A140238 Numbers k such that Sum_{i=1..k} d(i) is coprime to d(k), where d(k) is the number of positive divisors of k.

1, 2, 3, 4, 9, 10, 11, 12, 13, 14, 15, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 93, 94, 95, 97, 98, 99, 100, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135

Offset

1,2

Comments

Sum_{k=1..n} d(k) = Sum_{k=1..n} floor(n/k) = A006218(n).

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Maple

N:= 1000: # for terms <= N
T:= map(numtheory:-tau, [$1..N]):
S:= ListTools:-PartialSums(T):
select(t -> igcd(T[t], S[t])=1, [$1..N]); # Robert Israel, Oct 24 2023

Prog

(Python)
from math import gcd, isqrt
from sympy import divisor_count
def A140238_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: gcd(divisor_count(n), -(s:=isqrt(n))**2+(sum(n//k for k in range(1, s+1))<<1))==1, count(max(startvalue, 1)))
A140238_list = list(islice(A140238_gen(), 30)) # Chai Wah Wu, Oct 23 2023
(PARI) isok(k) = gcd(sum(i=1, k, k\i), numdiv(k)) == 1; \\ Michel Marcus, Oct 29 2023

Crossrefs

Cf. A000005, A006218, A140237.

Sequence in context: A113234 A372072 A316534 * A135210 A037468 A265746

Adjacent sequences: A140235 A140236 A140237 * A140239 A140240 A140241

Keyword

nonn

Author

Leroy Quet, May 14 2008

Extensions

Extended by Ray Chandler, Jun 25 2009

Name edited by Michel Marcus, Oct 29 2023

Status

approved