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A372000
a(n) = product of primes p such that floor(n/p) is odd.
4
1, 2, 6, 3, 15, 10, 70, 35, 105, 42, 462, 77, 1001, 286, 4290, 2145, 36465, 24310, 461890, 46189, 969969, 176358, 4056234, 676039, 3380195, 520030, 1560090, 111435, 3231615, 430882, 13357342, 6678671, 220396143, 25928958, 907513530, 151252255, 5596333435, 589087730, 22974421470, 2297442147
OFFSET
1,2
COMMENTS
The only primes in the sequence are 2 and 3.
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.
For n >= 1, A055773(n) | a(n), where A055773(n) = A034386(n) / A034386(floor(n/2)).
LINKS
Michael De Vlieger, Plot prime(i) | a(n) at (x,y) = (n,i) for n = 1..2048.
FORMULA
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).
EXAMPLE
a(1) = 1 since n = 1 is the empty product.
a(2) = 2 since for n = 2, floor(n/p) = floor(2/2) = 1 is odd.
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.
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.
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.
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).
Prime factor
1111
n b(n) 23571379 b(n)
----------------------------
1 1 . 0
2 2 x 1
3 6 xx 3
4 3 .x 2
5 15 .xx 6
6 10 x.x 5
7 70 x.xx 13
8 35 ..xx 12
9 105 .xxx 14
10 42 xx.x 11
11 462 xx.xx 27
12 77 ...xx 24
13 1001 ...xxx 56
14 286 x...xx 49
15 4290 xxx.xx 55
16 2145 .xx.xx 54
17 36465 .xx.xxx 118
18 24310 x.x.xxx 117
19 461890 x.x.xxxx 245
20 46189 ....xxxx 240
----------------------------
01234567
Power of 2
MATHEMATICA
Table[Times @@ Select[Prime@ Range@ PrimePi[n], OddQ@ Quotient[n, #] &], {n, 40}] (* or *)
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}]
PROG
(PARI) a(n) = vecprod(select(x->((n\x) % 2), primes([1, n]))); \\ Michel Marcus, Apr 16 2024
(SageMath)
print([prod(p for p in prime_range(n + 1) if is_odd(n//p)) for n in range(1, 41)])
# Peter Luschny, Apr 16 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michael De Vlieger, Apr 15 2024
STATUS
approved