A382837 Numbers k such that k - A071324(k) > A000010(k).

60, 70, 84, 120, 140, 154, 168, 180, 200, 210, 220, 240, 252, 260, 264, 280, 286, 300, 312, 336, 340, 350, 360, 374, 390, 396, 408, 418, 420, 442, 456, 468, 480, 490, 494, 504, 510, 520, 528, 540, 560, 570, 588, 598, 600, 624, 630, 646, 660, 672, 680, 700

Offset: 1

Author

Shreyansh Jaiswal, Apr 06 2025

Keywords

• nonn

Comments

This sequence lists the counterexamples to Atanassov's conjecture that n - A071324(n) <= A000010(n) for all n. (See Atanassov link, page 2).

Every positive integer multiple of 210 (4th primorial) is a term within the sequence. This can be proved by a slight modification of the proof of Theorem 2 (see Jaiswal link). - Ivan N. Ianakiev, Apr 13 2025

It appears that odd terms within the sequence are extremely uncommon.

It appears that this sequence probably has a nonzero natural density. Computations of the ratio f(n)/n, where f(n) is the number of terms <=n (see links) up to n = 10^6, suggest that the natural density of this sequence (if it exists) is around 0.08. Since every multiple of 210 is a term within the sequence, this value, if it exists, is >= 1/210, which is approximately 0.0047.

Conjecture: For each even natural number n, infinitely many pairs (k,k+n) exist, such that both k and k+n are terms in the sequence. (Example - n = 876 has pairs (84, 960), (180, 1056), (264, 1140), (300, 1176)...)

Examples

			60 - A071324(60) = 60 - 40 = 20 > A000010(60) = 16.
		

Links

• Shreyansh Jaiswal, Table of n, a(n) for n = 1..10000
• Krassimir Atanassov, A remark on an arithmetic function. Part 2, Notes on Number Theory and Discrete Mathematics 15(3) (2009), 21-22.
• Shreyansh Jaiswal, Logarithmic plot of the ratio f(n)/n, upto n = 10^6
• Shreyansh Jaiswal, Plot of the ratio f(n)/n, from 10^5 <= n <= 10^6 [An enlarged version of the previous graph.]
• Shreyansh Jaiswal, On two Conjectures by Atanassov on the Alternating sum-of-divisors function, Jun 14 2025.

Crossrefs

Cf. A000010, A071324.

Programs

Mathematica

f[n_]:=Plus@@(-(d=Divisors[n])*(-1)^(Range[Length[d],1,-1]));
fQ[n_]:=n-f[n]>EulerPhi[n]; Select[Range[700], fQ[#]&] (* Ivan N. Ianakiev, Apr 13 2025, thanks to Amiram Eldar at A071324 *)

PARI

isok(k) = my(d=Vecrev(divisors(k))); k - sum(i=1, #d, (-1)^(i+1)*d[i]) > eulerphi(k); \\ Michel Marcus, Apr 12 2025

Python

from sympy import divisors;  from functools import lru_cache
from sympy import totient
cached_divisors = lru_cache()(divisors)
def c(n):  return sum(d if i%2==0 else -d for i, d in enumerate(reversed(cached_divisors(n))))
for n in range(1, 701):
    if n - c(n) > totient(n):
        print(n, end=", ")