A045763 -id:A045763 - cp's OEIS frontend

A360543 a(n) = number of numbers k < n, gcd(k, n) > 1, such that omega(k) > omega(n) and rad(n) | rad(k), where omega(n) = A001221(n) and rad(n) = A007947(n).

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 6, 0, 0, 0, 0, 11, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 5, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 1, 26, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 3, 23, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 7, 0, 3, 1, 4

Offset: 1

Michael De Vlieger, Mar 06 2023

nonn

			a(4) = 0 since k = 1..3 are prime powers.
a(8) = 1 since only k = 6 is such that p = 3, q = 5, but gcd(6, 10) = 2.
a(9) = 1 since the following satisfies definition: {6},
a(16) = 4, i.e., {6, 10, 12, 14},
a(25) = 3, i.e., {10, 15, 20},
a(27) = 6, i.e., {6, 12, 15, 18, 21, 24},
a(32) = 11, i.e., {6, 10, 12, 14, 18, 20, 22, 24, 26, 28, 30},
a(36) = 1, i.e., {30},
a(40) = 1, i.e., {30},
a(45) = 1, i.e., {30}, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Diagram showing k <= n, n = 1..36, where a(n) is the number of numbers k in row n shown in blue. Numbers k in green in row n are counted by A360480(n). Together, blue and green numbers are counted by A243823(n) and appear in row n of A272619. Dots at (k, n) in red are divisors, and in yellow and magenta in row n are counted by A243822(n).
Michael De Vlieger, Plot k < n at (x, y) = (k, -n) for n = 1..2^10, where black represents k such that gcd(k, n) > 1, such that omega(k) > omega(n) and rad(n) | rad(k).

Cf. A000005, A000010, A001221, A007947, A010846, A045763, A051953, A243822, A243823, A246547, A272619, A360480, A360765.

nn = 120; rad[n_] := rad[n] = Times @@ FactorInteger[n][[All, 1]]; c = Select[Range[4, nn], CompositeQ]; Table[Function[{q, r}, Count[TakeWhile[c, # <= n &], _?(And[PrimeNu[#] > q, Divisible[rad[#], r]] &)]] @@ {PrimeNu[n], rad[n]}, {n, nn}]

a(n) = A243823(n) - A360480(n).
a(n) = A045763(n) - A243822(n) - A360480(n).
a(n) = A051953(n) - A000005(n) - A243822(n) - A360480(n).
a(n) = A051953(n) - A010846(n) - A360480(n).
a(n) = A243823(n) = A045763(n) for n in A246547.
For prime power n = p^e, n > 1, a(n) = p^(e-1) - e.
For n in A360765, a(n) > 0.

A379336 Numbers k such that there exists a divisor pair (d, d/k) such that one neither divides nor is coprime to the other.

Original entry on oeis.org

24, 40, 48, 54, 56, 60, 72, 80, 84, 88, 90, 96, 104, 108, 112, 120, 126, 132, 135, 136, 140, 144, 150, 152, 156, 160, 162, 168, 176, 180, 184, 189, 192, 198, 200, 204, 208, 216, 220, 224, 228, 232, 234, 240, 248, 250, 252, 260, 264, 270, 272, 276, 280, 288, 294

Offset: 1

Michael De Vlieger, Dec 24 2024

Both divisors d and d/k are composite, since primes p either divide or are coprime to another number, and all numbers smaller than p are coprime to p.
Proper subset of A126706; contains A378769, which in turn contains A378984.
Consider composite k, m, k != m. Define a "neutral" relation to be such that 1 < gcd(k,m) and not equal to either k or m. Then neither k nor m divides the other, and k and m are not coprime. If k is neutral to m, then m is neutral to k, since order does not matter. Then either the squarefree kernel of one divides the other or it does not. Thus, there are 3 kinds of neutral relation:
Type A: Though gcd(k,m) > 1, k has a factor P that does not divide m, and m has a factor Q that does not divide k.
Type B: rad(k) = rad(m), yet neither k divides m nor m divides k, where rad = A007947 is the squarefree kernel.
Type C: Squarefree kernel of one number divides the other, while the other has a factor that does not divide the former.
A378769, subset of this sequence, contains numbers k that have all 3 types of neutral relation between at least 1 divisor pair (d, k/d) for each.

			a(1) = 24 = 2^3 * 3 = 4*6, both composite; gcd(4,6) = 2, 4 does not divide 6 (type C).
a(2) = 40 = 2^3 * 5 = 4*10, gcd(4,10) = 2 (type C).
a(3) = 48 = 2^4 * 3 = 6*8, gcd(6,8) = 2 (type C).
a(6) = 60 = 2^2 * 3 * 5 = 6*10, gcd(6,10) = 2 (type A).
a(12) = 96 = 2^5 * 3 = 6*16 = 8*12, both type C.
a(38) = 216 = 2^3 * 3^3 = 4*54 (type C) = 9*24 (type C) = 12*18 (type B)
a(1605) = 5400 = 2^3 * 3^3 * 5^2 = 4*1350 (type C) = 24*225 (type A) = 60*90 (type B) = A378769(1).
a(10475) = 32400 = 2^4 * 3^4 * 5^2 = 8*4050 (type C) = 48*675 (type A) = 120*270 (type B) = A378984(1) = A378769(14), etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Chart of (d, a(n)/d) for a(n) = 1..144, showing only the smallest d for each type of neutral relation, where type A is shown in gray, type B in black, and type C in either blue or gold.

Cf. A045763, A085986, A126706, A375055, A376271, A376936, A378767, A378769, A378984.

nn = 300; mm = Floor@ Sqrt[nn]; p = 2; q = 3;
Complement[
  Select[Range[nn], And[#2 > #1 > 1, #2 > 3] & @@ {PrimeNu[#], PrimeOmega[#]} &],
  Union[Reap[
    While[p <= mm, q = NextPrime[p];
      While[p*q <= mm, If[p != q, Sow[p*q]]; q = NextPrime[q]];
        p = NextPrime[p] ] ][[-1, 1]] ]^2 ]

This sequence is A376271 \ A085986 = {k : bigomega(k) > omega(k) > 1, bigomega(k) > 3} \ { k^2 : bigomega(k) = omega(k) = 2 }, where bigomega = A001222 and omega = A001221.

Union of A375055, A376936, and A378767.

A381096 Number of k <= n such that k is neither coprime to n and rad(k) != rad(n), where rad = A007947.

Original entry on oeis.org

0, 0, 0, 0, 0, 3, 0, 1, 1, 5, 0, 6, 0, 7, 6, 4, 0, 10, 0, 10, 8, 11, 0, 13, 3, 13, 6, 14, 0, 21, 0, 11, 12, 17, 10, 20, 0, 19, 14, 21, 0, 29, 0, 22, 19, 23, 0, 28, 5, 28, 18, 26, 0, 33, 14, 29, 20, 29, 0, 42, 0, 31, 25, 26, 16, 45, 0, 34, 24, 45, 0, 42, 0, 37

Offset: 1

Michael De Vlieger, Feb 14 2025

Number of k <= n in the cototient of n that do not share the same squarefree kernel as n.
Define a number k "neutral" to n to be such that 1 < gcd(k,n) < k, that is, k neither divides n nor is coprime to n. A045763(n) is the number of k < n such that k is neutral to n.
Define quality Q(k) to be true if k is such that 1 < gcd(k,n) and rad(k) != rad(n).
Then for k <= n and n > 1, a(n) = A045763(n), but admitting divisors k | n such that rad(k) != rad(n), and eliminating occasional nondivisors k such that rad(k) = rad(n), i.e., k listed in row n of A359929 for n = A360768(i).

			a(6) = 3 since {2, 3, 4} are neither coprime to 6 and do not have the squarefree kernel 6.
a(8) = 1 since only 6 is neither coprime to 8 and does not share the squarefree kernel 2 with 8.
a(10) = 5 since {2, 4, 5, 6, 8} are neither coprime to 10 nor have the squarefree kernel 10.
a(12) = 6 since {2, 3, 4, 8, 9, 10} are neither coprime to 12 nor have the squarefree kernel 6.
a(14) = 7 since {2, 4, 6, 7, 8, 10, 12} are neither coprime to 14 nor have the squarefree kernel 14, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.

Cf. A000010, A005361, A008479, A355432, A359929, A381094.

{0}~Join~Table[n - EulerPhi[n] - DivisorSigma[0, n/rad[n]], {n, 2, 120}]

a(1) = 0, a(p) = a(4) = 0.
a(n) = A045763(n) - A005361(n).
For n > 1, a(n) = n - phi(n) - tau(n/rad(n)) = A000010(n) - A005361(n).
For n > 1, a(n) = n - A000010(n) - A008479(n) + A355432(n).

A073760 Integers m such that A073758(m) = 4.

Original entry on oeis.org

6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 114, 118, 122, 126, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 218, 222, 226, 230

Offset: 1

Labos Elemer, Aug 08 2002

Essentially the same as A016825.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A016825, A045763, A073758, A073759.

a(n) = 4*n+2; \\ Altug Alkan, Jul 23 2018

a(n) = 4*n + 2. - Max Alekseyev, Mar 03 2007
a(n) = 8*n - a(n-1), with a(1)=6. - Vincenzo Librandi, Aug 08 2010
a(1)=6, a(2)=10; for n>2, a(n) = 2*a(n-1) - a(n-2). - Harvey P. Dale, Mar 06 2012

Definition simplified by Michel Marcus, Jul 26 2018

A073813 Difference between n and largest unrelated number belonging to n, when n runs over composite numbers. For primes and for 4, unrelated set is empty.

Original entry on oeis.org

0, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 5, 2, 3, 2, 2, 2, 3, 2, 5, 2, 2, 3, 2, 2, 2, 3, 2, 2, 7, 2, 3, 2, 2, 5, 2, 3, 2, 2, 2, 3, 2, 5, 2, 2, 3, 2, 2, 2, 3, 2, 7, 2, 2, 3, 2, 2, 5, 2, 3, 2, 2, 7, 2, 3, 2, 5, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 5, 2, 3, 2, 7, 2, 11, 2, 3, 2, 5, 2, 2, 3, 2, 2, 7, 2, 3, 2, 2

Offset: 1

Labos Elemer, Aug 15 2002

nonn

From Michael De Vlieger, Mar 28 2016 (Start):
a(0) = 0 since 4 is the smallest composite and "unrelated" numbers k with respect to n must be composite and smaller than n. Unrelated numbers k cannot be prime since primes p must either divide or be coprime to n; k cannot equal 1 since 1 is both a divisor of and coprime to n.
The test for unrelated numbers k that belong to n is 1 < gcd(k, n) < k.
(End)

			composite[1]=4, URS[4]={}, a(1)=0 by convention; n=14, c[14]=24, URS[24]={9,10,14,15,16,18,20,21,22}, a(14)=24-Max[URS[24]]=2.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A045763, A073759, A002808.

Cf. A056608. [From R. J. Mathar, Sep 23 2008]

c[x_] := FixedPoint[x+PrimePi[ # ]+1&, x]; tn[x_] := Table[j, {j, 1, x}]; di[x_] := Divisors[x]; rrs[x_] := Flatten[Position[GCD[tn[x], x], 1]]; rs[x_] := Union[rrs[x], di[x]]; urs[x_] := Complement[tn[x], rs[x]]; Table[c[w]-Max[urs[c[w]]], {w, 1, 128}]
Prepend[Function[k, k - SelectFirst[Range[k - 2, 2, -1], 1 < GCD[#, k] < # &]] /@ Select[Range[6, 138], ! PrimeQ@ # &], 0] (* Michael De Vlieger, Mar 28 2016, Version 10 *)

See program.

A074845 Numbers k such that S(k) = largest difference between consecutive divisors of k (ordered by size), where S(k) is the Kempner function (A002034).

Original entry on oeis.org

6, 8, 9, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514

Offset: 1

Jason Earls, Sep 10 2002

It appears that terms > 6 are simply given by: composite k such that k^2 doesn't divide A000254(k). - Benoit Cloitre, Mar 09 2004
It appears that A011776(a(k)) = 2. - Gionata Neri, Jul 31 2017
It appears that this sequence consists of the numbers k such that A045763(k) > 0 and k does not divide A070251(k). - Isaac Saffold, Jun 01 2018

Michael De Vlieger, Table of n, a(n) for n = 1..670 (a(n) < 10^4, from b-file at A002034).

Cf. A002034, A060681.

Select[Range@ 514, Function[n, Module[{m = 1}, While[! Divisible[m!, n], m++]; m] == Max@ Differences@ Divisors@ n]] (* Michael De Vlieger, Jul 31 2017 *)

K(n) = my(s=1); while(s!%n>0, s++); s;
dd(n) = my(vd=divisors(n)); vecmax(vector(#vd-1, k, vd[k+1] - vd[k]));
isok(n) = K(n) == dd(n); \\ Michel Marcus, Aug 03 2017

A300155 Numbers n for which A243822(n) = A000005(n).

Original entry on oeis.org

34, 38, 46, 50, 54, 58, 62, 105, 249, 267, 268, 284, 291, 292, 303, 309, 316, 321, 324, 327, 332, 339, 356, 363, 381, 385, 388, 393, 404, 411, 412, 417, 428, 436, 447, 452, 453, 455, 471, 484, 489, 500, 501, 507, 508, 519, 537, 543, 573, 579, 591, 595, 597

Offset: 1

Michael De Vlieger, Feb 26 2018

nonn

Indices of zeros in A299990, i.e., A010846(n) - 2*A000005(n) = 0.
Composite numbers m have nondivisors k in the cototient such that k | n^e with e > 1. These k appear in row n of A272618 and are enumerated by A243822(n). These nondivisors k are a kind of "regular" number along with divisors d of n; both are listed in row n of A162306 and are together enumerated by A045763(n). Divisors of n are listed in row n of A027750.
This sequence lists numbers that have an equal number of nondivisors k in the cototient of n as divisors d.
The smallest odd term is 105.

			34 is the first term since it is the smallest number for which A243822(34) = A000005(34). For n = 34, there are 4 divisors {1, 2, 17, 34} and 4 nondivisors 1 <= m <= n such that m | n^e with e > 1: {4, 8, 16, 32}.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000005, A010846, A027750, A045763, A162306, A243822, A272618, A299990, A299991, A299992.

Select[Range@ 600, Function[n, Count[Range[n], _?(PowerMod[n, Floor@ Log2@ n, #] == 0 &)] == 2 DivisorSigma[0, n]]]

A300858 a(n) = A243823(n) - A243822(n).

Original entry on oeis.org

0, 0, 0, 0, 0, -1, 0, 1, 1, -1, 0, -1, 0, 1, 2, 4, 0, -1, 0, 3, 4, 3, 0, 3, 3, 5, 6, 7, 0, -5, 0, 11, 6, 7, 6, 6, 0, 9, 8, 11, 0, 1, 0, 13, 12, 13, 0, 13, 5, 13, 12, 17, 0, 13, 10, 19, 14, 19, 0, 5, 0, 21, 18, 26, 12, 11, 0, 23, 18, 15, 0, 25, 0, 25, 24, 27

Offset: 1

Michael De Vlieger, Mar 14 2018

sign

Consider numbers in the cototient of n, listed in row n of A121998. For composite n > 4, there are nondivisors m in the cototient, listed in row n of A133995. Of these m, there are two species. The first are m that divide n^e with integer e > 1, while the last do not divide n^e. These are listed in row n of A272618 and A272619, and counted by A243822(n) and A243823(n), respectively. This sequence is the difference between the latter and the former species of nondivisors in the cototient of n.
Since A045763(n) = A243822(n) + A243823(n), this sequence examines the balance of the two components among nondivisors in the cototient of n.
For positive n < 6 and for p prime, a(n) = a(p) = 0, thus a(A046022(n)) = 0.
For prime powers p^e, a(p^e) = A243823(p^e), since A243822(p^e) = 0, thus a(n) = A243823(n) for n in A000961.
Value of a(n) is generally nonnegative; a(n) is negative for n = {6, 10, 12, 18, 30}; a(30) = -5, but a(n) = -1 for the rest of the aforementioned numbers. These five numbers are a subset of A295523.

			a(6) = -1 since the only nondivisor in the cototient of 6 is 4, which divides 6^e with e > 1; therefore 0 - 1 = -1.
a(8) = 1 since the only nondivisor in the cototient of 8 is 6, and 6 does not divide 8^e with e > 1, therefore 1 - 0 = 1.
Some values of a(n) and related sequences:
   n  a(n) A243823(n) A243822(n)    A272619(n)       A272618(n)
  -------------------------------------------------------------
   1   0          0          0      -                -
   2   0          0          0      -                -
   3   0          0          0      -                -
   4   0          0          0      -                -
   5   0          0          0      -                -
   6  -1          0          1      -                {4}
   7   0          0          0      -                -
   8   1          1          0      {6}              -
   9   1          1          0      {6}              -
  10  -1          1          2      {6}              {4,8}
  11   0          0          0      -                -
  12  -1          1          2      {10}             {8,9}
  13   0          0          0      -                -
  14   1          3          2      {6,10,12}        {4,8}
  15   2          3          1      {6,10,12}        {9}
  16   4          4          0      {6,10,12,14}     -
  17   0          0          0      -                -
  18  -1          3          4      {10,14,15}       {4,8,12,16}
  19   0          0          0      -                -
  20   3          5          2      {6,12,14,15,18}  {8,16}
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000005, A000010, A000961, A010846, A046022, A121998, A133995, A173540, A243822, A243823, A272618, A272619, A295523, A300859, A300861.

f[n_] := Count[Range@ n, _?(PowerMod[n, Floor@ Log2@ n, #] == 0 &)]; Array[#1 - #3 + 1 - 2 #2 + #4 & @@ {#, f@ #, EulerPhi@ #, DivisorSigma[0, #]} &, 76]

a(n) = 1 + n + numdiv(n) - eulerphi(n) - 2*sum(k=1, n, if(gcd(n,k)-1, 0, moebius(k)*(n\k))); \\ Michel Marcus, Mar 17 2018

a(n) = 1 + n - A000010(n) - 2*A010846(n) + A000005(n).

A376281 Number of pairs (d, k/d), d | k, d < k/d, such that gcd(d, k/d) is not in {1, d, k/d}, where k is in A379336.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 2, 1, 3, 3, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 4, 1, 1, 1, 1, 1, 1, 2, 5, 1, 1, 1, 1, 1, 3, 1, 3, 1, 2, 1, 1

Offset: 1

Michael De Vlieger, Jan 08 2025

nonn

Number of ways to write k = A379336(n) as a product of numbers i and j that are neither coprime nor does one number divide the other. Both i and j are necessarily composite.

Both i and j = k/i appear in row k of A133995.

			Let s(n) = A379336(n).
a(1) = 1 since s(1) = 24 = 4*6.
a(2) = 1 since s(2) = 40 = 4*10.
a(3) = 1 since s(3) = 48 = 6*8.
a(12) = 2 since s(12) = 96 = 6*16 = 8*12.
a(16) = 3 since s(16) = 120 = 4*30 = 6*20 = 10*12.
a(44) = 4 since s(44) = 240 = 6*40 = 8*30 = 10*24 = 12*20.
a(75) = 5 since s(75) = 360 = 4*90 = 10*36 = 12*30 = 15*24 = 18*20.
a(105) = 6 since s(105) = 480 = 6*80 = 8*60 = 10*48 = 12*40 = 16*30 = 20*24, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A045763, A133995, A379336, A379552, A379752, A379772.

nn = 500; mm = Floor@ Sqrt[nn]; p = 2; q = 3;
s = Complement[
  Select[Range[nn],
    And[#2 > #1 > 1, #2 > 3] & @@ {PrimeNu[#], PrimeOmega[#]} &],
  Union[Reap[
    While[p <= mm, q = NextPrime[p];
      While[p*q <= mm, If[p != q, Sow[p*q]]; q = NextPrime[q]];
        p = NextPrime[p]] ][[-1, 1]] ]^2 ];
Table[k = s[[n]];
  1/2*DivisorSum[k, 1 &, ! MemberQ[{1, #1, #2}, GCD[#1, #2]] & @@ {#, k/#} &],
  {n, Length[s]}]

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

cp's OEIS Frontend

A360543 a(n) = number of numbers k < n, gcd(k, n) > 1, such that omega(k) > omega(n) and rad(n) | rad(k), where omega(n) = A001221(n) and rad(n) = A007947(n).

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A379336 Numbers k such that there exists a divisor pair (d, d/k) such that one neither divides nor is coprime to the other.

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A381096 Number of k <= n such that k is neither coprime to n and rad(k) != rad(n), where rad = A007947.

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A073760 Integers m such that A073758(m) = 4.

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A073813 Difference between n and largest unrelated number belonging to n, when n runs over composite numbers. For primes and for 4, unrelated set is empty.

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A074845 Numbers k such that S(k) = largest difference between consecutive divisors of k (ordered by size), where S(k) is the Kempner function (A002034).

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A300155 Numbers n for which A243822(n) = A000005(n).

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A300858 a(n) = A243823(n) - A243822(n).

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