A371906 -id:A371906 - cp's OEIS frontend

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

Michael De Vlieger, Apr 15 2024

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)).

			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

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.

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

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}]

a(n) = vecprod(select(x->((n\x) % 2), primes([1, n]))); \\ Michel Marcus, Apr 16 2024

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

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).

A371907 a(n) = sum of 2^(k-1) such that floor(n/prime(k)) is even.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 3, 1, 4, 4, 7, 7, 14, 8, 9, 9, 10, 10, 15, 5, 20, 20, 23, 19, 50, 48, 57, 57, 62, 62, 63, 45, 108, 96, 99, 99, 226, 192, 197, 197, 206, 206, 223, 217, 472, 472, 475, 467, 470, 404, 437, 437, 438, 418, 427, 297, 808, 808, 815, 815, 1838, 1828

Offset: 1

Michael De Vlieger, Apr 17 2024

This is a transform of A372007(n) = s(n). Write the prime indices of k factors prime(k) | s(n) instead as 2^(k-1) and take the sum for all primes p | s(n). Hence, s(14) = 105 = 3*5*7 becomes a(14) = 2^1 + 2^2 + 2^3 = 2 + 4 + 8 = 14.

			a(1) = 0 since n = 1 is the empty product.
a(2) = 0 since for n = prime(1) = 2, floor(2/2) = 1 is odd. Therefore a(2) = 0.
a(3) = 0 since for n = 3 and prime(1) = 2, floor(3/2) = 1 is odd, and for prime(2) = 3, floor(3/3) = 1 is odd. Hence a(3) = 0.
a(4) = 1 since for n = 4 and prime(1) = 2, floor(4/2) = 2 is even, but for prime(2) = 3, floor(4/3) = 1 is odd. Therefore, a(4) = 2^(1-1) = 1.
a(8) = 1 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^(1-1) + 2^(2-1) = 1 + 2 = 3, etc.
Table relating a(n) with b(n), s(n), and t(n), diagramming powers of 2 with "x" that sum to a(n) or b(n), or prime factors with "x" that produce s(n) or t(n). Sequences s(n) = A372007(n), t(n) = A372000(n), c(n) = A034386(n), b(n) = A371906(n), and c(n) = A357215(n) = a(n) + b(n). Column A (at top) shows powers of 2 that sum to a(n), with B same for b(n), while column S represents prime factors of s(n), T same of t(n).
      [A] 2^k     [B] 2^k
   n   0123  a(n)  012345  b(n)   c(n)   s(n)   t(n)  v(n)
  --------------------------------------------------------
   1   .       0   .         0   2^0-1     1      1   P(0)
   2   .       0   x         1   2^1-1     1      2   P(1)
   3   .       0   xx        3   2^2-1     1      6   P(2)
   4   x       1   .x        2   2^2-1     2      3   P(2)
   5   x       1   .xx       6   2^3-1     2     15   P(3)
   6   .x      2   x.x       5   2^3-1     3     10   P(3)
   7   .x      2   x.xx     13   2^4-1     3     70   P(4)
   8   xx      3   ..xx     12   2^4-1     6     35   P(4)
   9   x       1   .xxx     14   2^4-1     2    105   P(4)
  10   ..x     4   xx.x     11   2^4-1     5     42   P(4)
  11   ..x     4   xx.xx    27   2^5-1     5    462   P(5)
  12   xxx     7   ...xx    24   2^5-1    30     77   P(5)
  13   xxx     7   ...xxx   56   2^6-1    30   1001   P(6)
  14   .xxx   14   x...xx   49   2^6-1   105    286   P(6)
  15   ...x    8   xxx.xx   55   2^6-1     7   4290   P(6)
  16   x..x    9   .xx.xx   54   2^6-1    14   2145   P(6)
  --------------------------------------------------------
       2357        [T] 11
       [S]         235713

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, "Tiger Stripe" Factors of Primorials, ResearchGate, 2024.
Plot powers 2^(i-1) that sum to a(n) at (x,y) = (n,i) for n = 1..2048, 12X vertical exaggeration.

Cf. A357215, A371906, A372007.

Table[Total[2^(-1 + Select[Range@ PrimePi[n], EvenQ@ Quotient[n, Prime[#]] &])], {n, 50}]

a(n) = my(vp=primes([1, n])); vecsum(apply(x->2^(x-1), Vec(select(x->(((n\x) % 2)==0), vp, 1)))); \\ Michel Marcus, Apr 30 2024

a(n) = A357215(n) - A371906(n).

A372007 a(n) = product of those prime(k) such that floor(n/prime(k)) is even.

Original entry on oeis.org

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, 429, 143, 2002, 2002, 15015, 15015, 30030, 910, 7735, 221, 1326, 1326, 12597, 323, 3230, 3230, 33915, 33915, 746130, 49742, 572033, 572033, 3432198, 490314, 1225785, 24035

Offset: 1

Michael De Vlieger, Apr 17 2024

The only primes in the sequence are 2, 3, 5, and 7.

			a(1) = 1 since n = 1 is the empty product.
a(2) = 1 since for n = 2, floor(n/p) = floor(2/2) = 1 is odd.
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.
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.
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.
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.
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).
       [A]          [B] 11
   n   2357   a(n)  235713    b(n)  c(n)  s(n) t(n)   v(n)
  --------------------------------------------------------
   1   .        1   .           1   P(0)    0    0   2^0-1
   2   .        1   x           2   P(1)    0    1   2^1-1
   3   .        1   xx          6   P(2)    0    3   2^2-1
   4   x        2   .x          3   P(2)    1    2   2^2-1
   5   x        2   .xx        15   P(3)    1    6   2^3-1
   6   .x       3   x.x        10   P(3)    2    5   2^3-1
   7   .x       3   x.xx       70   P(4)    2   13   2^4-1
   8   xx       6   ..xx       35   P(4)    3   12   2^4-1
   9   x        2   .xxx      105   P(4)    1   14   2^4-1
  10   ..x      5   xx.x       42   P(4)    4   11   2^4-1
  11   ..x      5   xx.xx     462   P(5)    4   27   2^5-1
  12   xxx     30   ...xx      77   P(5)    7   24   2^5-1
  13   xxx     30   ...xxx   1001   P(6)    7   56   2^6-1
  14   .xxx   105   x...xx    286   P(6)   14   49   2^6-1
  15   ...x     7   xxx.xx   4290   P(6)    8   55   2^6-1
  16   x..x    14   .xx.xx   2145   P(6)    9   54   2^6-1
  --------------------------------------------------------
       0123         012345
     [S] 2^k        [T] 2^k

Michael De Vlieger, Table of n, a(n) for n = 1..7591
Michael De Vlieger, Plot prime(i) | a(n) at (x,y) = (n,i) for n = 1..2048, 12X vertical exaggeration.
Michael De Vlieger, "Tiger Stripe" Factors of Primorials, ResearchGate, 2024.

Cf. A005117, A034386, A371907, A372000.

Table[Times @@ Select[Prime@ Range@ PrimePi[n], EvenQ@ Quotient[n, #] &], {n, 51}] (* or *)
Table[Product[Prime[i], {j, PrimePi[n]}, {i, 1 + PrimePi[Floor[n/(2  j + 1)]], PrimePi[Floor[n/(2  j)]]}], {n, 51}]

a(n) = my(vp=primes([1, n])); vecprod(select(x->(((n\x) % 2)==0), vp)); \\ Michel Marcus, Apr 30 2024

a(n) = A034386(n) / A372000(n).

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).

cp's OEIS Frontend

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

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A371907 a(n) = sum of 2^(k-1) such that floor(n/prime(k)) is even.

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A372007 a(n) = product of those prime(k) such that floor(n/prime(k)) is even.

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Mathematica

PARI

Formula