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%I #27 May 05 2024 19:52:35
%S 1,1,1,2,2,3,3,6,2,5,5,30,30,105,7,14,14,21,21,210,10,55,55,330,66,
%T 429,143,2002,2002,15015,15015,30030,910,7735,221,1326,1326,12597,323,
%U 3230,3230,33915,33915,746130,49742,572033,572033,3432198,490314,1225785,24035
%N a(n) = product of those prime(k) such that floor(n/prime(k)) is even.
%C The only primes in the sequence are 2, 3, 5, and 7.
%H Michael De Vlieger, <a href="/A372007/b372007.txt">Table of n, a(n) for n = 1..7591</a>
%H Michael De Vlieger, <a href="/A372007/a372007.png">Plot prime(i) | a(n) at (x,y) = (n,i)</a> for n = 1..2048, 12X vertical exaggeration.
%H Michael De Vlieger, <a href="https://doi.org/10.13140/RG.2.2.10066.36806">"Tiger Stripe" Factors of Primorials</a>, ResearchGate, 2024.
%F a(n) = A034386(n) / A372000(n).
%F a(n) = Product_{k = 1..pi(n)} Product_{j = 1+floor(n/(2*k+1))..floor(n/(2*k))} prime(j), where pi(x) = A000720(n).
%e a(1) = 1 since n = 1 is the empty product.
%e a(2) = 1 since for n = 2, floor(n/p) = floor(2/2) = 1 is odd.
%e a(3) = 1 since for n = 3 and p = 2, floor(3/2) = 1 is odd, and for p = 3, floor(3/3) = 1 is odd.
%e a(4) = 2 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(4) = 2.
%e a(5) = 2 since for n = 5, though floor(5/2) = 2 is even, floor(5/3) and floor(5/5) are both odd. Therefore, a(5) = 2.
%e a(8) = 6 since for n = 8, both floor(8/2) and floor(8/3) are even, but both floor(8/5) and floor(8/7) are odd. Therefore, a(8) = 2*3 = 6, etc.
%e Table relating a(n) with b(n), s(n), and t(n), diagramming prime factors with "x" that produce a(n) or b(n), or powers of 2 with "x" that sum to s(n) or t(n). Sequences b(n) = A372000(n), c(n) = A034386(n), s(n) = A371907(n), t(n) = A371906(n), and v(n) = A357215(n) = s(n) + t(n). Column A represents prime factors of a(n), B same of b(n), while column S (at bottom) shows powers of 2 that sum to s(n), with T same for t(n). P(n) = A002110(n).
%e [A] [B] 11
%e n 2357 a(n) 235713 b(n) c(n) s(n) t(n) v(n)
%e --------------------------------------------------------
%e 1 . 1 . 1 P(0) 0 0 2^0-1
%e 2 . 1 x 2 P(1) 0 1 2^1-1
%e 3 . 1 xx 6 P(2) 0 3 2^2-1
%e 4 x 2 .x 3 P(2) 1 2 2^2-1
%e 5 x 2 .xx 15 P(3) 1 6 2^3-1
%e 6 .x 3 x.x 10 P(3) 2 5 2^3-1
%e 7 .x 3 x.xx 70 P(4) 2 13 2^4-1
%e 8 xx 6 ..xx 35 P(4) 3 12 2^4-1
%e 9 x 2 .xxx 105 P(4) 1 14 2^4-1
%e 10 ..x 5 xx.x 42 P(4) 4 11 2^4-1
%e 11 ..x 5 xx.xx 462 P(5) 4 27 2^5-1
%e 12 xxx 30 ...xx 77 P(5) 7 24 2^5-1
%e 13 xxx 30 ...xxx 1001 P(6) 7 56 2^6-1
%e 14 .xxx 105 x...xx 286 P(6) 14 49 2^6-1
%e 15 ...x 7 xxx.xx 4290 P(6) 8 55 2^6-1
%e 16 x..x 14 .xx.xx 2145 P(6) 9 54 2^6-1
%e --------------------------------------------------------
%e 0123 012345
%e [S] 2^k [T] 2^k
%t Table[Times @@ Select[Prime@ Range@ PrimePi[n], EvenQ@ Quotient[n, #] &], {n, 51}] (* or *)
%t Table[Product[Prime[i], {j, PrimePi[n]}, {i, 1 + PrimePi[Floor[n/(2 j + 1)]], PrimePi[Floor[n/(2 j)]]}], {n, 51}]
%o (PARI) a(n) = my(vp=primes([1, n])); vecprod(select(x->(((n\x) % 2)==0), vp)); \\ _Michel Marcus_, Apr 30 2024
%Y Cf. A005117, A034386, A371907, A372000.
%K nonn,easy
%O 1,4
%A _Michael De Vlieger_, Apr 17 2024
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