A377038 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the squarefree numbers.

1, 2, 1, 3, 1, 0, 5, 2, 1, 1, 6, 1, -1, -2, -3, 7, 1, 0, 1, 3, 6, 10, 3, 2, 2, 1, -2, -8, 11, 1, -2, -4, -6, -7, -5, 3, 13, 2, 1, 3, 7, 13, 20, 25, 22, 14, 1, -1, -2, -5, -12, -25, -45, -70, -92, 15, 1, 0, 1, 3, 8, 20, 45, 90, 160, 252, 17, 2, 1, 1, 0, -3, -11, -31, -76, -166, -326, -578

Offset

0,2

Comments

Row n is the k-th differences of A005117 = the squarefree numbers.

Links

Table of n, a(n) for n=0..77.

Formula

A(i,j) = sum_{k=0..j} (-1)^(j-k) binomial(j,k) A005117(i+k).

Example

Array form:
        n=1:  n=2:  n=3:  n=4:  n=5:  n=6:  n=7:  n=8:  n=9:
  ----------------------------------------------------------
  k=0:   1     2     3     5     6     7    10    11    13
  k=1:   1     1     2     1     1     3     1     2     1
  k=2:   0     1    -1     0     2    -2     1    -1     0
  k=3:   1    -2     1     2    -4     3    -2     1     1
  k=4:  -3     3     1    -6     7    -5     3     0    -2
  k=5:   6    -2    -7    13   -12     8    -3    -2     3
  k=6:  -8    -5    20   -25    20   -11     1     5    -5
  k=7:   3    25   -45    45   -31    12     4   -10    10
  k=8:  22   -70    90   -76    43    -8   -14    20   -19
  k=9: -92   160  -166   119   -51    -6    34   -39    28
Triangle form:
   1
   2   1
   3   1   0
   5   2   1   1
   6   1  -1  -2  -3
   7   1   0   1   3   6
  10   3   2   2   1  -2  -8
  11   1  -2  -4  -6  -7  -5   3
  13   2   1   3   7  13  20  25  22
  14   1  -1  -2  -5 -12 -25 -45 -70 -92
  15   1   0   1   3   8  20  45  90 160 252

Mathematica

nn=9;
t=Table[Take[Differences[NestList[NestWhile[#+1&, #+1, !SquareFreeQ[#]&]&, 1, 2*nn], k], nn], {k, 0, nn}]
Table[t[[j, i-j+1]], {i, nn}, {j, i}]

Crossrefs

Row k=0 is A005117.

Row k=1 is A076259.

Row k=2 is A376590.

The version for primes is A095195, noncomposites A376682, composites A377033.

A version for partitions is A175804, cf. A053445, A281425, A320590.

Triangle row-sums are A377039, absolute version A377040.

Column n = 1 is A377041, for primes A007442 or A030016.

First position of 0 in each row is A377042.

For nonsquarefree instead of squarefree numbers we have A377046.

For prime-powers instead of squarefree numbers we have A377051.

A000040 lists the primes, differences A001223, seconds A036263.

A005117 lists the squarefree numbers, complement A013929 (differences A078147).

A073576 counts integer partitions into squarefree numbers, factorizations A050320.

Cf. A007674, A053797, A053806, A061398, A072284, A075526, A112925, A112926, A120992, A373198, A376311, A376591, A376592.

Sequence in context: A375037 A334318 A199056 * A376682 A350004 A144966

Adjacent sequences: A377035 A377036 A377037 * A377039 A377040 A377041

Keyword

sign,tabl

Author

Gus Wiseman, Oct 18 2024

Status

approved