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 A381801 #7 Mar 14 2025 20:11:52
%S A381801 0,0,1,0,1,0,1,2,0,1,0,1,2,3,4,0,1,0,1,2,4,0,1,3,0,1,2,4,5,6,8,0,1,0,
%T A381801 1,2,3,4,6,8,9,0,1,0,1,2,4,7,8,0,1,3,5,6,9,10,12,0,1,2,4,8,0,1,0,1,2,
%U A381801 3,4,6,8,9,10,12,14,16,0,1,0,1,2,4,5,8,10,12,16
%N A381801 Irregular triangle read by rows: row n lists the residues r mod n of numbers k such that rad(k) | n, where rad = A007947.
%C A381801 Define S(p,n) to be the set of residues r (mod n) taken by the power range of prime divisor p, i.e., {p^m, m >= 1}.
%C A381801 Define T(n) to be the union of the tensor product of distinct terms in S(p,n) for all p|n, where the products are expressed mod n.
%C A381801 Row n of this triangle is T(n), a superset of row n of A381799.
%C A381801 For n > 1, the intersection of row n of this triangle and row n of A038566 is {1}.
%H A381801 Michael De Vlieger, <a href="/A381801/b381801.txt">Table of n, a(n) for n = 1..22906</a> (rows n = 1..500, flattened)
%H A381801 Michael De Vlieger, <a href="/A381801/a381801.png">Plot k in row n at (x,y) = (k,-n)</a>, n = 1..36, showing reduced residues mod n in gray and labeling terms in row n. The number n appears on the left in red italic, and row length A381800(n) in blue at right.
%H A381801 Michael De Vlieger, <a href="/A381801/a381801_1.png">Plot k in row n at (x,y) = (k,-n)</a>, n = 1..5000.
%F A381801 Row 1 is {0} since 1 is the empty product and the only number that has zero prime factors is 1, congruent to 0 (mod 1).
%F A381801 Row n begins with {0,1} for n > 1.
%F A381801 For prime p, row p = {0,1}.
%F A381801 For prime power p^m, m > 0, row p = union of {0} and {p^i, i < m}.
%F A381801 Row n is a subset of row n of A121998, considering n in A121998 instead as n mod n = 0.
%F A381801 Row n is a superset of row n of A162306, considering n in A162306 instead as n mod n = 0.
%e A381801 Table of c(n) = A381800(n) and T(n) for select n:
%e A381801 n c(n) T(n)
%e A381801 -----------------------------------------------------------------------------
%e A381801 1 1 {0}
%e A381801 2 2 {0, 1}
%e A381801 3 2 {0, 1}
%e A381801 4 3 {0, 1, 2}
%e A381801 5 2 {0, 1}
%e A381801 6 5 {0, 1, 2, 3, 4}
%e A381801 8 4 {0, 1, 2, 4}
%e A381801 9 3 {0, 1, 3}
%e A381801 10 7 {0, 1, 2, 4, 5, 6, 8}
%e A381801 11 2 {0, 1}
%e A381801 12 8 {0, 1, 2, 3, 4, 6, 8, 9}
%e A381801 14 6 {0, 1, 2, 4, 7, 8}
%e A381801 15 8 {0, 1, 3, 5, 6, 9, 10, 12}
%e A381801 16 5 {0, 1, 2, 4, 8}
%e A381801 18 12 {0, 1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 16}
%e A381801 20 9 {0, 1, 2, 4, 5, 8, 10, 12, 16}
%e A381801 24 11 {0, 1, 2, 3, 4, 6, 8, 9, 12, 16, 18}
%e A381801 28 9 {0, 1, 2, 4, 7, 8, 14, 16, 21}
%e A381801 30 19 {0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 25, 27}
%e A381801 36 16 {0, 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 20, 24, 27, 28, 32}
%e A381801 For n = 10, we have S(2,10) = {1, 2, 4, 6, 8} and S(5,10) = {1, 5}. Therefore we have the following distinct products:
%e A381801 1 2 4 8 6
%e A381801 5 0
%e A381801 Hence T(10) = {0, 1, 2, 4, 5, 6, 8}; terms in A003592 belong to these residues (mod 10).
%e A381801 For n = 12, we have S(2,12) = {1, 2, 4, 8} and S(3,12) = {1, 3, 9}. Therefore we have the following distinct products:
%e A381801 1 2 4 8
%e A381801 3 6 0
%e A381801 9
%e A381801 Thus T(12) = {0, 1, 2, 3, 4, 6, 8, 9}, terms in A003586 belong to these residues (mod 12).
%e A381801 For n = 30, we have {1, 2, 4, 8, 16}, {1, 3, 9, 21, 27}, and {1, 5, 25}. Therefore we have the following distinct products:
%e A381801 1 2 4 8 16 5 10 20 25
%e A381801 3 6 12 24 15 0
%e A381801 9 18
%e A381801 27
%e A381801 Thus T(30) = {0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 25, 27}; terms in A051037 belong to these residues (mod 30).
%t A381801 Table[Union@ Flatten@ Mod[TensorProduct @@ Map[(p = #; NestWhileList[Mod[p*#, n] &, 1, UnsameQ, All]) &, FactorInteger[n][[All, 1]] ], n], {n, 30}]
%Y A381801 Cf. A007947, A038566, A121998, A162306, A381799, A381800.
%K A381801 nonn,tabf
%O A381801 1,8
%A A381801 _Michael De Vlieger_, Mar 07 2025