A381094 -id:A381094 - 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).

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).

A381499 a(n) = sum of numbers k < n such that 1 < gcd(k,n) < k and rad(k) does not divide n, where rad = A007947.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 6, 6, 6, 0, 10, 0, 28, 28, 42, 0, 39, 0, 65, 65, 80, 0, 102, 45, 126, 96, 159, 0, 111, 0, 210, 148, 210, 138, 253, 0, 280, 221, 338, 0, 342, 0, 411, 366, 444, 0, 547, 140, 563, 403, 601, 0, 700, 344, 708, 512, 750, 0, 751, 0, 868, 703, 930

Offset: 1

Michael De Vlieger, Mar 02 2025

nonn

Analogous to A066760(n), the sum of row n of A133995, and A381497(n), sum of row n of A381094.

			Table of n and a(n) for select n, showing prime power decomposition of the latter and row n of A272619:
 n   a(n)  Factor(a(n))  Row n of A272619
-----------------------------------------------------
 8     6   2 * 3         {6}
 9     6   2 * 3         {6}
10     6   2 * 3         {6}
12    10   2 * 5         {10}
14    28   2^2 * 7       {6,10,12}
15    28   2^2 * 7       {6,10,12}
16    42   2 * 3 * 7     {6,10,12,14}
18    39   3 * 13        {10,14,15}
20    65   5 * 13        {6,12,14,15,18}
21    65   5 * 13        {6,12,14,15,18}
22    80   2^4 * 5       {6,10,12,14,18,20}
24   102   2 * 3 * 17    {10,14,15,20,21,22}
25    45   3^2 * 5       {10,15,20}
26   126   2 * 3^2 * 7   {6,10,12,14,18,20,22,24}
27    96   2^5 * 3       {6,12,15,18,21,24}
28   159   3 * 53        {6,10,12,18,20,21,22,24,26}

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Log log scatterplot of a(n), n = 8..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, A121998, A243823, A272619, A381497.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; If[PrimeQ[n], 0, Total@ Select[Range[n], And[1 < GCD[#, n] < #, ! Divisible[n, rad[#]]] &]], {n, 120}]

a(n) is the sum of row n of A272619.

a(n) = 0 for prime n, n = 4, and n = 6.

A381674 a(n) = product of numbers k < n such that 1 < gcd(k,n) and rad(k) != rad(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 24, 1, 6, 6, 1920, 1, 17280, 1, 322560, 97200, 10080, 1, 58060800, 1, 1393459200, 51438240, 40874803200, 1, 536481792000, 3000, 25505877196800, 9797760, 535623421132800, 1, 40999294770610176000000, 1, 41845579776000, 51855036710400, 23310331287699456000

Offset: 1

Michael De Vlieger, Mar 15 2025

nonn

Terms are in A055932.

The only squarefree terms are 1 and 6.

			Table of n and a(n) for select n, showing exponents of prime factors of the latter and row n of A381094:
                                             1  1  1
   n                       a(n)  2  3  5  7  1  3  7   Row n of A381094
  ---------------------------------------------------------------------------------------
   6                        24   3, 1                  {2,3,4}
   8                         6   1, 1                  {6}
   9                         6   1, 1                  {6}
  10                      1920   7, 1, 1               {2,4,5,6,8}
  12                     17280   7, 3, 1               {2,3,4,8,9,10}
  14                    322560  10, 2, 1, 1            {2,4,6,7,8,10,12}
  15                     97200   4, 5, 2               {3,5,6,9,10,12}
  16                     10080   5, 2, 1, 1            {6,10,12,14}
  18                  58060800  12, 4, 2, 1            {2,3,4,8,9,10,14,15,16}
  20                1393459200  15, 5, 2, 1            {2,4,5,6,8,12,14,15,16,18}
  24              536481792000  15, 5, 3, 2, 1         {2,3,4,8,9,10,14,15,16,20,21,22}
  25                      3000   3, 1, 3               {10,15,20}
  30   40999294770610176000000  25,13, 6, 3, 1, 1      {2,3,4,5,6,8,9,10,12,14,..,28}
  32            41845579776000  16, 6, 3, 2, 1, 1      {6,10,12,14,18,20,22,24,26,28,30}
  36   11358323143857930240000  25,10, 4, 3, 2, 1, 1   {2,3,4,8,9,10,14,15,16,20,..,34}
a(n) = 6 for n = 8 or 9, since 6 is the only number less than n that shares a factor with n but rad(6) != rad(n).
a(6) = (2*3)*(4) = 24.
a(10) = (2*4*6*8)*(5) = 1920.
a(12) = (2*4*8*10)*(3*9) = 17280, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..599
Michael De Vlieger, Log log scatterplot of log_10(a(n)), n = 1..2^14, showing n that are prime powers in gold, n that are squarefree in green, and other n in blue and magenta, where magenta additionally represents powerful n that are not prime powers.
Michael De Vlieger, Plot prime(i)^m | a(n) at (x,y) = (n, i), n = 1..2048, 2X vertical exaggeration, with a color function representing m, where black represents m = 1, red m = 2, ..., magenta represents the largest m in the dataset, i.e., m = 2035.

Cf. A055932, A070251, A381094, A381497, A381675.

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

a(n) is the product of row n of A381094.
a(n) = 1 for prime n and n = 4.
a(2*p) = p * 2^(p-1) * (p-1)! = A381675(n) for odd prime p = prime(n), n > 1.

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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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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A381499 a(n) = sum of numbers k < n such that 1 < gcd(k,n) < k and rad(k) does not divide n, where rad = A007947.

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A381674 a(n) = product of numbers k < n such that 1 < gcd(k,n) and rad(k) != rad(n).

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