A382744 If k appears, 5*k does not.

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84

Offset: 1

Author

Jan Snellman, Apr 04 2025

Keywords

• nonn

Comments

Also: numbers with an even number of 5's in their prime factorization.

Natural density 5/6.

Examples

			5 is removed since 5 = 5*1, 10 is removed, 15 is removed, 20 is removed, but 25 remains.
		

Links

• Jan Snellman, Table of n, a(n) for n = 1..8333
• Jan Snellman, Greedy Regular Convolutions, arXiv:2504.02795 [math.NT], 2025.

Crossrefs

Cf. A003159, A007417, A382745, A382746.

Programs

Maple

select(t -> padic:-ordp(t,5)::even, [$1..100]); # Robert Israel, Apr 04 2025

Mathematica

Select[Range[100], EvenQ[IntegerExponent[#, 5]] &] (* Amiram Eldar, Apr 04 2025 *)

Python

def ok(n):
    c = 0
    while n and n%5 == 0: n //= 5; c += 1
    return c&1 == 0
print([k for k in range(1, 82) if ok(k)]) # Michael S. Branicky, Apr 04 2025

Python

from sympy import integer_log
def A382744(n):
    def f(x): return n+x-sum((k:=x//5**m)-k//5 for m in range(0,integer_log(x,5)[0]+1,2))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Apr 10 2025

SageMath

[ for  in range(1,100) if (valuation(_,5) % 2) == 0]

Formula

a(n) ~ (6/5)*n.