A056171 -id:A056171 - cp's OEIS frontend

A056672 Number of unitary and squarefree divisors of n! Also, number of divisors of the special squarefree part of n!, A055773(n).

1, 2, 4, 2, 4, 2, 4, 4, 4, 2, 4, 4, 8, 4, 4, 4, 8, 8, 16, 16, 16, 8, 16, 16, 16, 8, 8, 8, 16, 16, 32, 32, 32, 16, 16, 16, 32, 16, 16, 16, 32, 32, 64, 64, 64, 32, 64, 64, 64, 64, 64, 64, 128, 128, 128, 128, 128, 64, 128, 128, 256, 128, 128, 128, 128, 128, 256, 256, 256, 256

Offset: 1

Labos Elemer, Aug 10 2000

The divisor d=1 is counted here as being free of prime divisors and also unitary.

			n=11: 11! = 2*2*2*2*2*2*2*2*3*3*3*3*5*5*7*11, has 540 divisors, 32 are unitary and 32 are squarefree. Only 4 divisors, {1,7,11,77} have both properties, so a(11)=4.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000142, A007913, A055229, A055231, A055773, A056171.

rad[n_] := Times @@ First /@ FactorInteger[n]; p[n_] := Denominator[n/rad[n]^2]; a[n_] := DivisorSigma[0, p[n!]]; Array[a, 70] (* Amiram Eldar, Sep 22 2019 *)

a(n) = my(f=n!); sumdiv(f, d, issquarefree(d) && (gcd(d, f/d) == 1)); \\ Michel Marcus, Sep 05 2017

a(n) = 1 << (primepi(n) - primepi(n>>1)); \\ Kevin Ryde, Jun 03 2023

a(n) = A000005(A055231(n!)).
a(n) = A000005(A007913(n!)/A055229(n!)).
a(n) = A000005(A055773(n)).
a(n) = 2^A056171(n). - Kevin Ryde, Jun 03 2023

A063955 Sum of the unitary prime divisors of n!.

Original entry on oeis.org

0, 2, 5, 3, 8, 5, 12, 12, 12, 7, 18, 18, 31, 24, 24, 24, 41, 41, 60, 60, 60, 49, 72, 72, 72, 59, 59, 59, 88, 88, 119, 119, 119, 102, 102, 102, 139, 120, 120, 120, 161, 161, 204, 204, 204, 181, 228, 228, 228, 228, 228, 228, 281, 281, 281, 281, 281, 252, 311, 311

Offset: 1

Labos Elemer, Sep 04 2001

			Prime divisors of 20! which have exponent 1 (i.e., unitary prime divisors) are {11, 13, 17, 19}, so a(20) = 11 + 13 + 17 + 19= 60. (The sum of all its prime divisors (unitary and non-unitary) is A034387(20).)

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)

Cf. A010051, A034387, A034444, A056169, A056170, A056171, A056172, A063956.

a:= n-> add(`if`(i[2]=1, i[1], 0), i=ifactors(n!)[2]):
seq(a(n), n=1..60);  # Alois P. Heinz, Jun 24 2018

a[n_] := Select[FactorInteger[n!], #[[2]] == 1&][[All, 1]] // Total;
Array[a, 60] (* Jean-François Alcover, Jan 01 2022 *)

a(n) = my(f=factor(n!)~); sum(i=1, length(f), if (f[2, i]==1, f[1, i])); \\ Harry J. Smith, Sep 04 2009

a(n) = Sum_{k=floor(n/2)+1..n} k*c(k), where c is the prime characteristic (A010051). - Wesley Ivan Hurt, Dec 23 2023

a(n) = A063956(n!). - Amiram Eldar, Jul 24 2024

A063960 Sum of non-unitary prime divisors of n!: sum of those prime divisors for which the exponent in the prime factorization exceeds 1.

Original entry on oeis.org

0, 0, 0, 2, 2, 5, 5, 5, 5, 10, 10, 10, 10, 17, 17, 17, 17, 17, 17, 17, 17, 28, 28, 28, 28, 41, 41, 41, 41, 41, 41, 41, 41, 58, 58, 58, 58, 77, 77, 77, 77, 77, 77, 77, 77, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 129, 129, 129, 129, 160, 160, 160, 160

Offset: 1

Labos Elemer, Sep 04 2001

Sum of the prime numbers among the smallest parts of the partitions of n into two parts. For example, a(8)=5; the partitions of 8 into two parts are (7,1), (6,2), (5,3) and (4,4). The prime numbers among the smallest parts are 2 and 3, so 2 + 3 = 5. - Wesley Ivan Hurt, Nov 01 2017

Number of distinct rectangles with integer length and prime width such that L + W = n, W <= L. For a(14)=17; the rectangles are 2 X 12, 3 X 11, 5 X 9, and 7 X 7. The sum of the lengths are then 2+3+5+7 = 17. - Wesley Ivan Hurt, Nov 08 2017

			20! = (2^18)*(3^8)*(5^4)*(7^2)*11*13*17*19, the non-unitary prime divisors are {2, 3, 5, 7}, so a(20) = 2 + 3 + 5 + 7 = 17.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A010051, A034387, A034444, A056169, A056170, A056171, A056172, A063958.

seq(add(j, j=select(isprime, [$1..iquo(n,2)])), n=1..65); # Peter Luschny, Nov 28 2022

Join[{0,0,0},Table[Total[Transpose[Select[FactorInteger[n!], Last[#]>1&]][[1]]],{n,4,70}]] (* Harvey P. Dale, Jun 19 2013 *)

{ for (n=1, 1000, f=factor(n!)~; a=0; for (i=1, length(f), if (f[2, i]>1, a+=f[1, i])); write("b063960.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 04 2009

a(n) = Sum_{i=1..floor(n/2)} i * A010051(i). - Wesley Ivan Hurt, Oct 31 2017
a(n) = A034387(floor(n/2)) for n >= 2. - Georg Fischer, Nov 28 2022
a(n) = A063958(n!). - Amiram Eldar, Jul 24 2024

A064028 Sum of the unitary divisors of n!.

Original entry on oeis.org

1, 3, 12, 36, 216, 1020, 8160, 61920, 507744, 4383392, 52600704, 624249600, 8739494400, 109190390400, 1583122968000, 25318378008000, 455730804144000, 8193040840252800, 163860816805056000, 3256371347261760000, 67204676251838361600, 1366492477414792734720

Offset: 1

Labos Elemer, Sep 11 2001

nonn

			n=6, 6! = 720, sum of the 8 unitary ones of its 30 divisors is 1020, a(6) = 720+1+16+45+9+80+5+144 = 1020.

Amiram Eldar, Table of n, a(n) for n = 1..450
Charles R. Wall, Problem H-374, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 22, No. 3 (1984), p. 280; Bounds of Joy, Solution to Problem H-374 by the proposer, ibid., Vol. 24, No. 2 (1986), p. 188.

Cf. A034448, A046656, A056657, A056171, A056172, A000203, A000142, A062569.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma/@ (Range[17]!) (* Amiram Eldar, Jun 23 2019 *)

valp(n,p)=my(s); while(n\=p, s+=n); s
a(n)=my(s=1); forprime(p=2,n, s*=p^valp(n,p)+1); s \\ Charles R Greathouse IV, Jan 26 2023

a(n) = usigma(n!) = A034448(A000142(n)).

a(n)/n! <= 2 (while usigma(n)/n and sigma(n!)/n! are unbounded; Wall, 1984). - Amiram Eldar, Feb 08 2022

A338363 a(n) = n + pi(n) - pi(floor(n/2)), where pi = A000720.

Original entry on oeis.org

1, 3, 5, 5, 7, 7, 9, 10, 11, 11, 13, 14, 16, 16, 17, 18, 20, 21, 23, 24, 25, 25, 27, 28, 29, 29, 30, 31, 33, 34, 36, 37, 38, 38, 39, 40, 42, 42, 43, 44, 46, 47, 49, 50, 51, 51, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 64, 66, 67, 69, 69, 70, 71, 72, 73, 75, 76, 77, 78, 80, 81

Offset: 1

N. J. A. Sloane, Nov 04 2020

nonn

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A000720, A056171, A095117, A282029, A283312, A338362.

A338363[n_] := PrimePi[n] - PrimePi[n/2] + n;
Array[A338363, 100] (* Paolo Xausa, Jan 30 2024 *)

a(n) = n + primepi(n) - primepi(n\2); \\ Michel Marcus, Nov 04 2020

a(n) = n + A056171(n). - Alois P. Heinz, Nov 04 2020

A064138 Sum of non-unitary divisors of n!.

Original entry on oeis.org

0, 0, 0, 24, 144, 1398, 11184, 97200, 973296, 10950696, 131408352, 1593191808, 22304685312, 333297226080, 5103130001760, 81686161277280, 1470350902991040, 26490792085668288, 529815841713365760, 10635027891469974720

Offset: 1

Labos Elemer, Sep 11 2001

nonn

			For n = 6, 6! = 720, the sum of its 30 divisors is 2418, the sum of the 8 unitary divisors is 1020, so the remaining 22 divisors give a(6) = 1398.

Harry J. Smith, Table of n, a(n) for n = 1..100

Cf. A034448, A048105, A046656, A056657, A056171, A056172, A000203, A000142, A062569, A063955, A063960.

f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := p^e + 1; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n!]) - Times @@ f2 @@@ fct; a[1] = 0; Array[a, 20] (* Amiram Eldar, Apr 01 2024 *)

usigma(n)= { local(f,s=1); f=factor(n); for(i=1, matsize(f)[1], s*=1 + f[i, 1]^f[i, 2]); return(s) }
{ n=0; f=1; for (n=1, 100, f*=n; write("b064138.txt", n, " ", sigma(f) - usigma(f)); ) } \\ Harry J. Smith, Sep 08 2009

a(n) = sigma(n!) - usigma(n!) = A000203(n!) - A034448(A000142(n)) = A062569(n) - A034448(n!) = A048105(n!).

Term corrected and more terms added by Harry J. Smith, Sep 08 2009

A080361 a(n) is the difference between the largest and the smallest positive integers x such that the number of unitary-prime-divisors of x! equals n. Same as the difference between the largest and the smallest positive integers x such that the number of primes in (x/2,x] equals n.

Original entry on oeis.org

8, 13, 15, 21, 15, 15, 13, 9, 25, 27, 5, 23, 39, 37, 27, 21, 7, 45, 37, 39, 39, 1, 21, 17, 11, 35, 27, 53, 35, 17, 19, 27, 29, 9, 11, 11, 5, 21, 43, 27, 11, 69, 61, 63, 15, 5, 1, 5, 33, 29, 27, 5, 5, 17, 57, 43, 47, 17, 25, 51, 47, 11, 3, 25, 27, 23, 77, 57, 35, 19, 29, 37, 27, 23, 9

Offset: 1

Labos Elemer, Feb 21 2003

nonn

J. Sondow, Ramanujan Prime in MathWorld [From _Jonathan Sondow_, Aug 10 2008]

Cf. A056171, A056169, A000720, A000142, A080359, A080360.

Cf. A104272 Ramanujan primes. [From Jonathan Sondow, Aug 10 2008]

a(n)=Max{x; Pi[x]-Pi[x/2]=n}-Min{x; Pi[x]-Pi[x/2]=n}=A080360[n]-A080359[n].

Definition corrected by Jonathan Sondow, Aug 10 2008

Typo in formula corrected by Daniel Forgues, Aug 06 2009

A080362 a(n) is the number of positive integers x such that the number of unitary-prime-divisors of x! equals n. Same as the number of positive integers x such that the number of primes in (x/2,x] equals n.

Original entry on oeis.org

4, 10, 7, 14, 7, 10, 12, 5, 14, 16, 3, 10, 18, 16, 15, 11, 7, 16, 19, 14, 9, 2, 14, 14, 8, 11, 18, 19, 24, 10, 14, 16, 20, 10, 11, 3, 6, 13, 18, 21, 9, 31, 37, 10, 15, 6, 2, 6, 21, 12, 7, 6, 6, 16, 15, 34, 14, 10, 15, 29, 22, 9, 4, 14, 16, 17, 25, 36, 12, 15, 13, 19, 19, 8, 10, 5, 12

Offset: 1

Labos Elemer, Feb 21 2003

nonn

			n=5,a(5)=7 because in 7 factorials 5 primes arise with exponent 1: in factorials of 31,32,33,37,41,46; e.g. in 37! these are {19,23,29,31,37}, or 10 numbers x, exist such ones that number of unitary prime divisors of x! equals 2, namely in factorials of {3,5,7,8,9,11,12,13,15,16}.

J. Sondow, Ramanujan Prime in MathWorld [From _Jonathan Sondow_, Aug 10 2008]

Cf. A056171, A056169, A000720, A000142, A080359, A080360, A080361.

Cf. A104272 Ramanujan primes. [From Jonathan Sondow, Aug 10 2008]

a(n)=Card{x; Pi[x]-Pi[x/2]=n}, where Pi()=A000720().

Definition corrected by Jonathan Sondow, Aug 10 2008

A129843 a(n) = number of positive integers that are <= n and are coprime to n!! (n!! = A006882(n)).

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 3, 3, 4, 2, 4, 3, 4, 3, 4, 3, 5, 4, 5, 5, 5, 4, 5, 5, 5, 4, 5, 4, 5, 5, 5, 6, 6, 5, 6, 5, 6, 5, 6, 5, 6, 6, 6, 7, 6, 6, 6, 7, 6, 7, 6, 7, 6, 8, 6, 8, 6, 7, 6, 8, 6, 8, 6, 8, 7, 8, 7, 9, 7, 9, 7, 10, 7, 10, 7, 10, 7, 10, 7, 11

Offset: 1

Leroy Quet, Jun 03 2007

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A006882, A056171, A070939.

f:= proc(n) local t;
     if n::odd then ilog2(n)+1
     else 1+numtheory:-pi(n) - numtheory:-pi(n/2)
     fi
end proc:
f(2):= 1:
map(f, [$1..100]); # Robert Israel, Dec 08 2022

a[n_]:=Module[{},co=0;For[i=1,iStefan Steinerberger, Jun 05 2007 *)
Table[Total[Boole[CoprimeQ[n!!,Range[n]]]],{n,80}] (* Harvey P. Dale, Dec 12 2022 *)

From Robert Israel, Dec 08 2022: (Start)
If n is odd, a(n) = A070939(n).
If n > 2 is even, a(n) = 1 + A056171(n). (End)

More terms from Stefan Steinerberger, Jun 05 2007

A182910 Number of unitary prime divisors of the swinging factorial (A056040) n$ = n! / floor(n/2)!^2.

Original entry on oeis.org

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

Offset: 0

Peter Luschny, Mar 14 2011

nonn

A prime divisor of n is unitary iff its exponent is 1 in the prime power factorization of n. A unitary prime divisor of the swinging factorial n$ can be smaller than n/2. For n >= 30 the swinging factorial has more unitary prime divisors than the factorial and it never has fewer unitary prime divisors. Thus a(n) >= PrimePi(n) - PrimePi(n/2).

			16$ = 2*3*3*5*11*13. So 16$ has one non-unitary prime divisor and a(16) = 4.

Cf. A056171.

UnitaryPrimeDivisor := proc(f,n) local k, F; F := f(n):
add(`if`(igcd(iquo(F,k),k)=1,1,0),k=numtheory[factorset](F)) end;
A056040 := n -> n!/iquo(n,2)!^2;
A182910 := n -> UnitaryPrimeDivisor(A056040,n);
seq(A182910(i), i=1..LEN);

Table[Function[m, If[m == 1, 0, Count[FactorInteger[m][[All, -1]], 1]]][n!/Floor[n/2]!^2], {n, 0, 67}] (* Michael De Vlieger, Aug 02 2017 *)

from sympy import factorint, factorial
def a056169(n): return 0 if n==1 else sum(1 for i in factorint(n).values() if i==1)
def a056040(n): return factorial(n)//factorial(n//2)**2
def a(n): return a056169(a056040(n))
print([a(n) for n in range(68)]) # Indranil Ghosh, Aug 02 2017

cp's OEIS Frontend

A056672 Number of unitary and squarefree divisors of n! Also, number of divisors of the special squarefree part of n!, A055773(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A063955 Sum of the unitary prime divisors of n!.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A063960 Sum of non-unitary prime divisors of n!: sum of those prime divisors for which the exponent in the prime factorization exceeds 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A064028 Sum of the unitary divisors of n!.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A338363 a(n) = n + pi(n) - pi(floor(n/2)), where pi = A000720.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A064138 Sum of non-unitary divisors of n!.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A080361 a(n) is the difference between the largest and the smallest positive integers x such that the number of unitary-prime-divisors of x! equals n. Same as the difference between the largest and the smallest positive integers x such that the number of primes in (x/2,x] equals n.

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions