A048657 -id:A048657 - cp's OEIS frontend

A056171 a(n) = pi(n) - pi(floor(n/2)), where pi is A000720.

0, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3, 2, 2, 2, 3, 3, 4, 4, 4, 3, 4, 4, 4, 3, 3, 3, 4, 4, 5, 5, 5, 4, 4, 4, 5, 4, 4, 4, 5, 5, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 6, 7, 7, 8, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 10, 9, 9, 9, 9, 9, 10, 10, 10, 9, 10, 10, 10, 9, 9, 9, 10, 10, 10, 10, 10, 9, 9, 9, 10, 10

Offset: 1

Labos Elemer, Jul 27 2000

Also, the number of unitary prime divisors of n!. A prime divisor of n is unitary iff its exponent is 1 in the prime power factorization of n. In general, gcd(p, n/p) = 1 or p. Here we count the cases when gcd(p, n/p) = 1.
A unitary prime divisor of n! is >= n/2, hence their number is pi(n) - pi(n/2). - Peter Luschny, Mar 13 2011
See also the references and links mentioned in A143227. - Jonathan Sondow, Aug 03 2008
From Robert G. Wilson v, Mar 20 2017: (Start)
First occurrence of k is at n = A080359(k).
The last occurrence of k is at n = A080360(k).
The number of times k appears is A080362(k). (End)
Lev Schnirelmann proved that for every n, a(n) > (1/log_2(n))*(n/3 - 4*sqrt(n)) - 1 - (3/2)*log_2(n). - Arkadiusz Wesolowski, Nov 03 2017

			10! = 2^8 * 3^2 * 5^2 * 7. The only unitary prime divisor is 7, so a(10) = 1.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 214.

Daniel Forgues, Table of n, a(n) for n=1..100000
Ethan Berkove and Michael Brilleslyper, Subgraphs of Coprime Graphs on Sets of Consecutive Integers, Integers, Vol. 22 (2022), #A47, see p. 8.

Cf. A000720, A001221, A034444, A048105, A048656, A048657, A056169, A056172.
Cf. A014085, A060715, A104272, A143223, A143224, A143225, A143226, A143227.
Cf. A080359, A080360, A080362.

A056171 := proc(x)
     numtheory[pi](x)-numtheory[pi](floor(x/2)) ;
end proc:
seq(A056171(n),n=1..130) ; # N. J. A. Sloane, Sep 01 2015
A056171 := n -> nops(select(isprime,[$iquo(n,2)+1..n])):
seq(A056171(i),i=1..98); # Peter Luschny, Mar 13 2011

s=0; Table[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; s, {k,100}]
Table[PrimePi[n]-PrimePi[Floor[n/2]],{n,100}] (* Harvey P. Dale, Sep 01 2015 *)

A056171=n->primepi(n)-primepi(n\2) \\ M. F. Hasler, Dec 31 2016

from sympy import primepi
[primepi(n) - primepi(n//2) for n in range(1,151)] # Indranil Ghosh, Mar 22 2017

[prime_pi(n)-prime_pi(floor(n/2)) for n in range(1,99)] # Stefano Spezia, Apr 22 2025

a(n) = A000720(n) - A056172(n). - Robert G. Wilson v, Apr 09 2017

a(n) = A056169(n!). - Amiram Eldar, Jul 24 2024

Definition simplified by N. J. A. Sloane, Sep 01 2015

A056172 Number of non-unitary prime divisors of n!.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14

Offset: 1

Labos Elemer, Jul 27 2000

A non-unitary prime divisor for n! cannot exceed n/2.

			10! = 2^8 * 3^4 * 5^2 * 7. The non-unitary prime divisors are 2, 3, and 5 because their exponents exceed 1, so a(10) = 3.  The only unitary prime divisor of 10! is 7.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000720, A001221, A034444, A048105, A048656, A048657, A056170, A056171.

with(numtheory); A056172:=n->pi(floor(n/2)); seq(A056172(k),k=1..100); # Wesley Ivan Hurt, Sep 30 2013

Table[PrimePi[Floor[n/2]], {n,100}] (* Wesley Ivan Hurt, Sep 30 2013 *)

a(n) = pi(n/2).
A prime divisor of x is non-unitary iff its exponent is at least 2 in the prime power factorization of x. In general, GCD(p, x/p) = 1 or p. Cases are counted when GCD(p, n/p) > 1.
a(n) = A000720(n) - A056171(n). - Robert G. Wilson v, Apr 09 2017
a(n) = A056170(n!). - Amiram Eldar, Jul 24 2024

Example corrected by Jon E. Schoenfield, Sep 30 2013

A064030 Product of unitary divisors of n!.

Original entry on oeis.org

1, 2, 36, 576, 207360000, 268738560000, 416336312719673760153600000000, 6984964247141514123629140377600000000, 300679807141675805997423113304381849600000000

Offset: 1

Labos Elemer, Sep 13 2001

nonn

			n = 6 has 8 unitary divisors:{16,45,9,80,5,144,1,720}, a(6) = 720^4 = 268738560000

Amiram Eldar, Table of n, a(n) for n = 1..17

Cf. A048656, A048657, A061537, A061538, A000142.

a[n_] := (n!)^(2^(PrimePi[n]-1)); Array[a, 10] (* Amiram Eldar, Jul 16 2019 *)

a(n)=(n!)^(A034444(n!)/2)

A064031 Product of non-unitary divisors of n!.

Original entry on oeis.org

1, 1, 1, 576, 207360000, 26956124946896309452800000000000, 2841003716170671644367609186370356458508919205193278721884160000000000000000000000

Offset: 1

Labos Elemer, Sep 13 2001

nonn

			n=6: 720 has 22 non-unitary divisors: a(6)=720^11=26956124946896309452800000000000

Amiram Eldar, Table of n, a(n) for n = 1..10

Cf. A048656, A048657, A061537, A061538, A000142, A048105, A000005, A034444.

a[n_] := (n!)^(DivisorSigma[0, n!]/2 - 2^(PrimePi[n]-1)); Array[a, 10] (* Amiram Eldar, Jul 16 2019 *)

a(n) = A061538(n!) = (n!)^(A048105(n!)/2) = (n!)^((A000005(n!)-A034444(n!))/2)

cp's OEIS Frontend

A056171 a(n) = pi(n) - pi(floor(n/2)), where pi is A000720.

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A056172 Number of non-unitary prime divisors of n!.

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A064030 Product of unitary divisors of n!.

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A064031 Product of non-unitary divisors of n!.

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