A372000 a(n) = product of primes p such that floor(n/p) is odd.

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

Author

Michael De Vlieger, Apr 15 2024

Keywords

• nonn
• easy

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)).

Examples

			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
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..3384
• Michael De Vlieger, Plot prime(i) | a(n) at (x,y) = (n,i) for n = 1..2048.

Crossrefs

Cf. A005117, A008336, A034386, A055773, A260850, A371906.

Programs

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}]

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

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).