A381096 -id:A381096 - cp's OEIS frontend

A381240 Indices of records in A381096.

1, 6, 10, 12, 14, 18, 22, 24, 28, 30, 42, 54, 60, 66, 78, 84, 90, 102, 114, 120, 126, 132, 138, 150, 168, 180, 198, 204, 210, 240, 252, 264, 270, 294, 300, 330, 360, 378, 390, 420, 450, 462, 480, 504, 510, 540, 546, 570, 600, 630, 660, 690, 714, 750, 780, 810, 840

Offset: 1

Michael De Vlieger, Feb 18 2025

nonn

Let f(k) = A381096(k) = k - phi(k) - tau(k/rad(k)) = k - A000010(k) - A005361(k), where phi = A000010, tau = A000005, and rad = A007947. This sequence contains k such that f(k) > f(j) for j < k as k increases.
Apart from a(1) = 1, terms are in A024619.
Conjecture 1: For i > 1, A002110(i) is in this sequence.
Conjecture 2: Intersection with A001694 (i.e., in A286708) is {900, 1800}.

			Let g(n) = A067255(n) be the exponents of prime power factors p^m | n, writing "." for m = 0 and ending at the pi(gpf(n))-th term. Example: for n = 84, g(84) = {2, 1, 0, 1}, therefore we write "21.1" for concision in the table below.
Table of first 12 terms.
   n  a(n)  g(a(n)) f(a(n))
  --------------------------
   1    1   .           0
   2    6   11          3
   3   10   1.1         5
   4   12   21          6
   5   14   1..1        7
   6   18   12         10
   7   22   1...1      11
   8   24   31         13
   9   28   2..1       14
  10   30   111        21
  11   42   11.1       29
  12   54   13         33

Michael De Vlieger, Table of n, a(n) for n = 1..1782 (a(n) < 2^28).
Michael De Vlieger, Plot p^m | a(n) at (x,y) = (n,pi(p)), n = 1..1781, 4X vertical exaggeration for clarity, with a color function representing m = 1 in black, m = 2 in red, m = 3 in orange, ..., largest m in the dataset in magenta.

Cf. A000010, A005361, A024619, A381096, A381241.

r = 0; nn = 2^20; f[x_] := x - EulerPhi[x] - DivisorSigma[0, x/Apply[Times, FactorInteger[x][[All, 1]] ] ]; {1}~Join~Reap[Monitor[Do[If[# > r, r = #; Sow[n]] &[f[n] ], {n, nn}], n]]

A381241 Records in A381096.

Original entry on oeis.org

0, 3, 5, 6, 7, 10, 11, 13, 14, 21, 29, 33, 42, 45, 53, 58, 64, 69, 77, 85, 88, 90, 93, 108, 117, 128, 136, 138, 161, 172, 176, 181, 195, 208, 216, 249, 258, 267, 293, 322, 326, 341, 347, 354, 381, 390, 401, 425, 434, 484, 498, 513, 521, 547, 586, 590, 645, 652

Offset: 1

Michael De Vlieger, Feb 24 2025

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..1782

Cf. A381096, A381240.

rad[x_] := Times @@ FactorInteger[x][[All, 1]]; nn = Length[s];
r = 0; nn = 2^10; f[x_] := x - EulerPhi[x] - DivisorSigma[0, x/rad[x]]; {0}~Join~Reap[Do[If[# > r, r = #; Sow[#]] &[f[n]], {n, nn}]][[-1, -1]]

A381094 Triangle read by rows where row n contains k < n that are neither coprime to n nor have the same squarefree kernel as n, or 0 if there are no such k.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 3, 4, 0, 6, 6, 2, 4, 5, 6, 8, 0, 2, 3, 4, 8, 9, 10, 0, 2, 4, 6, 7, 8, 10, 12, 3, 5, 6, 9, 10, 12, 6, 10, 12, 14, 0, 2, 3, 4, 8, 9, 10, 14, 15, 16, 0, 2, 4, 5, 6, 8, 12, 14, 15, 16, 18, 3, 6, 7, 9, 12, 14, 15, 18, 2, 4, 6, 8, 10, 11, 12, 14, 16, 18, 20

Offset: 1

Michael De Vlieger, Feb 14 2025

Let rad(k) = A007947(k), the squarefree kernel of k.
Let T(n) be row n of this sequence and let S(n) be row n of A133995.
T(n) contains numbers k < n such that k and n share at least one prime factor p, but not all distinct prime p | n.
T(n) is a superset of S(n), since S(n) does not contain any divisor d | n, while T(n) allows d | n such that rad(d) != rad(n).

			Table begins:
   n   row n
  ---------------------------
   1:  0;
   2:  0;
   3:  0;
   4:  0;
   5:  0;
   6:  2, 3, 4;
   7:  0;
   8:  6;
   9:  6;
  10:  2, 4, 5, 6, 8;
  11:  0;
  12:  2, 3, 4, 8, 9, 10;
  13:  0;
  14:  2, 4, 6, 7, 8, 10, 12;
  15:  3, 5, 6, 9, 10, 12;
  16:  6, 10, 12, 14;
From _Michael De Vlieger_, Mar 03 2025: (Start)
Row 10 is the union of {2, 4, 6, 8, 10} and {5, 10} without 10.
Row 12 is the union of {2, 4, 6, 8, 10, 12} and {3, 6, 9, 12} without {6, 12}.
Row 30 is the union of {2, 4, ..., 30}, {3, 6, ..., 30}, and {5, 10, ..., 30} without 30.
Row 84 is the union of {2, 4, ..., 84}, {3, 6, ..., 84}, and {7, 14, ..., 84} without {42, 84}, etc. (End)

Michael De Vlieger, Table of n, a(n) for n = 1..11873 (rows n = 1..250, flattened)

Cf. A007947, A121998, A133995, A369609, A381096.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; Select[Range[n], Nor[CoprimeQ[#, n], rad[#] == r] &], {n, 120}]

T(n) = { k < n : 1 < gcd(k,n), rad(k) != rad(n) }.
T(n) = S(n) \ { k : k | n, rad(k) = rad(n) }.
For prime p, T(p) = {}, but we write 0 to signify the empty set.
T(4) = 0, since k < 4 is either coprime to 4 or rad(k) = 2.
Let U(n) be row n of A121998 and let R(n) be row n of A369609. T(n) = U(n) \ R(n). - Michael De Vlieger, Mar 03 2025

A381497 a(n) = sum of numbers k < n such that 1 < gcd(k,n) and rad(k) != rad(n), where rad = A007947.

Original entry on oeis.org

0, 0, 0, 0, 0, 9, 0, 6, 6, 25, 0, 36, 0, 49, 45, 42, 0, 81, 0, 100, 84, 121, 0, 144, 45, 169, 96, 196, 0, 315, 0, 210, 198, 289, 175, 354, 0, 361, 273, 430, 0, 609, 0, 484, 435, 529, 0, 648, 140, 655, 459, 676, 0, 801, 385, 826, 570, 841, 0, 1260, 0, 961, 798

Offset: 1

Michael De Vlieger, Mar 02 2025

nonn

Analogous to A066760(n), the sum of row n of A133995, and A381499(n), sum of row n of A272619.

			Table of n and a(n) for select n, showing prime power decomposition of both and row n of A381094:
   n   Factor(n) a(n)  Factor(a(n))  Row n of A381094
  -------------------------------------------------------------------
   6   2 * 3       9   3^2           {2,3,4}
   8   2^3         6   2 * 3         {6}
   9   3^2         6   2 * 3         {6}
  10   2 * 5      25   5^2           {2,4,5,6,8}
  12   2^2 * 3    36   2^2 * 3^2     {2,3,4,8,9,10}
  14   2 * 7      49   7^2           {2,4,6,7,8,10,12}
  15   3 * 5      45   3^2 * 5       {3,5,6,9,10,12}
  16   2^4        42   2 * 3 * 7     {6,10,12,14}
  18   2 * 3^2    81   3^4           {2,3,4,8,9,10,14,15,16}
  20   2^2 * 5   100   2^2 * 5^2     {2,4,5,6,8,12,14,15,16,18}
  21   3 * 7      84   2^2 * 3 * 7   {3,6,7,9,12,14,15,18}
  22   2 * 11    121   11^2          {2,4,6,8,10,11,12,14,16,18,20}
  24   2^3 * 3   144   2^4 * 3^2     {2,3,4,8,9,10,14,15,16,20,21,22}
a(6) = (2+4) + (3) = 9,
a(n) = 6 for n in {8, 9} since 6 is the only number less than n that shares a factor with n but does not have the same squarefree kernel as n.
a(10) = (2+4+6+8) + (5) = 25.
a(12) = (2+4+8+10) + (3+9) = 36.
a(14) = (2+4+6+8+10+12) + (7) = 49.
a(15) = (3+6+9+12) + (5+10) = 45.
a(16) = (6+10+12+14) = 42, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Log log scatterplot of a(n), n = 6..2^14, ignoring a(n) = 0, showing a(n) for prime power n in gold, a(n) for squarefree n in green, otherwise blue.

Cf. A007947, A038566, A066760, A067392, A121998, A369609, A381094, A381096, A381498, A381499.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; Total@ Select[Range[n], Nor[CoprimeQ[#, n], rad[#] == r] &], {n, 120}]

a(n) is the sum of row n of A381094.
a(n) = 0 for prime n and n = 4.
a(n) = A067392(n) - A381498(n).

cp's OEIS Frontend

A381240 Indices of records in A381096.

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A381241 Records in A381096.

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A381094 Triangle read by rows where row n contains k < n that are neither coprime to n nor have the same squarefree kernel as n, or 0 if there are no such k.

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A381497 a(n) = sum of numbers k < n such that 1 < gcd(k,n) and rad(k) != rad(n), where rad = A007947.

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