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 A381800 #11 Mar 14 2025 20:16:41
%S A381800 1,2,2,3,2,5,2,4,3,7,2,8,2,6,8,5,2,12,2,9,9,13,2,11,3,15,4,9,2,19,2,6,
%T A381800 9,11,12,16,2,21,6,12,2,24,2,16,15,14,2,16,3,28,20,17,2,31,8,12,21,31,
%U A381800 2,28,2,8,13,7,10,32,2,13,15,35,2,20,2,39,29,24
%N A381800 a(n) = number of distinct residues r mod n of numbers k such that rad(k) | n, where rad = A007947.
%H A381800 Michael De Vlieger, <a href="/A381800/b381800.txt">Table of n, a(n) for n = 1..5000</a>
%H A381800 Michael De Vlieger, <a href="/A381800/a381800.png">Log log scatterplot of a(n)</a>, n = 1..5000, showing a(n) for prime n in red, a(n) for proper prime power n in gold, a(n) such that n is squarefree and composite in green, and a(n) such that n is neither squarefree nor prime power in blue and magenta, where the latter color also signifies n is powerful but not a prime power.
%H A381800 Michael De Vlieger, <a href="/A381800/a381800.txt">Faster code for A381800 and A381801</a>, 2025.
%F A381800 a(n) = length of row n of A381801.
%F A381800 a(1) = 1 since 1 is the empty product.
%F A381800 A010846(n) <= a(n) <= A051953(n).
%F A381800 a(n) >= 2 for n > 1.
%F A381800 For prime p, a(p) = A010846(p^m) = A000005(p^m) = A381798(p) = 2.
%F A381800 For prime power p^m, m > 0, a(p^m) = A010846(p^m) = A000005(p^m) = A381798(p^m) = m+1.
%F A381800 For n in A024619, a(n) > A381798(n).
%e A381800 n a(n) row n of A381801
%e A381800 ----------------------------------------------
%e A381800 1 1 {0}
%e A381800 2 2 {0,1}
%e A381800 3 2 {0,1}
%e A381800 4 3 {0,1,2}
%e A381800 6 5 {0,1,2,3,4}
%e A381800 8 4 {0,1,2,4}
%e A381800 10 7 {0,1,2,4,5,6,8}
%e A381800 12 8 {0,1,2,3,4,6,8,9}
%e A381800 14 6 {0,1,2,4,7,8}
%e A381800 15 8 {0,1,3,5,6,9,10,12}
%e A381800 18 12 {0,1,2,3,4,6,8,9,10,12,14,16}
%e A381800 20 9 {0,1,2,4,5,8,10,12,16}
%e A381800 21 9 {0,1,3,6,7,9,12,15,18}
%e A381800 22 13 {0,1,2,4,6,8,10,11,12,14,16,18,20}
%e A381800 24 11 {0,1,2,3,4,6,8,9,12,16,18}
%e A381800 26 15 {0,1,2,4,6,8,10,12,13,14,16,18,20,22,24}
%e A381800 28 9 {0,1,2,4,7,8,14,16,21}
%e A381800 30 19 {0,1,2,3,4,5,6,8,9,10,12,15,16,18,20,21,24,25,27}
%e A381800 36 16 {0,1,2,3,4,6,8,9,12,16,18,20,24,27,28,32}
%t A381800 Table[CountDistinct@ Flatten@ Mod[TensorProduct @@ Map[(p = #; NestWhileList[Mod[p*#, n] &, 1, UnsameQ, All]) &, FactorInteger[n][[All, 1]]], n], {n, 120}]
%Y A381800 Cf. A000005, A010846, A024619, A051953, A381798, A381801.
%K A381800 nonn
%O A381800 1,2
%A A381800 _Michael De Vlieger_, Mar 07 2025