A333261 Numbers k such that A071324(k) = A071324(k+1).

1, 5, 51, 68, 87, 116, 171, 176, 591, 2108, 2403, 7143, 8787, 9308, 18548, 19371, 27387, 32127, 34887, 37928, 40131, 140667, 180548, 192428, 200996, 433311, 521727, 934449, 2476671, 2617563, 3960896, 8198156, 9670748, 11892512, 16585748, 19113651, 25367396, 25643012

Offset: 1

Author

Amiram Eldar, Mar 13 2020

Keywords

• nonn
• hard

Comments

From Shreyansh Jaiswal, Jun 14 2025: (Start)

If the density of the set containing all even terms exists, then it is less than 0.15. (Proposition 3 in Jaiswal.)

Let k denote any even term. Then, the least prime factor of k+1 is either 3 or 5. (Theorem 11 in Jaiswal.)

Each even term satisfies at least one of three specific congruences. (Theorem 2 in Jaiswal.)

10519952096 and 16159802432 are also terms of this sequence.

Conjecture: There are infinitely many terms of this sequence. (Conjecture 15 in Jaiswal.) (End)

Examples

			1 is a term since A071324(1) = A071324(2) = 1.
5 is a term since A071324(5) = A071324(6) = 4.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..89 (terms below 10^10)
• Shreyansh Jaiswal, On the Even Solutions of A071324(n) = A071324(n+1), Jun 14, 2025.

Crossrefs

Cf. A002961, A071324, A206368.

Programs

Mathematica

f[n_] := Plus @@ (-(d = Divisors[n])*(-1)^(Range[Length[d],1,-1])); seq = {}; f1 = f[1]; Do[f2 = f[n]; If[f1 == f2, AppendTo[seq, n-1]]; f1 = f2, {n, 2, 50000}]; seq
SequencePosition[Table[Total[Times@@@Partition[Riffle[Reverse[Divisors[n]],{1,-1},{2,-1,2}],2]],{n,2565*10^4}],{x_,x_}][[All,1]] (* Harvey P. Dale, Nov 06 2022 *)

Python

from sympy import divisors;  from functools import lru_cache
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(1,2201):
    if c(n) == c(n+1):
        print(n, end=", ") # Shreyansh Jaiswal, Apr 14 2025