A360768 -id:A360768 - cp's OEIS frontend

A361098 Intersection of A360765 and A360768.

36, 48, 50, 54, 72, 75, 80, 96, 98, 100, 108, 112, 135, 144, 147, 160, 162, 189, 192, 196, 200, 216, 224, 225, 240, 242, 245, 250, 252, 270, 288, 294, 300, 320, 324, 336, 338, 350, 352, 360, 363, 375, 378, 384, 392, 396, 400, 405, 416, 432, 441, 448, 450, 468, 480, 484, 486, 490, 500, 504, 507, 525

Offset: 1

Michael De Vlieger, Mar 15 2023

nonn

Numbers k that are neither prime powers nor squarefree, such that rad(k) * A053669(k) < k and k/rad(k) >= A119288(k), where rad(k) = A007947(k).
Numbers k such that A360480(k), A360543(k), A361235(k), and A355432(k) are positive.
Subset of A126706. All terms are neither prime powers nor squarefree.
From Michael De Vlieger, Aug 03 2023: (Start)
Superset of A286708 = A001694 \ {{1} U A246547}, which in turn is a superset of A303606. We may write k in A286708 as m*rad(k)^2, m >= 1. Since omega(k) > 1, it is clear both k/rad(k) > A053669(k) and k/rad(k) >= A119288(k). Also superset of A359280 = A286708 \ A303606.
This sequence contains {A002182 \ A168263}. (End)

			For prime p, A360480(p) = A360543(p) = A361235(p) = A355432(p) = 0, since k < p is coprime to p.
For prime power n = p^e > 4, e > 0, A360543(n) = p^(e-1) - e, but A360480(n) = A361235(n) = A355432(n) = 0, since the other sequences require omega(n) > 1.
For squarefree composite n, A360480(n) >= 1 and A361235(n) >= 1 (the latter for n > 6), but A360543(n) = A355432(n) = 0, since the other sequences require at least 1 prime power factor p^e | n with e > 0.
For n = 18, A360480(n) = | {10, 14, 15} | = 3,
            A360543(n) = | {} | = 0,
            A361235(n) = | {4, 8, 16} | = 3,
            A355432(n) = | {12} | = 1.
Therefore 18 is not in the sequence.
For n = 36, A360480(n) = | {10, 14, 15, 20, 21, 22, 26, 28, 33, 34} | = 10,
            A360543(n) = | {30} | = 1,
            A361235(n) = | {8, 16, 27, 32} | = 4,
            A355432(n) = | {24} | = 1.
Therefore 36 is the smallest term in the sequence.
Table pertaining to the first 12 terms:
Key: a = A360480, b = A360543, c = A243823; d = A361235, e = A355432, f = A243822;
g = A046753 = f + c, tau = A000005, phi = A000010.
    n |  a + b =  c | d + e = f | g + tau + phi - 1 =  n
  ------------------------------------------------------
   36 | 10 + 1 = 11 | 4 + 1 = 5 | 16 +  9 + 12 - 1 =  36
   48 | 16 + 2 = 18 | 3 + 2 = 5 | 23 + 10 + 16 - 1 =  48
   50 | 18 + 1 = 19 | 4 + 2 = 6 | 25 +  6 + 20 - 1 =  50
   54 | 19 + 2 = 21 | 4 + 4 = 8 | 29 +  8 + 18 - 1 =  54
   72 | 27 + 4 = 31 | 4 + 2 = 6 | 37 + 12 + 24 - 1 =  72
   75 | 25 + 2 = 27 | 2 + 1 = 3 | 30 +  6 + 40 - 1 =  75
   80 | 32 + 3 = 35 | 3 + 1 = 4 | 39 + 10 + 32 - 1 =  80
   96 | 38 + 7 = 45 | 4 + 4 = 8 | 53 + 12 + 32 - 1 =  96
   98 | 41 + 3 = 44 | 5 + 2 = 7 | 51 +  6 + 42 - 1 =  98
  100 | 42 + 4 = 46 | 4 + 2 = 6 | 52 +  9 + 40 - 1 = 100
  108 | 44 + 8 = 52 | 5 + 4 = 9 | 61 + 12 + 36 - 1 = 108
  112 | 48 + 3 = 51 | 3 + 1 = 4 | 55 + 10 + 48 - 1 = 112

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Diagram showing k = 1..n for n = 1..54 in blue for k counted by A360480(n), in green for k counted by A360543(n), in gold for k counted by A361235(n), and in magenta for k counted by A355432(n). Red dots indicate k | n such that k > 1, while gray dots indicate gcd(k, n) = 1.
Michael De Vlieger, 1016 X 1016 pixel bitmap read left to right in rows, then top to bottom where the k-th pixel is black if A126706(k) is in this sequence, else white (1032256 pixels total).

Cf. A000005, A000010, A001694, A002182, A007947, A045763, A053669, A119288, A126706, A168263, A243822, A243823, A355432, A360480, A360543, A361235.

Subsets: A095990\{18}, A286708, A303606, A359280.

nn = 2^16;
a053669[n_] := If[OddQ[n], 2, p = 2; While[Divisible[n, p], p = NextPrime[p]]; p];
s = Select[Range[nn], Nor[PrimePowerQ[#], SquareFreeQ[#]] &];
Reap[ Do[n = s[[j]];
    If[And[#1*a053669[n] < n, n/#1 >= #2] & @@ {Times @@ #, #[[2]]} &@
      FactorInteger[n][[All, 1]], Sow[n]], {j, Length[s]}]][[-1, -1]]

A359929 Irregular triangle read by rows, where row n lists k < t such that rad(k) = rad(t) but k does not divide t, where t = A360768(n) and rad(k) = A007947(k).

Original entry on oeis.org

12, 18, 24, 18, 36, 20, 40, 12, 24, 36, 48, 48, 54, 45, 50, 60, 18, 36, 54, 72, 28, 56, 40, 80, 24, 48, 72, 96, 98, 90, 84, 75, 54, 96, 108, 63, 60, 90, 120, 50, 100, 12, 24, 36, 48, 72, 96, 108, 144, 126, 120, 150, 147, 18, 36, 54, 72, 108, 144, 162, 56, 112, 132, 80, 160, 48, 96, 144, 162, 192, 98, 196

Offset: 1

Michael De Vlieger, Mar 29 2023

			Table of some of the first rows of the sequence, showing both even and odd b(n) = A360768(n) with both a single and multiple terms in the row:
   n   b(n)  row n of this sequence
  ---------------------------------
   1    18   12;
   2    24   18;
   3    36   24;
   4    48   18, 36;
   5    50   20, 40;
   6    54   12, 24, 36, 48;
  ...
   8    75   45;
  ...
  18   135   75;
  ...
  23   162   12, 24, 36, 48, 72, 96, 108, 144;
  ...
  56   375   45, 135, 225;
  57   378   84, 168, 252, 294, 336;
  58   384   18, 36, 54, 72, 108, 144, 162, 216, 288, 324

Michael De Vlieger, Table of n, a(n) for n = 1..14675 (rows n = 1..3000, flattened)
Michael De Vlieger, Plot (k, t) at (x, -y), where k = A126706(i) and t = A360768(j) for i = 1..48 and j = 1..108, showing k in dark blue, t in dark red, and for t and nondivisor k such that rad(k) = rad(t), we highlight in large black dots. This sequence counts the number of black dots in row n.

Cf. A007947, A126706, A355432, A359382, A360768.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
s = Select[Range[2^7], Nor[SquareFreeQ[#], PrimePowerQ[#]] &];
t = Select[s, #1/#2 >= #3 & @@ {#1, Times @@ #2, #2[[2]]} & @@
      {#, FactorInteger[#][[All, 1]]} &];
Flatten@ Map[Function[{n, k},
    Select[TakeWhile[s, # < n &],
      And[rad[#] == k, ! Divisible[n, #]] &]] @@ {#, rad[#]} &, t]

Row lengths are in A359382.

A359382 a(n) = number of k < t such that rad(k) = rad(t) and k does not divide t, where t = A360768(n) and rad(k) = A007947(k).

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 2, 1, 1, 1, 4, 2, 2, 4, 1, 1, 1, 1, 3, 1, 3, 2, 8, 1, 2, 1, 7, 2, 1, 2, 5, 2, 1, 1, 3, 3, 1, 6, 1, 1, 5, 5, 4, 5, 1, 1, 4, 8, 3, 3, 1, 2, 1, 4, 2, 3, 5, 10, 2, 1, 3, 3, 1, 1, 1, 6, 1, 3, 7, 1, 1, 7, 3, 14, 3, 6, 3, 2, 1, 1, 2, 7, 2, 1, 1, 2, 2, 8, 4, 6, 4, 8, 1, 1, 2, 1, 6, 9, 2, 1

Offset: 1

Michael De Vlieger, Mar 29 2023

nonn

This sequence contains nonzero values in A355432.

			Table relating a(n) to b(n) = A360768(n) and row n of A359929.
n  b(n)   row n of A359929   a(n)
---------------------------------
1   18    12                   1
2   24    18                   1
3   36    24                   1
4   48    18, 36               2
5   50    20, 40               2
6   54    12, 24, 36, 48       4

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Scatterplot of a(n), n = 1..2^16, highlighting records (A360768(n) in A360589) in red.

Cf. A007947, A010846, A013929, A020639, A024619, A027750, A126706, A162306, A243822, A272618, A355432, A359929, A360589, A360768.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
s = Select[Range[671], Nor[SquareFreeQ[#], PrimePowerQ[#]] &];
t = Select[s, #1/#2 >= #3 & @@ {#1, Times @@ #2, #2[[2]]} & @@
      {#, FactorInteger[#][[All, 1]]} &];
Map[Function[{n, k},
    Count[TakeWhile[s, # < n &],
      _?(And[rad[#] == k, ! Divisible[n, #]] &)]] @@ {#, rad[#]} &, t]

a(n) = A355432(A360768(n)) = length of row n in A359929.

A362844 a(n) is the largest k < A360768(n) such that rad(k) = rad(A360768(n)) and n mod k != 0, where rad(n) = A007947(n).

Original entry on oeis.org

12, 18, 24, 36, 40, 48, 54, 45, 50, 60, 72, 56, 80, 96, 98, 90, 84, 75, 108, 63, 120, 100, 144, 126, 150, 147, 162, 112, 132, 160, 192, 196, 135, 156, 180, 176, 175, 200, 168, 198, 240, 216, 252, 270, 204, 234, 250, 288, 294, 208, 228, 280, 242, 300, 297, 225, 336, 324, 224, 264, 320, 375, 306, 276

Offset: 1

Michael De Vlieger, May 19 2023

nonn

Largest nondivisor less than m = A360768(n) that shares the same squarefree kernel as m.

a(n) is in A126706, not a permutation of A126706.

			A360768(1) = 18; the smallest nondivisor k < 18 such that rad(k) = rad(18) = 6 is a(1) = 12.
A360768(2) = 24; the smallest nondivisor k < 24 such that rad(k) = rad(24) = 6 is a(2) = 18.
A360768(5) = 50; the smallest nondivisor k < 50 such that rad(k) = rad(50) = 10 is a(5) = 40.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^12, showing records in red.

Cf. A007947, A126706, A360768, A362041.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; s = Select[Select[Range[414],  Nor[SquareFreeQ[#], PrimePowerQ[#]] &], #1/#2 >= #3 & @@ {#1, Times @@ #2, #2[[2]]} & @@ {#, FactorInteger[#][[All, 1]]} &]; Table[Function[r, SelectFirst[Range[m - 1, 1, -1], r == rad[#] &] ][rad[m]], {m, s}]

A243822 Number of k < n such that rad(k) | n but k does not divide n, where rad = A007947.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 2, 1, 0, 0, 4, 0, 2, 1, 3, 0, 3, 0, 3, 0, 2, 0, 10, 0, 0, 2, 4, 1, 5, 0, 4, 2, 3, 0, 11, 0, 3, 2, 4, 0, 5, 0, 6, 2, 3, 0, 8, 1, 3, 2, 4, 0, 14, 0, 4, 2, 0, 1, 14, 0, 4, 2, 12, 0, 6, 0, 5, 3, 4, 1, 15, 0, 4, 0, 5, 0, 16, 1, 5, 3, 3, 0, 20, 1, 4, 3, 5, 1, 8, 0, 7, 2, 6

Offset: 1

Michael De Vlieger, Jun 11 2014

nonn

Former name: number of "semidivisors" of n, numbers m < n that do not divide n but divide n^e for some integer e > 1. See ACM Inroads paper.

			From _Michael De Vlieger_, Aug 11 2024: (Start)
Let S(n) = row n of A162306 and let D(n) = row n of A027750.a(2) = 0 since S(2) \ D(2) = {1, 2} \ {1, 2} is null.
a(10) = 2 since S(10) \ D(10) = {1, 2, 4, 5, 8, 10} \ {1, 2, 5, 10} = {4, 8}.a(16) = 0 since S(16) \ D(16) = {1, 2, 4, 8, 16} \ {1, 2, 4, 8, 16} is null, etc.Table of a(n) and S(n) \ D(n):
   n  a(n)  row n of A272618.
  ---------------------------
   6    1   {4}
  10    2   {4, 8}
  12    2   {8, 9}
  14    2   {4, 8}
  15    1   {9}
  18    4   {4, 8, 12*, 16}
  20    2   {8, 16}
  21    1   {9}
  22    3   {4, 8, 16}
  24    3   {9, 16, 18*}
  26    3   {4, 8, 16}
  28    2   {8, 16}
  30   10   {4, 8, 9, 12, 16, 18, 20, 24, 25, 27}
Terms in A272618 marked with an asterisk are counted by A355432. All other terms are counted by A361235. (End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Exploring Number Bases as Tools, ACM Inroads, March 2012, Vol. 3, No. 1, pp. 4-12.
Michael De Vlieger, Regular and coregular numbers, ResearchGate, 2024.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20

Cf. A000005, A000961, A024619, A027750, A010846, A045763, A162306, A243823, A272618, A304570, A355432, A361235.

Table[Count[Range[n], ?(And[Divisible[n, Times @@ FactorInteger[#][[All, 1]]], ! Divisible[n, #]] &)], {n, 120}] (* _Michael De Vlieger, Aug 11 2024 *)

a(n) = A010846(n) - A000005(n) = card({row n of A162306} \ {row n of A027750}).
a(n) = A045763(n) - A243823(n).
a(n) = (Sum_{1<=k<=n, gcd(n,k)=1} mu(k)*floor(n/k)) - tau(n). - Michael De Vlieger, May 10 2016, after Benoit Cloitre at A010846.
From Michael De Vlieger, Aug 11 2024" (Start)
a(n) = 0 for n in A000961, a(n) > 0 for n in A024619.
a(n) = A051953(n) - A000005(n) + 1 = n - A000010(n) - A000005(n) - A243823(n) + 1.
a(n) = A355432(n) + A361235(n).
a(n) = A355432(n) for n in A360768.
a(n) = A361235(n) for n not in A360768.
a(n) = number of terms in row n of A272618.
a(n) = sum of row n of A304570. (End)

New name from David James Sycamore, Aug 11 2024

A355432 a(n) = number of k < n such that rad(k) = rad(n) and k does not divide n, where rad(k) = A007947(k).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Michael De Vlieger, Feb 22 2023

nonn

a(n) = 0 for prime powers and squarefree numbers.

			a(1) = 18, since 18/6 >= 3. We note that rad(12) = rad(18) = 6, yet 12 does not divide 18.
a(2) = 24, since 24/6 >= 3. rad(18) = rad(24) = 6 and 24 mod 18 = 6.
a(3) = 36, since 36/6 >= 3. rad(24) = rad(36) = 6 and 36 mod 24 = 12.
a(6) = 54, since 54/6 >= 3. m in {12, 24, 36, 48} are such that rad(m) = rad(54) = 6, but none divides 54, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Plot (k, n) at (x, -y), k = 1..n, n = 1..54, showing k in A126706 in dark blue, n in A360768 in dark red, and for n and nondivisor k such that rad(k) = rad(n), we highlight in large black dots. This sequence counts the number of black dots in row n.
Michael De Vlieger, Condensation of the above plot, showing k = 1..n and only n in A360768 and n <= 54.

Cf. A005361, A007947, A008479, A010846, A013929, A020639, A024619, A027750, A126706, A162306, A243822, A272618, A360589, A360768.

rad[n_] := rad[n] = Times @@ FactorInteger[n][[All, 1]]; Table[Which[PrimePowerQ[n], 0, SquareFreeQ[n], 0, True, r = rad[n]; Count[Select[Range[n], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], _?(And[rad[#] == r, Mod[n, #] != 0] &)]], {n, 120}]

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
a(n) = my(rn=rad(n)); sum(k=1, n-1, if (n % k, rad(k)==rn)); \\ Michel Marcus, Feb 23 2023

a(n) > 0 for n in A360768.
a(n) < A243822(n) < A010846(n).
a(n) = A008479(n) - A005361(n). - Amiram Eldar, Oct 25 2024

A364998 Numbers k neither squarefree nor prime power such that rad(k)A119288(k) <= k but rad(k)A053669(k) > k.

Original entry on oeis.org

18, 24, 90, 120, 126, 150, 168, 180, 198, 234, 264, 306, 312, 342, 408, 414, 456, 522, 552, 558, 630, 666, 696, 738, 744, 774, 840, 846, 888, 954, 984, 990, 1032, 1050, 1062, 1098, 1128, 1170, 1206, 1260, 1272, 1278, 1314, 1320, 1386, 1416, 1422, 1464, 1470, 1494

Offset: 1

Michael De Vlieger, Aug 16 2023

nonn

Subset of A126706, numbers that are neither squarefree nor prime powers.
For k in this sequence, let p = A119288(k), q = A053669(k), and r = A007947(k).
A355432(k) > 0, A360543(k) = 0. There exist nondivisors m < k such that rad(m) = rad(k); however, m < k, gcd(m,k) > 1 such that both omega(k) > omega(m) and rad(m) | k do not exist.

			Let b(n) = A126706(n), S = A360768, and T = A363082.
b(1) = 12 is not in the sequence since p*r = 3*6 = 18 and q*r = 5*6 = 30; both exceed 12, thus 12 is not in S.
b(2) = a(1) = 18 since p*r = 3*6 = 18 and q*r = 5*6 = 30. Indeed, 18 does not exceed 18 and 30 is larger than 18, hence 18 is in both S and T.
b(6) = 36 is not in the sequence since p*r = 3*6 = 18 and q*r = 5*6, and both do not exceed 36, therefore 36 is in S but not T.
b(7) = 40 is not in the sequence since p*r = 5*10 = 50 and q*r = 3*10 = 30. Though 50 > 40, 30 < 40, thus 40 is neither in S nor T, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated plot of b(n) = A126706(n), with n = 20*(y-1) + x at (x, -y), for x = 1..20 and y = 1..20, thus showing 400 terms. Terms in this sequence are colored black, those in A364999 in blue, in A364997 in green, and in A361098 in red.
Michael De Vlieger, Plot of b(n), with n = 120*(y-1) + x at (x, -y), for x = 1..120 and y = 1..120, thus showing 14400 terms, using the same color scheme as described immediately above.
Michael De Vlieger, Plot of b(n), with n = 1016*(y-1) + x at (x, -y), for x = 1..1016 and y = 1..1016, thus showing 1032256 terms. Terms in this sequence are colored black, else white. Demonstrates a strong quasiperiodic pattern approximately mod 169.

Cf. A007947, A053669, A119288, A126706, A355432, A360432, A360768, A361098, A363082, A364997, A364999.

Select[Select[Range[1500], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], Function[{k, f}, Function[{p, q, r}, And[p r <= k, q r > k]] @@ {f[[2, 1]], SelectFirst[Prime@ Range[PrimePi[f[[-1, 1]]] + 1], ! Divisible[k, #] &], Times @@ f[[All, 1]]}] @@ {#, FactorInteger[#]} &]

Intersection of A363082 and A360768.

A360767 Numbers k that are neither prime power nor squarefree, such that k/rad(k) < q, where rad(k) = A007947(k) and prime q = A119288(k).

Original entry on oeis.org

12, 20, 28, 40, 44, 45, 52, 56, 60, 63, 68, 76, 84, 88, 92, 99, 104, 116, 117, 124, 132, 136, 140, 148, 152, 153, 156, 164, 171, 172, 175, 176, 184, 188, 204, 207, 208, 212, 220, 228, 232, 236, 244, 248, 260, 261, 268, 272, 275, 276, 279, 280, 284, 292, 296, 297, 304, 308, 315, 316, 325, 328, 332, 333

Offset: 1

Michael De Vlieger, Feb 28 2023

Proper subsequence of A126706.

Numbers k such that there does not exist j such that 1 < j < k and rad(j) = rad(k), but j does not divide k.

			a(1) = 12, since 12/6 < 3.
a(2) = 20, since 20/10 < 5.
a(3) = 28, since 28/14 < 7.
a(4) = 40, since 40/10 < 5, etc.

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

Cf. A007947, A013929, A024619, A119288, A126706, A355432, A360768.

Select[Select[Range[120], Nor[SquareFreeQ[#], PrimePowerQ[#]] &], #1/#2 < #3 & @@ {#1, Times @@ #2, #2[[2]]} & @@ {#, FactorInteger[#][[All, 1]]} &]

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
f(n) = if (isprimepower(n) || (n==1), 1, my(f=factor(n)[, 1]); f[2]); \\ A119288
isok(k) = !isprimepower(k) && !issquarefree(k) && (k/rad(k) < f(k)); \\ Michel Marcus, Mar 01 2023

This sequence is { k in A126706 : k/A007947(k) < A119288(k) } = A126706 \ A360768.

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).

A360589 Numbers k that set records in A355432.

Original entry on oeis.org

1, 18, 48, 54, 162, 384, 486, 1350, 1458, 2250, 2430, 3750, 6000, 6750, 7290, 11250, 12150, 14580, 15000, 15360, 18750, 21870, 30720, 33750, 36450, 37500, 43740, 56250, 61440, 65610, 93750, 122880, 168750, 182250, 187500, 196830, 245760, 281250, 328050, 360150, 375000, 393660

Offset: 1

Michael De Vlieger, Feb 22 2023

nonn

Subset of A055932.
For n > 1, subset of A360768, which is in turn a subset of A126706.
Conjecture: for n > 2, subset of A364702. - Michael De Vlieger, Oct 04 2024

			Let rad(m) = A007947(m).
a(1) = 1 since 1 is the empty product.
a(2) = 18 since {12} is a nondivisor k < 18 such that rad(k) = rad(18).
a(3) = 48 since {18, 36} are nondivisors k < 48 such that rad(k) = rad(48).
a(4) = 54 since {12, 24, 36, 48} are nondivisors k < 54 such that rad(k) = rad(54), etc.
Table shows prime decomposition of a(n) = Product p^e, noting multiplicity e in the pi(p)-th position. For example, a(n) = 1350 = 2 * 3^3 * 5^2, hence we write 1.3.2.
a(n) = A055932(i) and has A360912(n) nondivisors k < a(n) such that rad(k) = rad(a(n)).
   n    a(n) A067255(a(n))  i  A360912(n)
  ----------------------------------------
   1      1      0          1          0
   2     18      1.2        8          1
   3     48      4.1       13          2
   4     54      1.3       14          4
   5    162      1.4       25          8
   6    384      7.1       37         10
   7    486      1.5       42         14
   8   1350      1.3.2     65         16
   9   1458      1.6       67         21
  10   2250      1.2.3     81         23
  11   2430      1.5.1     85         26
  12   3750      1.1.4     99         33
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..2071
Michael De Vlieger, Plot p^m | a(n) at (x,y) = (n, pi(p)), n = 1..2071, 24X vertical exaggeration, with a color function that represents m = 1 in black, m = 2 in red, m = 3 in orange, ... m = 34 in magenta. (Represents column "A067255(a(n))" in table in Example below.)

Cf. A007947, A055932, A126706, A355432, A360768, A360912, A364702.

rad[n_] := rad[n] = Times @@ FactorInteger[n][[All, 1]]; t = Select[Range[2^14], Nor[SquareFreeQ[#], PrimePowerQ[#]] &]; s = Select[t, #1/#2 >= #3 & @@ {#1, Times @@ #2, #2[[2]]} & @@ {#, FactorInteger[#][[All, 1]]} &]; t = Table[m = s[[n]]; r = rad[m]; Count[TakeWhile[t, # < m &], _?(And[rad[#] == r, Mod[m, #] != 0] &)], {n, Length[s]}]; {1}~Join~Map[s[[FirstPosition[t, #][[1]]]] &, Union@ FoldList[Max, t]]

cp's OEIS Frontend

A361098 Intersection of A360765 and A360768.

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A359929 Irregular triangle read by rows, where row n lists k < t such that rad(k) = rad(t) but k does not divide t, where t = A360768(n) and rad(k) = A007947(k).

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A359382 a(n) = number of k < t such that rad(k) = rad(t) and k does not divide t, where t = A360768(n) and rad(k) = A007947(k).

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A362844 a(n) is the largest k < A360768(n) such that rad(k) = rad(A360768(n)) and n mod k != 0, where rad(n) = A007947(n).

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A243822 Number of k < n such that rad(k) | n but k does not divide n, where rad = A007947.

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A355432 a(n) = number of k < n such that rad(k) = rad(n) and k does not divide n, where rad(k) = A007947(k).

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A364998 Numbers k neither squarefree nor prime power such that rad(k)*A119288(k) <= k but rad(k)*A053669(k) > k.

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A360767 Numbers k that are neither prime power nor squarefree, such that k/rad(k) < q, where rad(k) = A007947(k) and prime q = A119288(k).

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A364998 Numbers k neither squarefree nor prime power such that rad(k)A119288(k) <= k but rad(k)A053669(k) > k.