A355432 -id:A355432 - cp's OEIS frontend

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

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).

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.

A361235 a(n) = number of k < n, such that k does not divide n, omega(k) < omega(n) and rad(k) | rad(n), where omega(n) = A001221(n) and rad(n) = A007947(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 2, 1, 0, 0, 3, 0, 2, 1, 3, 0, 2, 0, 3, 0, 2, 0, 10, 0, 0, 2, 4, 1, 4, 0, 4, 2, 3, 0, 11, 0, 3, 2, 4, 0, 3, 0, 4, 2, 3, 0, 4, 1, 3, 2, 4, 0, 14, 0, 4, 2, 0, 1, 14, 0, 4, 2, 12, 0, 4, 0, 5, 2, 4, 1, 15, 0, 3, 0, 5, 0, 16, 1, 5, 3, 3, 0, 19, 1, 4, 3, 5, 1, 4, 0, 5

Offset: 1

Michael De Vlieger, Mar 06 2023

nonn

a(n) = 0 for prime powers, since the definition implies omega(n) >= 2.

			a(6) = 1 since k = 4 is such that rad(4)|rad(6) = 2|6 and omega(4) < omega(6).
a(10) = 2 since k = 4 is such that rad(4)|rad(10) = 2|10 and omega(4) < omega(10), and k = 8 is such that rad(8)|rad(10) = 2|10 and omega(8) < omega(10).
a(12) = 2 since the following satisfies definition: {8, 9}.
a(14) = 2, i.e., {4, 8}.
a(15) = 1, i.e., {9}.
a(18) = 3, i.e., {8, 9, 16}.
a(30) = 10, i.e., {4, 8, 9, 12, 16, 18, 20, 24, 25, 27}, 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 gold. Numbers k in magenta in row n are counted by A355432(n). Together, gold and magenta numbers are counted by A243822(n) and appear in row n of A272618. Dots at (k, n) in red are divisors, and in green and blue in row n are counted by A243823(n).
Michael De Vlieger, Plot k < n at (x, y) = (k, -n) for n = 1..2^10, where black represents k such that k mod n != 0, such that omega(k) < omega(n) and rad(k) | rad(n).

Cf. A000961, A001221, A007947, A010846, A013929, A045763, A051953, A243822, A243823, A272618, A355432.

nn = 2^10;
rad[n_] := rad[n] = Times @@ FactorInteger[n][[All, 1]];
{0}~Join~Table[
   If[PrimePowerQ[n], 0,
    q = PrimeNu[n]; r = rad[n];
    Count[ DeleteCases[ Range[n],
     _?(Or[Divisible[n, #], CoprimeQ[#, n], ! Divisible[r, rad[#]]] &)],
     _?(PrimeNu[#] < q &)]],
   {n, 2, nn}]

a(n) = A243822(n) - A355432(n).
a(n) = A045763(n) - A243823(n) - A355432(n).
a(n) = A051953(n) - A000005(n) - A243823(n) - A355432(n) + 1.
a(n) = A010846(n) - A000005(n) - A355432(n).
a(n) = 0 for n in A000961.
a(n) > 0 for n in A013929.
a(n) = A243822(n) for n not in A360768.

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.

A361487 Odd numbers k that are neither prime powers nor squarefree, such that k/rad(k) >= q, where rad(k) = A007947(k) and prime q = A119288(k).

Original entry on oeis.org

75, 135, 147, 189, 225, 245, 363, 375, 405, 441, 507, 525, 567, 605, 675, 735, 825, 845, 847, 867, 875, 891, 945, 975, 1029, 1053, 1083, 1089, 1125, 1183, 1215, 1225, 1275, 1323, 1375, 1377, 1425, 1445, 1485, 1521, 1539, 1575, 1587, 1617, 1625, 1701, 1715, 1725, 1755, 1805, 1815, 1859, 1863, 1875, 1911

Offset: 1

Michael De Vlieger, Mar 29 2023

nonn

Odd terms in A360768, which itself is a proper subsequence of A126706.

Odd numbers k such that there exists j such that 1 < j < k and rad(j) = rad(k), but j does not divide k.

			a(1) = 75, since 75/15 >= 5. We note that rad(45) = rad(75) = 15, yet 45 does not divide 75.
a(2) = 135, since 135/15 >= 5. Note: rad(75) = rad(135) = 15, yet 45 does not divide 135.
a(3) = 147, since 147/21 >= 7. Note: rad(63) = rad(147) = 21, yet 147 mod 63 = 21.
Chart below shows k < a(n) such that rad(k) = rad(n), yet k does not divide n:
      75 | 45   .
     135 |  .   .  75   .   .
     147 |  .  63   .   .   .   .
     189 |  .   .   .   .   .   . 147   .   .   .
a(n) 225 |  .   .   .   .   . 135   .   .   .   .   .   .
     245 |  .   .   .   .   .   .   .   .   . 175   .   .   .
     363 |  .   .   .  99   .   .   .   .   .   .   .   .   .   .   .   .   . 297
     375 | 45   .   .   .   . 135   .   .   .   .   .   . 225   .   .   .   .   .
     ----------------------------------------------------------------------------
         | 45  63  75  99 117 135 147 153 171 175 189 207 225 245 261 275 279 297
                                        k in A360769

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, 1020 pixel square bitmap of indices n = 1..1040400, read left to right, top to bottom, such that A360768(n) in this sequence appears in black, else white. There is a faint pattern apparently related to that mentioned in A360768.
Michael De Vlieger, Chart showing k < a(n), n = 1..36, rows n contain k such that rad(k) = rad(n), yet k does not divide n. These k are in A360769, the number of k in row a(n) given by A355432(a(n)).

Cf. A005408, A007947, A013929, A024619, A119288, A126706, A355432, A360768, A360769.

Select[Select[Range[1, 2000, 2], Nor[SquareFreeQ[#], PrimePowerQ[#]] &], #1/#2 >= #3 & @@ {#1, Times @@ #2, #2[[2]]} & @@ {#, FactorInteger[#][[All, 1]]} &]

is(k) = { if (k%2, my (f = factor(k)); #f~ > 1 && k/vecprod(f[,1]~) >= f[2, 1], 0); } \\ Rémy Sigrist, Mar 29 2023

This sequence is { odd k in A126706 : k/A007947(k) >= A119288(k) }.

cp's OEIS Frontend

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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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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A361235 a(n) = number of k < n, such that k does not divide n, omega(k) < omega(n) and rad(k) | rad(n), where omega(n) = A001221(n) and rad(n) = A007947(n).

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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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A361487 Odd numbers k that are neither prime powers nor squarefree, such that k/rad(k) >= q, where rad(k) = A007947(k) and prime q = A119288(k).

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