu, Jul 10 2022

Status

approved