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 A332821 #25 Jan 18 2021 02:40:57
%S A332821 2,5,9,11,12,16,17,21,23,28,30,31,39,40,41,47,49,52,54,57,59,66,67,70,
%T A332821 72,73,75,76,83,87,88,91,96,97,100,102,103,109,111,116,126,127,128,
%U A332821 129,130,133,135,136,137,138,148,149,154,157,159,165,167,168,169,172,175,179,180,183,184,186,190,191,197,203,211,212
%N A332821 One part of a 3-way classification of the positive integers. Numbers n for which A048675(n) == 1 (mod 3).
%C A332821 The positive integers are partitioned between A332820, this sequence and A332822.
%C A332821 For each prime p, the terms include exactly one of p and p^2. The primes alternate between this sequence and A332822. This sequence has the primes with odd indexes, those in A031368.
%C A332821 The terms are the even numbers in A332822 halved. The terms are also the numbers m such that 5m is in A332822, and so on for alternate primes: 11, 17, 23 etc. Likewise, the terms are the numbers m such that 3m is in A332820, and so on for alternate primes: 7, 13, 19 etc.
%C A332821 The numbers that are half of the even terms of this sequence are in A332820, which consists exactly of those numbers. The numbers that are one third of the terms that are multiples of 3 are in A332822, which consists exactly of those numbers. For larger primes, an alternating pattern applies as described in the previous paragraph.
%C A332821 If we take each odd term of this sequence and replace each prime in its factorization by the next smaller prime, the resulting number is in A332822, which consists entirely of those numbers.
%C A332821 The product of any 2 terms of this sequence is in A332822, the product of any 3 terms is in A332820, and the product of a term of A332820 and a term of this sequence is in this sequence. So if a number k is present, k^2 is in A332822, k^3 is in A332820, and k^4 is in this sequence.
%C A332821 If k is an even number, exactly one of {k/2, k, 2k} is in the sequence (cf. A191257 / A067368 / A213258); and generally if k is a multiple of a prime p, exactly one of {k/p, k, k*p} is in the sequence.
%F A332821 {a(n) : n >= 1} = {2 * A332820(k) : k >= 1} U {A003961(A332822(k)) : k >= 1}.
%F A332821 {a(n) : n >= 1} = {A332822(k)^2 : k >= 1} U {A331590(2, A332820(k)) : k >= 1}.
%t A332821 Select[Range@ 212, Mod[Total@ #, 3] == 1 &@ Map[#[[-1]]*2^(PrimePi@ #[[1]] - 1) &, FactorInteger[#]] &] (* _Michael De Vlieger_, Mar 15 2020 *)
%o A332821 (PARI) isA332821(n) = { my(f = factor(n)); (1==((sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3)); };
%Y A332821 Positions of ones in A332823; equivalently, numbers in row 3k+1 of A277905 for some k >= 0.
%Y A332821 Cf. A048675, A332820, A332822.
%Y A332821 Comparable 2 or 3-way classifications: A000379/A000028, A001969/A000069, A003159/A036554, A005843/A005408, A028260/A026424, A191257/A067368/A213258, A325431/A325432, A329609/A329604/A332812.
%Y A332821 Subsequences: intersection of A026478 and A066208, A031368 (prime terms), A033431\{0}, A052934\{1}, A069486, A099800, A167747\{1}, A244725\{0}, A244728\{0}, A338911 (semiprime terms).
%K A332821 nonn
%O A332821 1,1
%A A332821 _Antti Karttunen_ and _Peter Munn_, Feb 25 2020