A381497 -id:A381497 - cp's OEIS frontend

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.

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

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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