cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A250269 Primitive part of n! (for n>=1): n! = Product_{d|n} a(d).

Original entry on oeis.org

1, 2, 6, 12, 120, 60, 5040, 1680, 60480, 15120, 39916800, 55440, 6227020800, 8648640, 1816214400, 518918400, 355687428096000, 147026880, 121645100408832000, 55870214400, 1689515283456000, 14079294028800, 25852016738884976640000, 771008958720
Offset: 1

Views

Author

Matthew Vandermast, Dec 16 2014

Keywords

Comments

The title is analogous to the title of A061446.
For any integer sequence a, the sequence b such that b(n) = Product_{d|n} a(d) is a divisibility sequence. Not every divisibility sequence b corresponds to some integer sequence a such that b(n) = Product_{d|n} a(d), however.
This sequence is not itself a divisibility sequence; a(15) does not divide a(30), for example.

Examples

			The divisors of 10 are 1, 2, 5 and 10. 10! = a(1) * a(2) * a(5) * a(10) = 1 * 2 * 120 * 15120 = 3628800.
Between 1 and 10 inclusive, 4 integers are coprime to 10: 1, 3, 7 and 9. Let b(n) = lcm (1...n) = A003418(n), and let [x] denote the floor function. Then:
a(10) = b[10/1] * b[10/3] * b[10/7] * b[10/9]
"   "   = b(10) * b(3) * b(1) * b(1)
"   "   = 2520 * 6 * 1 * 1
"   "   = 15120.

Links

Crossrefs

Cf. A000142, A075071. Subsequence of A250270.
Cf. A000010 (comments on product formulas), A008683.

Programs

  • Mathematica
    Array[Product[(d!)^MoebiusMu[#/d], {d, Divisors[#]}] &, 24] (* Michael De Vlieger, Nov 11 2021 *)
  • PARI
    a(n)={my(r=1);fordiv(n,d,r*=d!^moebius(n/d));r} \\ Joerg Arndt, Jan 18 2015

Formula

a(n) = Product_{i = 1..n, gcd(n, i) = 1} lcm (1..floor(n/i)).
a(n) = Product_{i = 1..floor(n/2), gcd(n, i) = 1} lcm (1..floor(n/i)) (equivalent formula).
a(n) = n! iff n is prime.
a(n) = Product_{d|n} (d!)^moebius(n/d). - Joerg Arndt, Jan 18 2015
a(n) = Product_{k=1..n} (gcd(n,k)!)^(mu(n/gcd(n,k))/phi(n/gcd(n,k))) = Product_{k=1..n} ((n/gcd(n,k))!)^(mu(gcd(n,k))/phi(n/gcd(n,k))) where mu = A008683, phi = A000010. - Richard L. Ollerton, Nov 08 2021

A075422 Primitive numbers n such that the product of factorials of all proper divisors of n does not divide n!.

Original entry on oeis.org

24, 30, 36, 40, 54, 84, 100, 102, 112, 126, 132, 140, 156, 176, 198, 208, 220, 228, 234, 260, 272, 276, 294, 308, 340, 342, 348, 350, 364, 372, 380, 392, 414, 444, 460, 462, 476, 490, 492, 516, 522, 532, 546, 558, 564, 572, 580, 608, 620, 636, 644, 666, 708
Offset: 1

Views

Author

Robert G. Wilson v & Amarnath Murthy, Sep 14 2002

Keywords

Comments

If a number is in the sequence, all of its multiples also meet the criterion, but are not included. This is what the word "primitive" refers to.

Examples

			The product of the factorials of the proper divisors of 24, 1! * 2! * 3! * 4! * 6! * 8! * 12!, is divisible by 2^26 and therefore does not divide 24! (which is divisible by 2^22 only). 24 is the smallest number with this property. - _M. F. Hasler_, Dec 31 2016

Crossrefs

Cf. A075071. See A075460 for the odd terms of this sequence.
See A248693 for the list of all (also non-primitive) terms (and PARI code).

Programs

  • Mathematica
    f[n_] := n!/Apply[Times, Drop[Divisors[n], -1]! ]; a = {}; Do[b = f[n]; If[ !IntegerQ[b], If[ Select[n/a, IntegerQ] == {}, a = Append[a, n]]], {n, 1, 725}]; a

Formula

a(n) appears to be asymptotic to c*n with 12 < c < 15. - Benoit Cloitre, Sep 16 2002

Extensions

Edited by M. F. Hasler, Dec 30 2016

A248693 Numbers k such that the product of factorials of proper divisors of k does not divide k!.

Original entry on oeis.org

24, 30, 36, 40, 48, 54, 60, 72, 80, 84, 90, 96, 100, 102, 108, 112, 120, 126, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240, 252, 260, 264, 270, 272, 276, 280, 288, 294, 300, 306, 308, 312, 320
Offset: 1

Views

Author

Clark Kimberling, Oct 12 2014

Keywords

Comments

The least odd k in this sequence is 1575; see A248694 and A075460.
It seems that the property is satisfied iff v_2(n!) < v_2(P), where v_2 is the 2-adic valuation, and P = product_{d|n, dM. F. Hasler, Dec 30 2016

Examples

			Let Q(n) = n!/(product of the proper divisors of n).  Then Q(n) an integer for n = 1..23, as indicated by the following list of Q(n) for n = 1..24:  1, 2, 6, 12, 120, 60, 5040, 840, 60480, 15120, 39916800, 2310, 6227020800, 8648640, 1816214400, 10810800, 355687428096000, 2042040, 121645100408832000, 116396280, 1689515283456000, 14079294028800, 25852016738884976640000, 7436429/48.

Links

Crossrefs

Cf. A075422 (primitive terms = not a multiple of an earlier term), A248694 (odd terms), A075460 (odd primitive terms), A075071, A027750.

Programs

  • Mathematica
    d[n_] := Drop[Divisors[n], -1]!; p[n_] := Apply[Times, d[n]]
u = Select[Range[25000], ! IntegerQ[#!/p[#]] &]; (* A248693 *)
Select[u, OddQ[#] &]  (* A248694 *)
(* Second program *)
Select[Range@ 320, ! Divisible[#!, Times @@ Map[Factorial, Most@ Divisors@ #]] &] (* Michael De Vlieger, Dec 31 2016 *)
  • PARI
    is(n)=prod(i=2,n-1,i,Mod(n,prod(j=2,-1+#n=divisors(n),n[j]!))) \\ Returns nonzero (actually, Mod(n!,P) where P = product_{d|n, dM. F. Hasler, Dec 30 2016

A075460 Odd primitive numbers such that n! divided by product of factorials of all proper divisors of n is not an integer.

Original entry on oeis.org

1575, 2835, 3465, 4095, 5355, 5775, 5985, 6435, 6615, 6825, 7245, 8085, 9135, 9765, 11655, 12705, 12915, 13545, 14805, 15015, 15435, 16695, 18585, 19215, 19635, 21105, 21945, 22275, 22365, 22995, 23205, 24885, 25245, 25935, 26145, 26565
Offset: 1

Views

Author

Robert G. Wilson v, Sep 16 2002

Keywords

Comments

If a number is in the sequence, then all of its multiples would also meet the criterion, but are not included. This is meant by the word "primitive" in the definition.

Examples

			1575 = 3^2*5^2*7 is in the sequence, because the product of the factorials of its proper divisors { 1, 3, 5, 7, 9, 15, 21, 25, 35, 45, 63, 75, 105, 175, 225, 315, 525 } does not divide 1575!. (For example, the former's 2-adic valuation equals 1588 while the latter's 2-adic valuation equals only 1569.) This is the smallest odd number with this property. - _M. F. Hasler_, Dec 30 2016

Crossrefs

Cf. A075071. The first primitive n's with this property (most of which are even) are in A075422.

Programs

  • Mathematica
    f[n_] := n!/Apply[Times, Drop[Divisors[n], -1]! ]; a = {}; Do[b = f[n]; If[ !IntegerQ[b], If[ Select[n/a, IntegerQ] == {}, Print[n]; a = Append[a, n]]], {n, 1, 28213, 2}]; a

Extensions

Edited by M. F. Hasler, Dec 30 2016
Showing 1-4 of 4 results.

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