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

%I A382837 #53 Jun 19 2025 16:42:12
%S A382837 60,70,84,120,140,154,168,180,200,210,220,240,252,260,264,280,286,300,
%T A382837 312,336,340,350,360,374,390,396,408,418,420,442,456,468,480,490,494,
%U A382837 504,510,520,528,540,560,570,588,598,600,624,630,646,660,672,680,700
%N A382837 Numbers k such that k - A071324(k) > A000010(k).
%C A382837 This sequence lists the counterexamples to Atanassov's conjecture that n - A071324(n) <= A000010(n) for all n. (See Atanassov link, page 2).
%C A382837 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
%C A382837 It appears that odd terms within the sequence are extremely uncommon.
%C A382837 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.
%C A382837 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)...)
%H A382837 Shreyansh Jaiswal, <a href="/A382837/b382837.txt">Table of n, a(n) for n = 1..10000</a>
%H A382837 Krassimir Atanassov, <a href="https://nntdm.net/volume-15-2009/number-3/21-22/">A remark on an arithmetic function. Part 2</a>, Notes on Number Theory and Discrete Mathematics 15(3) (2009), 21-22.
%H A382837 Shreyansh Jaiswal, <a href="/A382837/a382837.png">Logarithmic plot of the ratio f(n)/n, upto n = 10^6</a>
%H A382837 Shreyansh Jaiswal, <a href="/A382837/a382837_1.png">Plot of the ratio f(n)/n, from 10^5 <= n <= 10^6</a> [An enlarged version of the previous graph.]
%H A382837 Shreyansh Jaiswal, <a href="/A382837/a382837.pdf">On two Conjectures by Atanassov on the Alternating sum-of-divisors function</a>, Jun 14 2025.
%e A382837 60 - A071324(60) = 60 - 40 = 20 > A000010(60) = 16.
%t A382837 f[n_]:=Plus@@(-(d=Divisors[n])*(-1)^(Range[Length[d],1,-1]));
%t A382837 fQ[n_]:=n-f[n]>EulerPhi[n]; Select[Range[700], fQ[#]&] (* _Ivan N. Ianakiev_, Apr 13 2025, thanks to _Amiram Eldar_ at A071324 *)
%o A382837 (Python)
%o A382837 from sympy import divisors;  from functools import lru_cache
%o A382837 from sympy import totient
%o A382837 cached_divisors = lru_cache()(divisors)
%o A382837 def c(n):  return sum(d if i%2==0 else -d for i, d in enumerate(reversed(cached_divisors(n))))
%o A382837 for n in range(1, 701):
%o A382837     if n - c(n) > totient(n):
%o A382837         print(n, end=", ")
%o A382837 (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
%Y A382837 Cf. A000010, A071324.
%K A382837 nonn
%O A382837 1,1
%A A382837 _Shreyansh Jaiswal_, Apr 06 2025