A385720 Numbers k >= 1 such that k/A000005(k) + (k+1)/A000005(k+1) is an integer.

1, 5, 6, 8, 10, 13, 22, 37, 45, 46, 58, 61, 62, 69, 73, 74, 77, 82, 89, 106, 114, 117, 126, 146, 149, 150, 154, 157, 166, 167, 178, 186, 193, 197, 198, 206, 221, 226, 233, 237, 258, 261, 262, 263, 266, 277, 278, 279, 280, 290, 293, 306, 309, 311, 312, 313

Offset: 1

Author

Ctibor O. Zizka, Jul 07 2025

Keywords

• nonn

Comments

k/A000005(k) + (k+1)/A000005(k+1) = (3*k + 1)/4 for k >= 5 from A256072.

k/A000005(k) + (k+1)/A000005(k+1) = (3*k + 2)/4 for k >= 6 from A077065.

k/A000005(k) + (k+1)/A000005(k+1) =< (3*k + 2)/4 for k >= 5.

Examples

			For k = 6: 6/A000005(6) + 7/A000005(7) = 6/4 + 7/2 = 5, thus k = 6 is a term.
		

Crossrefs

Cf. A000005, A005384, A005385, A077065, A256072.

Programs

Mathematica

Position[Plus @@@ Partition[Table[n/DivisorSigma[0, n], {n, 1, 320}], 2, 1], ?IntegerQ] // Flatten (* _Amiram Eldar, Jul 08 2025 *)

PARI

isok(k) = denominator(k/numdiv(k) + (k+1)/numdiv(k+1)) == 1; \\ Michel Marcus, Jul 08 2025

Python

from itertools import count, islice
from sympy import divisor_count
def A385720_gen(startvalue=1): # generator of terms >= startvalue
    m = max(startvalue,1)
    a, b = divisor_count(m), divisor_count(m+1)
    for k in count(m):
        if not (k*b+(k+1)*a)%(a*b):
            yield k
        a, b = b, divisor_count(k+2)
A385720_list = list(islice(A385720_gen(),30)) # Chai Wah Wu, Jul 13 2025