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

1,8

Comments

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

Links

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.

Formula

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

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

From Michael De Vlieger, Apr 29 2026: (Start)

a(n) = A000217(n) - A023896(n) - A244974(n) + 1.

a(n) = A000217(n) - A023896(n) - A000203(n) - A395300(n) + 1.

a(n) = A067392(n) - A395300(n) - A000203(n).

a(n) = A066760(n) - A395300(n). (End)

Example

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}

Mathematica

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}]

Crossrefs

Cf. A000203, A000217, A002808, A007947, A023896, A038566, A066760, A067392, A121998, A243823, A244974, A272619, A381497, A395300.

Sequence in context: A019180 A019103 A272619 * A172360 A175288 A349187

Adjacent sequences: A381496 A381497 A381498 * A381500 A381501 A381502

Keyword

nonn

Author

Michael De Vlieger, Mar 02 2025

Status

approved