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 A372000 #27 Apr 16 2024 05:35:37
%S A372000 1,2,6,3,15,10,70,35,105,42,462,77,1001,286,4290,2145,36465,24310,
%T A372000 461890,46189,969969,176358,4056234,676039,3380195,520030,1560090,
%U A372000 111435,3231615,430882,13357342,6678671,220396143,25928958,907513530,151252255,5596333435,589087730,22974421470,2297442147
%N A372000 a(n) = product of primes p such that floor(n/p) is odd.
%C A372000 The only primes in the sequence are 2 and 3.
%C A372000 We can approach the sequence in a manner akin to A260850, a variant of A008336. Set k = 1. Then for all prime factors p | n, if p | k, divide k by p, otherwise multiply k by p. Then we set a(n) = k. This accounts for the "toggling on or off" of prime factors as n increases.
%C A372000 For n >= 1, A055773(n) | a(n), where A055773(n) = A034386(n) / A034386(floor(n/2)).
%H A372000 Michael De Vlieger, <a href="/A372000/b372000.txt">Table of n, a(n) for n = 1..3384</a>
%H A372000 Michael De Vlieger, <a href="/A372000/a372000.png">Plot prime(i) | a(n) at (x,y) = (n,i)</a> for n = 1..2048.
%F A372000 a(n) = Product_{k = 1..floor(pi(n)/2)+1} Product_{j = 1+floor(n/(2*k))..floor(n/(2*k-1))} prime(j), where pi(x) = A000720(n).
%e A372000 a(1) = 1 since n = 1 is the empty product.
%e A372000 a(2) = 2 since for n = 2, floor(n/p) = floor(2/2) = 1 is odd.
%e A372000 a(3) = 6 since for n = 3 and p = 2, floor(3/2) = 1 is odd, and for p = 3, floor(3/3) = 1 is odd. Hence a(3) = 2*3 = 6.
%e A372000 a(4) = 3 since for n = 4 and p = 2, floor(4/2) = 2 is even, but for p = 3, floor(4/3) = 1 is odd. Therefore, a(n) = 3.
%e A372000 a(5) = 15 since for n = 5, though floor(5/2) = 2 is even, floor(5/3) and floor(5/5) are both odd. Therefore, a(n) = 3*5 = 15, etc.
%e A372000 Table relating a(n) with b(n), diagramming prime factors with "x" that produce a(n), or powers of 2 with "x" that sum to b(n), where b(n) = A371906(n).
%e A372000 Prime factor
%e A372000 1111
%e A372000 n b(n) 23571379 b(n)
%e A372000 ----------------------------
%e A372000 1 1 . 0
%e A372000 2 2 x 1
%e A372000 3 6 xx 3
%e A372000 4 3 .x 2
%e A372000 5 15 .xx 6
%e A372000 6 10 x.x 5
%e A372000 7 70 x.xx 13
%e A372000 8 35 ..xx 12
%e A372000 9 105 .xxx 14
%e A372000 10 42 xx.x 11
%e A372000 11 462 xx.xx 27
%e A372000 12 77 ...xx 24
%e A372000 13 1001 ...xxx 56
%e A372000 14 286 x...xx 49
%e A372000 15 4290 xxx.xx 55
%e A372000 16 2145 .xx.xx 54
%e A372000 17 36465 .xx.xxx 118
%e A372000 18 24310 x.x.xxx 117
%e A372000 19 461890 x.x.xxxx 245
%e A372000 20 46189 ....xxxx 240
%e A372000 ----------------------------
%e A372000 01234567
%e A372000 Power of 2
%t A372000 Table[Times @@ Select[Prime@ Range@ PrimePi[n], OddQ@ Quotient[n, #] &], {n, 40}] (* or *)
%t A372000 Table[Product[Prime[i], {j, 1 + Floor[PrimePi[n]/2]}, {i, 1 + PrimePi[Floor[n/(2 j)]], PrimePi[Floor[n/(2 j - 1)]]}], {n, 40}]
%o A372000 (PARI) a(n) = vecprod(select(x->((n\x) % 2), primes([1, n]))); \\ _Michel Marcus_, Apr 16 2024
%o A372000 (SageMath)
%o A372000 print([prod(p for p in prime_range(n + 1) if is_odd(n//p)) for n in range(1, 41)])
%o A372000 # _Peter Luschny_, Apr 16 2024
%Y A372000 Cf. A005117, A008336, A034386, A055773, A260850, A371906.
%K A372000 nonn,easy
%O A372000 1,2
%A A372000 _Michael De Vlieger_, Apr 15 2024