A369871 a(n) = least prime factor of A024411(n).

2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 5, 2, 3, 3, 2, 2, 5, 3, 2, 2, 3, 5, 2, 2, 2, 3, 2, 3, 5, 2, 7, 2, 3, 2, 3, 2, 7, 3, 2, 2, 3, 5, 3, 2, 2, 5, 2, 3, 2, 7, 2, 3, 2, 2, 5, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 2, 3, 7, 2, 5, 3, 2, 3, 5, 2, 7, 3, 2, 3, 2, 3, 2, 5, 2, 3

Offset

1,1

Comments

Conjecture: every prime occurs infinitely many times.

From Aditya P Bhattad, Apr 29 2026: (Start)

The conjecture is true. For an odd prime p, choose primes q,r > p and set m = p*q*r. Then lpf(m) = p. The two coprime odd factorizations 1*(p*q*r) and p*(q*r) give two primitive Pythagorean triples with short leg m, since in both cases the larger/smaller factor ratio is > 1 + sqrt(2). Hence every odd prime occurs infinitely often.

For p = 2, take m = 12*r with r >= 5 prime. The factorizations 12*r = 2*1*(6*r) = 2*3*(2*r) give two primitive triples with short leg 12*r, since 6*r > 1 + sqrt(2) and 2*r/3 > 1 + sqrt(2). Thus 2 also occurs infinitely often. (End)

Links

Table of n, a(n) for n=1..86.

Example

Factorizations of the first four terms of A024411 are given by 20 = 2 * 2 * 5; 28 = 2 * 2 * 7; 33 = 3 * 13; 36 = 2*2*3*3; so that (a(1), a(2), a(3), a(4)) = (2, 2, 3, 2).
Corner of array whose n-th row shows the numbers n such that a(n) = n-th prime:
   1    2    4    6    7    9   11
   3    5    8   10   14   15   19
  12   18   23   30   43   47   56
  32   38   51   69   77   94  107

Mathematica

readOEISbFile[bNumber_] := ToExpression[ToString[Map[StringReplace[#,
 FromCharacterCode[32] -> FromCharacterCode[44]] &,
 Map[Most[#] &, Cases[Map[{#, StringFreeQ[StringTake[#, 1],
 Except[Apply[Alternatives, Flatten[{Map[ToString[#] &, Range[10] - 1],
 FromCharacterCode[45], FromCharacterCode[8722], "+"}]]]]} &,
 Map[StringDrop[StringDrop[#, 1], -1] &, DeleteCases[
 Map[ToString[{#}] &, Import[If[Head[bNumber] === Integer,
"https://oeis.org/" <> "A" <> # <> "/b" <> # <> ".txt" &[
 StringTrim[StringReplace[ToString[PadLeft[IntegerDigits[bNumber], 6]],
 FromCharacterCode[{44, 32}] -> ""], "{" | "}"]],
 bNumber], "List"]], "{}"]]], {_, True}]]]]]
s = readOEISbFile["https://oeis.org/A024411/b024411.txt"];
(* s = 10000 terms of A024411 from b-file *)
t = Map[Last, s]; u = Take[t, 1000];
f[n_] := FactorInteger[n][[1, 1]]; Join[{1}, Array[f, 120, 2]]; v = Map[f, u]
(* Peter J. C. Moses, Feb 03 2024 *)

Crossrefs

Cf. A000040, A018252, A024411.

Sequence in context: A278399 A349197 A225416 * A295784 A275803 A060131

Adjacent sequences: A369868 A369869 A369870 * A369872 A369873 A369874

Keyword

nonn

Author

Clark Kimberling, Feb 03 2024

Status

approved