This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A343430 #22 Dec 24 2022 03:39:40
%S A343430 1,2,1,4,5,2,1,8,1,10,11,4,1,2,5,16,17,2,1,20,1,22,23,8,25,2,1,4,29,
%T A343430 10,1,32,11,34,5,4,1,2,1,40,41,2,1,44,5,46,47,16,1,50,17,4,53,2,55,8,
%U A343430 1,58,59,20,1,2,1,64,5,22,1,68,23,10,71,8,1,2,25,4,11,2,1,80,1,82,83,4,85
%N A343430 Part of n composed of prime factors of the form 3k-1.
%C A343430 Largest term of A004612 that divides n.
%C A343430 Modulo 6, the prime numbers are partitioned into 4 nonempty sets: {2}, {3}, primes of the form 6k-1 (A007528) and primes of the form 6k+1 (A002476). The modulo 3 partition is nearly the same, but unites the only even prime, 2, with primes of the form 6k-1 in the set of primes we use here.
%C A343430 A positive integer m is a Loeschian number (a term of A003136) if and only if a(A007913(m)) = 1, that is the squarefree part of m has no prime factors of the form 3k-1.
%F A343430 Completely multiplicative with a(p) = p if p is of the form 3k-1, otherwise a(p) = 1.
%F A343430 For k >= 1, a(n) = a(k*n) / gcd(k, a(k*n)).
%F A343430 a(n) = A006519(n) * A343431(n).
%F A343430 a(n) = (n / A038500(n)) / A248909(n) = A038502(n) / A248909(n).
%F A343430 A006519(a(n)) = a(A006519(n)) = A006519(n).
%F A343430 A343431(a(n)) = a(A343431(n)) = A343431(n).
%F A343430 A038500(a(n)) = a(A038500(n)) = 1.
%F A343430 A248909(a(n)) = a(A248909(n)) = 1.
%e A343430 n = 60 has prime factorization 60 = 2 * 2 * 3 * 5. Factors 2 = 3*1 - 1 and 5 = 3*2 - 1 have form 3k-1, whereas 3 does not (having form 3k). Multiplying the factors of form 3k-1, we get 2 * 2 * 5 = 20. So a(60) = 20.
%t A343430 f[p_, e_] := If[Mod[p, 3] == 2, p^e, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jun 11 2021 *)
%o A343430 (PARI) a(n) = {my(f = factor(n)); for (i=1, #f~, if ((f[i, 1] + 1) % 3, f[i, 1] = 1); ); factorback(f); } \\ after _Michel Marcus_ at A248909
%o A343430 (Python)
%o A343430 from math import prod
%o A343430 from sympy import factorint
%o A343430 def A343430(n): return prod(p**e for p, e in factorint(n).items() if p%3==2) # _Chai Wah Wu_, Dec 23 2022
%Y A343430 Equivalent sequences for prime factors of other forms: A006519 (2 only), A000265 (2k+1), A038500 (3 only), A038502 (3k+/-1), A170818 (4k+1), A097706 (4k-1), A248909 (6k+1), A343431 (6k-1).
%Y A343430 Range of terms: A004612 (closure under multiplication of A003627).
%Y A343430 Cf. A002476, A007528, squarefree part (A007913) of terms of A003136.
%Y A343430 First 28 terms are the same as A247503.
%K A343430 nonn,easy,mult
%O A343430 1,2
%A A343430 _Peter Munn_, Jun 08 2021