A357215 -id:A357215 - cp's OEIS frontend

A048656 a(n) is the number of unitary (and also of squarefree) divisors of n!.

1, 2, 4, 4, 8, 8, 16, 16, 16, 16, 32, 32, 64, 64, 64, 64, 128, 128, 256, 256, 256, 256, 512, 512, 512, 512, 512, 512, 1024, 1024, 2048, 2048, 2048, 2048, 2048, 2048, 4096, 4096, 4096, 4096, 8192, 8192, 16384, 16384, 16384, 16384, 32768, 32768, 32768, 32768

Offset: 1

Labos Elemer

nonn

Let K(n) be the field that is generated over the rationals Q by adjoining the square roots of the numbers 1,2,3,...,n, i.e., K(n) = Q(sqrt(1),sqrt(2),...,sqrt(n)); a(n) is the degree of this field over Q. - Avi Peretz (njk(AT)netvision.net.il), Mar 20 2001
For n>1, a(n) is the number of ways n! can be expressed as the product of two coprime integers p and q such that 0 < p/q < 1, if negative integers are considered as well. This is the answer to the 2nd problem of the International Mathematical Olympiad 2001. Example, for n = 3, the a(3) = 4 products are 3! = (-2)*(-3) = (-1)*(-6) = 1*6 = 2*3. - Bernard Schott, Jan 21 2021
a(n) = number of subsets S of {1,2,...,n} such that every number in S is a prime. - Clark Kimberling, Sep 17 2022

			For n = 7, n! = 5040 = 16*9*5*7 with 4 distinct prime factors, so a(7) = A034444(7!) = 16.
The subsets S of {1, 2, 3, 4} such that every number in S is a prime are these: {}, {2}, {3}, {2, 3}; thus, a(4) = 4. - _Clark Kimberling_, Sep 17 2022

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
International Mathematical Olympiad 2001, Hong Kong Preliminary Selection Contest, Problem 2.
Index to divisibility sequences.
Index to sequences related to Olympiads.

Cf. A000720, A001221, A034444, A357214, A357215.

Cf. A098882, A098990.

Table[2^PrimePi[n], {n, 1, 70}] (* Clark Kimberling, Sep 17 2022 *)

a(n)=2^primepi(n) \\ Charles R Greathouse IV, Apr 07 2012

A001221(n!) = A000720(n) so a(n) = A034444(n!) = 2^A000720(n).

Sum_{n>=1} 1/a(n) = A098882 + 1 = A098990 - 1. - Amiram Eldar, Mar 13 2025

A357214 a(n) = number of subsets S of {1, 2, ..., n} such that every number in S is a composite.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 8, 16, 32, 32, 64, 64, 128, 256, 512, 512, 1024, 1024, 2048, 4096, 8192, 8192, 16384, 32768, 65536, 131072, 262144, 262144, 524288, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 16777216, 33554432, 67108864, 134217728, 134217728

Offset: 1

Clark Kimberling, Sep 24 2022

			The subsets S of {1,2,3,4,5,6} such that every number in S is a composite are {}, {4}, {6}, and {4,6}, so a(6) = 4.

Cf. A000720, A048656, A089819, A357215.

(1/2) Table[2^(n - PrimePi[n]), {n, 50}]

a(n) = 1 << (n-primepi(n)-1); \\ Kevin Ryde, Sep 24 2022

from sympy import primepi
def a(n): return 2**(n-primepi(n)-1)
print([a(n) for n in range(1, 42)]) # Michael S. Branicky, Sep 24 2022

a(n) = (1/2)*(2^(n - A000720(n))).

a(n) = 2^A065855(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

A048656 a(n) is the number of unitary (and also of squarefree) divisors of n!.

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A357214 a(n) = number of subsets S of {1, 2, ..., n} such that every number in S is a composite.

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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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