A055067 -id:A055067 - cp's OEIS frontend

A280713 Partial sums of A055067 where A055067(n) is the product of non-divisors of n.

1, 2, 4, 7, 31, 51, 771, 1401, 14841, 51129, 3679929, 3957129, 482958729, 927745929, 6739632009, 27172044009, 20949961932009, 22047762636009, 6424421468364009, 6728534219386089, 122581010799226089, 2444896564058746089, 1126445624341666426089

Offset: 1

Jaroslav Krizek, Jan 07 2017

nonn

A055067(n) = the product of non-divisors of n.

Paolo Xausa, Table of n, a(n) for n = 1..400

Cf. A055067, A280714.

[&+[Factorial(k) / &*[d: d in Divisors(k)]: k in [1..n]]: n in [1..100]];

Accumulate[Array[#!/Times@@Divisors[#] &, 30]]

a(n) = Sum_{i=1..n} A055067(i).

A280714 Partial products of A055067.

Original entry on oeis.org

1, 1, 2, 6, 144, 2880, 2073600, 1306368000, 17557585920000, 637129677864960000, 2312016175036366848000000, 640890883720080890265600000000, 306987758727332698566646824960000000000, 136544225638605874463902854662848512000000000000

Offset: 1

Jaroslav Krizek, Jan 07 2017

nonn

A055067(n) = the product of non-divisors of n.

Cf. A000178, A055067, A092143, A280713.

[&*[Factorial(k) / &*[d: d in Divisors(k)]: k in [1..n]]: n in [1..100]];

FoldList[#1 #2 &, Table[Times @@ Complement[Range@ n, Divisors@ n], {n, 14}]] (* Michael De Vlieger, Jan 09 2017 *)

a(n) = Product_{i=1..n} A055067(i).

a(n) = A000178(n)/A092143(n). - Amiram Eldar, Aug 16 2025

A173540 Triangle read by rows in which row n lists the proper nondivisors of n, or zero if n <= 2.

Original entry on oeis.org

0, 0, 2, 3, 2, 3, 4, 4, 5, 2, 3, 4, 5, 6, 3, 5, 6, 7, 2, 4, 5, 6, 7, 8, 3, 4, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 9, 10, 5, 7, 8, 9, 10, 11, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 2, 3, 4, 5, 6, 7

Offset: 1

Omar E. Pol, May 24 2010

Define "proper nondivisors of n" as the positive numbers less than n that do not divide n.
Note that a(1) = 0 and a(2) = 0, by convention.
Row sums give A024816.
Row products give A055067, except the first two rows. - Reinhard Zumkeller, Feb 06 2012
T(n,1) = A199968(n). - Reinhard Zumkeller, Oct 02 2015
The n-th row has A049820(n) terms. - Michel Marcus, Dec 23 2015

			If written as a triangle:
  0;
  0;
  2;
  3;
  2, 3, 4;
  4, 5;
  2, 3, 4, 5,  6;
  3, 5, 6, 7;
  2, 4, 5, 6,  7,  8;
  3, 4, 6, 7,  8,  9;
  2, 3, 4, 5,  6,  7,  8,  9, 10;
  5, 7, 8, 9, 10, 11;
  2, 3, 4, 5,  6,  7,  8,  9, 10, 11, 12;
  3, 4, 5, 6,  8,  9, 10, 11, 12, 13;
  2, 4, 6, 7,  8,  9, 10, 11, 12, 13, 14;
  3, 5, 6, 7,  9, 10, 11, 12, 13, 14, 15;

Reinhard Zumkeller, Rows n=1..150 of triangle, flattened

Cf. A027750, A049820, A177235.

Cf. A199968, A024816 (row sums).

a173540 n k = a173540_row n !! (k-1)
a173540_row n = a173540_tabf !! (n-1)
a173540_tabf = [0] : [0] : map
               (\v -> [w | w <- [2 .. v - 1], mod v w > 0]) [3..]
-- Reinhard Zumkeller, Oct 02 2015, Feb 06 2012

Join[{0, 0}, Flatten[Table[Complement[Range[n], Divisors[n]], {n, 1, 20}]]] (* Geoffrey Critzer, Dec 13 2014 *)

A067391 a(n) is the least common multiple of numbers in {1,2,3,...,n-1} which do not divide n.

Original entry on oeis.org

1, 1, 2, 3, 12, 20, 60, 210, 840, 504, 2520, 27720, 27720, 51480, 360360, 180180, 720720, 4084080, 12252240, 232792560, 232792560, 21162960, 232792560, 5354228880, 5354228880, 2059318800, 26771144400, 80313433200, 80313433200

Offset: 1

Labos Elemer, Jan 22 2002

nonn

			For n=10: non-divisors = {3,4,6,7,8,9}, lcm(3,4,6,7,8,9) = 8*9*7 = 504 = a(10).
For n=18, a(18) = lcm(4,5,7,8,10,11,12,13,14,15,16,17) = 4084080.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A049820 [count], A007978 [min], A024816 [sum], A055067 [product].
Cf. A003418, A066574, A038610.
Cf. A173540.

a067391 n | n <= 2    = 1
          | otherwise = foldl lcm 1 $ a173540_row n
-- Reinhard Zumkeller, Apr 04 2012

a[n_] := LCM@@Select[Range[1, n-1], Mod[n, # ]!=0& ]
Join[{1,1},Table[LCM@@Complement[Range[n],Divisors[n]],{n,3,30}]] (* Harvey P. Dale, Mar 27 2013 *)

Let f(n) = lcm(1, 2, ..., n-1) = A003418(n-1). If n = 2*p^k for some prime p, then a(n) = f(n)/p; otherwise a(n) = f(n).

A070251 Unrelated-factorial numbers: product of numbers unrelated to n (numbers which have a common divisor with n but do not divide n).

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 6, 6, 192, 1, 720, 1, 23040, 6480, 10080, 1, 12902400, 1, 34836480, 2449440, 1857945600, 1, 50295168000, 3000, 980995276800, 9797760, 9564703948800, 1, 1518492398911488000, 1, 41845579776000, 1571364748800

Offset: 1

Amarnath Murthy, May 05 2002

nonn

a(p) = 1 if p is a prime. 4 is the only composite number such that a(4) = 1.
From Michael De Vlieger, Jan 15 2025: (Start)
Conjecture: a(n) is in A055932, and also often in A025487.
Conjectures: a(6) = 4 is likely the only powerful term that exceeds 1. a(8) = a(9) = 6 is likely the only squarefree number exceeding 1 that appears in the sequence.
Conjecture: For n = 2*p, p > 3, gcd(n, a(n)) > 1, rad(n) does not divide a(n), and rad(a(n)) does not divide n, since gpf(n) does not divide a(n). For composite n > 9 not an even squarefree semiprime, n divides a(n). (End)

			Table of a(n) for composite n <= 30, showing prime power decomposition by listing exponents of primes shown in the column heads:
   n                   a(n)   2  3  5  7 11 13
  ---------------------------------------------
   6                     4    2
   8                     6    1, 1
   9                     6    1, 1
  10                   192    6, 1
  12                   720    4, 2, 1
  14                 23040    9, 2, 1
  15                  6480    4, 4, 1
  16                 10080    5, 2, 1, 1
  18              12902400   13, 2, 2, 1
  20              34836480   12, 5, 1, 1
  21               2449440    5, 7, 1, 1
  22            1857945600   17, 4, 2, 1
  24           50295168000   10, 6, 3, 2, 1
  25                  3000    3, 1, 3
  26          980995276800   21, 5, 2, 1, 1
  27               9797760    7, 7, 1, 1
  28         9564703948800   19, 6, 2, 1, 1, 1
  30   1518492398911488000   22,10, 3, 3, 1, 1

Michael De Vlieger, Table of n, a(n) for n = 1..629
Michael De Vlieger, Log log scatterplot of log_10(a(n)), n = 1..10000, where gold represents proper prime power n, green represents squarefree composite n, bright green represents n in A002110, blue represents n in A332785, and purple represents powerful n that are not prime powers.
Michael De Vlieger, Plot p^m | a(n) at (x,y) = (n, pi(p)), n = 1..2048, with a color function showing m = 1 in black, m = 2 in red, ..., maximum m in magenta.

Cf. A045763, A066760, A133995.

A070251 := proc(n) local i;
remove(k->igcd(n,k)=1,{$1..n}); numtheory[divisors](n);
mul(i, i = %% minus % ) end:   # Peter Luschny, Oct 11 2011

a[n_] := Times @@ Complement[Range[n], Divisors[n]]/Times @@ Select[ Range[n], CoprimeQ[n, #]&];
Array[a, 33] (* Jean-François Alcover, Jun 03 2019 *)

a(n) = A055067(n)/A001783(n). - Vladeta Jovovic, May 06 2002
From Michael De Vlieger, Jan 15 2025: (Start)
Let S(n) = { k < n : 1 < gcd(k,n) < k } = row n of A133995 for composite n > 4.
a(n) = product of S(n).
pi(gpf(a(n))) <= pi(n/lpf(n)), i.e., A000720(A006530(a(n))) <= A000720(n/A020639(n)). (End)

More terms from Vladeta Jovovic, May 06 2002

A072046 Greatest common divisor of product of divisors of n and product of non-divisors < n.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 2, 3, 4, 1, 144, 1, 4, 45, 32, 1, 72, 1, 320, 63, 4, 1, 82944, 125, 4, 729, 448, 1, 162000, 1, 32768, 99, 4, 1225, 3359232, 1, 4, 117, 2560000, 1, 63504, 1, 704, 91125, 4, 1, 254803968, 343, 125000, 153, 832, 1, 8503056, 3025, 9834496, 171, 4, 1

Offset: 1

Reinhard Zumkeller, Jul 29 2002

			a(12) = GCD(A007955(12), A055067(12)) = GCD(1*2*3*4*6*12,5*7*8*9*10*11) = GCD(1728,277200) = 144;
a(13) = GCD(A007955(13), A055067(13)) = GCD(1*13,2*3*4*5*6*7*8*9*10*11*12) = GCD(13,479001600) = 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007955, A055067.

a072046 n = gcd (a007955 n) (a055067 n)
-- Reinhard Zumkeller, Feb 06 2012

a[n_] := (dd = Divisors[n]; GCD[Times @@ dd, Times @@ Complement[Range[n], dd]]); Array[a, 59]
a[n_] := GCD[(p = n^(DivisorSigma[0, n]/2)), n!/p]; Array[a, 60] (* Amiram Eldar, Jun 26 2022 *)

from math import isqrt, gcd, factorial
from sympy import divisor_count
def A072046(n): return gcd(p:=isqrt(n)**c if (c:=divisor_count(n)) & 1 else n**(c//2),factorial(n)//p) # Chai Wah Wu, Jun 25 2022

a(n) = GCD(A007955(n), A055067(n)).

cp's OEIS Frontend

A280713 Partial sums of A055067 where A055067(n) is the product of non-divisors of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A280714 Partial products of A055067.

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Mathematica

Formula

A173540 Triangle read by rows in which row n lists the proper nondivisors of n, or zero if n <= 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

A067391 a(n) is the least common multiple of numbers in {1,2,3,...,n-1} which do not divide n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

A070251 Unrelated-factorial numbers: product of numbers unrelated to n (numbers which have a common divisor with n but do not divide n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A072046 Greatest common divisor of product of divisors of n and product of non-divisors < n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Python

Formula