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 A381674 #11 Apr 02 2025 03:06:23
%S A381674 1,1,1,1,1,24,1,6,6,1920,1,17280,1,322560,97200,10080,1,58060800,1,
%T A381674 1393459200,51438240,40874803200,1,536481792000,3000,25505877196800,
%U A381674 9797760,535623421132800,1,40999294770610176000000,1,41845579776000,51855036710400,23310331287699456000
%N A381674 a(n) = product of numbers k < n such that 1 < gcd(k,n) and rad(k) != rad(n).
%C A381674 Terms are in A055932.
%C A381674 The only squarefree terms are 1 and 6.
%H A381674 Michael De Vlieger, <a href="/A381674/b381674.txt">Table of n, a(n) for n = 1..599</a>
%H A381674 Michael De Vlieger, <a href="/A381674/a381674.png">Log log scatterplot of log_10(a(n))</a>, n = 1..2^14, showing n that are prime powers in gold, n that are squarefree in green, and other n in blue and magenta, where magenta additionally represents powerful n that are not prime powers.
%H A381674 Michael De Vlieger, <a href="/A381674/a381674_1.png">Plot prime(i)^m | a(n) at (x,y) = (n, i)</a>, n = 1..2048, 2X vertical exaggeration, with a color function representing m, where black represents m = 1, red m = 2, ..., magenta represents the largest m in the dataset, i.e., m = 2035.
%F A381674 a(n) is the product of row n of A381094.
%F A381674 a(n) = 1 for prime n and n = 4.
%F A381674 a(2*p) = p * 2^(p-1) * (p-1)! = A381675(n) for odd prime p = prime(n), n > 1.
%e A381674 Table of n and a(n) for select n, showing exponents of prime factors of the latter and row n of A381094:
%e A381674 1 1 1
%e A381674 n a(n) 2 3 5 7 1 3 7 Row n of A381094
%e A381674 ---------------------------------------------------------------------------------------
%e A381674 6 24 3, 1 {2,3,4}
%e A381674 8 6 1, 1 {6}
%e A381674 9 6 1, 1 {6}
%e A381674 10 1920 7, 1, 1 {2,4,5,6,8}
%e A381674 12 17280 7, 3, 1 {2,3,4,8,9,10}
%e A381674 14 322560 10, 2, 1, 1 {2,4,6,7,8,10,12}
%e A381674 15 97200 4, 5, 2 {3,5,6,9,10,12}
%e A381674 16 10080 5, 2, 1, 1 {6,10,12,14}
%e A381674 18 58060800 12, 4, 2, 1 {2,3,4,8,9,10,14,15,16}
%e A381674 20 1393459200 15, 5, 2, 1 {2,4,5,6,8,12,14,15,16,18}
%e A381674 24 536481792000 15, 5, 3, 2, 1 {2,3,4,8,9,10,14,15,16,20,21,22}
%e A381674 25 3000 3, 1, 3 {10,15,20}
%e A381674 30 40999294770610176000000 25,13, 6, 3, 1, 1 {2,3,4,5,6,8,9,10,12,14,..,28}
%e A381674 32 41845579776000 16, 6, 3, 2, 1, 1 {6,10,12,14,18,20,22,24,26,28,30}
%e A381674 36 11358323143857930240000 25,10, 4, 3, 2, 1, 1 {2,3,4,8,9,10,14,15,16,20,..,34}
%e A381674 a(n) = 6 for n = 8 or 9, since 6 is the only number less than n that shares a factor with n but rad(6) != rad(n).
%e A381674 a(6) = (2*3)*(4) = 24.
%e A381674 a(10) = (2*4*6*8)*(5) = 1920.
%e A381674 a(12) = (2*4*8*10)*(3*9) = 17280, etc.
%t A381674 rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
%t A381674 Table[r = rad[n]; Times @@ Select[Range[n], Nor[CoprimeQ[#, n], rad[#] == r] &], {n, 120}]
%Y A381674 Cf. A055932, A070251, A381094, A381497, A381675.
%K A381674 nonn
%O A381674 1,6
%A A381674 _Michael De Vlieger_, Mar 15 2025