A140221 A number n is included if n is coprime to Sum_{k=1..n} d(k), where d(k) is the number of divisors of k.

1, 2, 3, 7, 9, 10, 11, 12, 13, 14, 17, 19, 23, 25, 27, 28, 29, 31, 32, 34, 35, 37, 41, 43, 45, 49, 50, 51, 52, 53, 54, 56, 58, 59, 61, 62, 65, 67, 69, 71, 73, 75, 77, 79, 81, 82, 83, 84, 86, 87, 88, 89, 92, 93, 94, 95, 97, 98, 101, 103, 107, 109, 111, 113, 115, 117, 119, 122

Offset: 1

Author

Leroy Quet, May 12 2008

Keywords

• nonn

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

Crossrefs

Cf. A006218, A050226, A140222.

Programs

Maple

N:= 500: # for terms <= N
S:= ListTools:-PartialSums(map(numtheory:-tau, [$1..N])):
select(t -> igcd(t,S[t])=1, [$1..N]); # Robert Israel, Feb 20 2024

Mathematica

With[{r = Range[200]}, PositionIndex[CoprimeQ[r, Accumulate[DivisorSigma[0, r]]]][True]] (* Paolo Xausa, Feb 21 2024 *)

Python

from math import gcd, isqrt
def A140221_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: gcd(n,-(s:=isqrt(n))**2+(sum(n//k for k in range(1,s+1))<<1))==1,count(max(startvalue,1)))
A140221_list = list(islice(A140221_gen(),30)) # Chai Wah Wu, Oct 23 2023

Extensions

More terms from Max Alekseyev, May 10 2009