A377046 -id:A377046 - cp's OEIS frontend

A377048 Antidiagonal-sums of the absolute value of the array A377046(n,k) = n-th term of k-th differences of nonsquarefree numbers (A013929).

4, 12, 13, 22, 28, 31, 39, 64, 85, 132, 395, 1103, 2650, 5868, 12297, 24694, 47740, 88731, 157744, 265744, 418463, 605929, 805692, 1104513, 2396645, 8213998, 21761334, 50923517, 110270883, 225997492, 444193562, 844498084, 1561942458, 2819780451, 4973173841

Offset: 1

Gus Wiseman, Oct 19 2024

nonn

These are the row-sums of the absolute value triangle version of A377046.

			The third antidiagonal of A377046 is (9, 1, -3), so a(3) = 13.

The version for primes is A376681, noncomposites A376684, composites A377035.
For squarefree instead of nonsquarefree numbers we have A377040.
The non-absolute version is A377047.
For leading column we have A377049.
For first position of 0 in each row we have A377050.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
Cf. A007674, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A376591, A376592.

nn=20;
t=Table[Take[Differences[NestList[NestWhile[#+1&, #+1,SquareFreeQ[#]&]&,4,2*nn],k],nn],{k,0,nn}];
Total/@Table[Abs[t[[j,i-j+1]]],{i,nn},{j,i}]

A377047 Antidiagonal-sums of the array A377046(n,k) = n-th term of k-th differences of nonsquarefree numbers (A013929).

Original entry on oeis.org

4, 12, 7, 22, 14, 17, 39, 0, 37, 112, -337, 1103, -2570, 5868, -12201, 24670, -47528, 88283, -155910, 259140, -393399, 512341, -456546, -191155, 2396639, -8213818, 21761218, -50922953, 110269343, -225991348, 444168748, -844390064, 1561482582, -2817844477

Offset: 1

Gus Wiseman, Oct 19 2024

sign

These are the row-sums of the triangle-version of A377046.

			The third antidiagonal of A377046 is (9, 1, -3), so a(3) = 7.

The version for primes is A140119, noncomposites A376683, composites A377034.
For squarefree instead of nonsquarefree numbers we have A377039.
The absolute value version is A377048.
For leading column we have A377049.
For first position of 0 in each row we have A377050.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
Cf. A007674, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A376311, A376591, A376592, A377051.

nn=20;
t=Table[Take[Differences[NestList[NestWhile[#+1&,#+1,SquareFreeQ[#]&]&,4,2*nn],k],nn],{k,0,nn}];
Total/@Table[t[[j,i-j+1]],{i,nn},{j,i}]

A377051 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the powers of primes.

Original entry on oeis.org

1, 2, 1, 3, 1, 0, 4, 1, 0, 0, 5, 1, 0, 0, 0, 7, 2, 1, 1, 1, 1, 8, 1, -1, -2, -3, -4, -5, 9, 1, 0, 1, 3, 6, 10, 15, 11, 2, 1, 1, 0, -3, -9, -19, -34, 13, 2, 0, -1, -2, -2, 1, 10, 29, 63, 16, 3, 1, 1, 2, 4, 6, 5, -5, -34, -97, 17, 1, -2, -3, -4, -6, -10, -16, -21, -16, 18, 115

Offset: 0

Gus Wiseman, Oct 20 2024

Row k of the array is the k-th differences of A000961.

			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     4     5     7     8     9    11
  k=1:   1     1     1     1     2     1     1     2     2
  k=2:   0     0     0     1    -1     0     1     0     1
  k=3:   0     0     1    -2     1     1    -1     1    -3
  k=4:   0     1    -3     3     0    -2     2    -4     6
  k=5:   1    -4     6    -3    -2     4    -6    10    -8
  k=6:  -5    10    -9     1     6   -10    16   -18     5
  k=7:  15   -19    10     5   -16    26   -34    23     9
  k=8: -34    29    -5   -21    42   -60    57   -14   -42
  k=9:  63   -34   -16    63  -102   117   -71   -28   104
Triangle form:
    1
    2    1
    3    1    0
    4    1    0    0
    5    1    0    0    0
    7    2    1    1    1    1
    8    1   -1   -2   -3   -4   -5
    9    1    0    1    3    6   10   15
   11    2    1    1    0   -3   -9  -19  -34
   13    2    0   -1   -2   -2    1   10   29   63
   16    3    1    1    2    4    6    5   -5  -34  -97

Row k=0 is A000961, exclusive A246655.
Row k=1 is A057820.
Row k=2 is A376596.
The version for primes is A095195, noncomposites A376682, composites A377033.
A version for partitions is A175804, cf. A053445, A281425, A320590.
For squarefree numbers we have A377038, nonsquarefree A377046.
Triangle row-sums are A377052, absolute version A377053.
Column n = 1 is A377054, for primes A007442 or A030016.
First position of 0 in each row is A377055.
A000040 lists the primes, differences A001223, seconds A036263.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
Cf. A025475, A053707, A093555, A174965, A361102, A376340, A376598, A376653.

nn=12;
t=Table[Take[Differences[NestList[NestWhile[#+1&, #+1,!PrimePowerQ[#]&]&,1,2*nn],k],nn],{k,0,nn}]
Table[t[[j,i-j+1]],{i,nn},{j,i}]

A(i,j) = Sum_{k=0..j} (-1)^(j-k)*binomial(j,k)*A000961(i+k).

A095195 T(n,0) = prime(n), T(n,k) = T(n,k-1)-T(n-1,k-1), 0<=k

Original entry on oeis.org

2, 3, 1, 5, 2, 1, 7, 2, 0, -1, 11, 4, 2, 2, 3, 13, 2, -2, -4, -6, -9, 17, 4, 2, 4, 8, 14, 23, 19, 2, -2, -4, -8, -16, -30, -53, 23, 4, 2, 4, 8, 16, 32, 62, 115, 29, 6, 2, 0, -4, -12, -28, -60, -122, -237, 31, 2, -4, -6, -6, -2, 10, 38, 98, 220, 457, 37, 6, 4, 8, 14, 20, 22, 12

Offset: 1

Reinhard Zumkeller, Jun 22 2004

T(n,0)=A000040(n); T(n,1)=A001223(n-1) for n>1; T(n,2)=A036263(n-2) for n>2; T(n,n-1)=A007442(n) for n>1.

Row k of the array (not the triangle) is the k-th differences of the prime numbers. - Gus Wiseman, Jan 11 2025

			Triangle begins:
   2;
   3,  1;
   5,  2,  1;
   7,  2,  0, -1;
  11,  4,  2,  2,  3;
  13,  2, -2, -4, -6, -9;
Alternative: array form read by antidiagonals:
     2,   3,   5,   7,  11,  13,  17,  19,  23,  29,  31,...
     1,   2,   2,   4,   2,   4,   2,   4,   6,   2,   6,...
     1,   0,   2,  -2,   2,  -2,   2,   2,  -4,   4,  -2,...
    -1,   2,  -4,   4,  -4,   4,   0,  -6,   8,  -6,   0,...
     3,  -6,   8,  -8,   8,  -4,  -6,  14, -14,   6,   4,...
    -9,  14, -16,  16, -12,  -2,  20, -28,  20,  -2,  -8,...
    23, -30,  32, -28,  10,  22, -48,  48, -22,  -6,  10,..,
   -53,  62, -60,  38,  12, -70,  96, -70,  16,  16, -12,...
   115,-122,  98, -26, -82, 166,-166,  86,   0, -28,  28,...
  -237, 220,-124, -56, 248,-332, 252, -86, -28,  56, -98,...
   457,-344,  68, 304,-580, 584,-338,  58,  84,-154, 308,...

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A036262-A036271.
Cf. A140119 (row sums).
Below, the inclusive primes (A008578) are 1 followed by A000040. See also A075526.
Rows of the array (columns of the triangle) begin: A000040, A001223, A036263.
Column n = 1 of the array is A007442, inclusive A030016.
The version for partition numbers is A175804, see A053445, A281425, A320590.
First position of 0 is A376678, inclusive A376855.
Absolute antidiagonal-sums are A376681, inclusive A376684.
The inclusive version is A376682.
For composite instead of prime we have A377033, see A377034-A377037.
For squarefree instead of prime we have A377038, nonsquarefree A377046.
Column n = 2 of the array is A379542.
Cf. A002808, A064113, A065890, A073783, A258026, A293467, A333254, A376683, A377051.

a095195 n k = a095195_tabl !! (n-1) !! (k-1)
a095195_row n = a095195_tabl !! (n-1)
a095195_tabl = f a000040_list [] where
   f (p:ps) xs = ys : f ps ys where ys = scanl (-) p xs
-- Reinhard Zumkeller, Oct 10 2013

A095195A := proc(n,k) # array, k>=0, n>=0
    option remember;
    if n =0 then
        ithprime(k+1) ;
    else
        procname(n-1,k+1)-procname(n-1,k) ;
    end if;
end proc:
A095195 := proc(n,k) # triangle, 0<=k=1
        A095195A(k,n-k-1) ;
end proc: # R. J. Mathar, Sep 19 2013

T[n_, 0] := Prime[n]; T[n_, k_] /; 0 <= k < n := T[n, k] = T[n, k-1] - T[n-1, k-1]; Table[T[n, k], {n, 1, 12}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Feb 01 2017 *)
nn=6;
t=Table[Differences[Prime[Range[nn]],k],{k,0,nn}];
Table[t[[j,i-j+1]],{i,nn},{j,i}] (* Gus Wiseman, Jan 11 2025 *)

A175804 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the n-th term of the k-th differences of partition numbers A000041.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, -1, 0, 1, 3, 2, 1, 1, 2, 5, -4, -2, -1, 0, 2, 7, 9, 5, 3, 2, 2, 4, 11, -21, -12, -7, -4, -2, 0, 4, 15, 49, 28, 16, 9, 5, 3, 3, 7, 22, -112, -63, -35, -19, -10, -5, -2, 1, 8, 30, 249, 137, 74, 39, 20, 10, 5, 3, 4, 12, 42, -539, -290, -153, -79, -40, -20, -10, -5, -2, 2, 14, 56

Offset: 0

Alois P. Heinz, Dec 04 2010

Odlyzko showed that the k-th differences of A000041(n) alternate in sign with increasing n up to a certain index n_0(k) and then stay positive.

Are there any zeros after the first four, which all lie in columns k = 1, 2? - Gus Wiseman, Dec 15 2024

			Square array A(n,k) begins:
   1,  0,  1, -1,  2,  -4,   9,  ...
   1,  1,  0,  1, -2,   5, -12,  ...
   2,  1,  1, -1,  3,  -7,  16,  ...
   3,  2,  0,  2, -4,   9, -19,  ...
   5,  2,  2, -2,  5, -10,  20,  ...
   7,  4,  0,  3, -5,  10, -20,  ...
  11,  4,  3, -2,  5, -10,  22,  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Gert Almkvist, On the differences of the partition function, Acta Arith., 61.2 (1992), 173-181.
Charles Knessl, Asymptotic Behavior of High-Order Differences of the Partition Function, Communications on Pure and Applied Mathematics, 44 (1991), 1033-1045.
A. M. Odlyzko, Differences of the partition function, Acta Arith., 49 (1988), 237-254.

Columns k=0-5 give: A000041, A002865, A053445, A072380, A081094, A081095.
Main diagonal gives A379378.
Cf. A119712, A155861.
For primes we have A095195 or A376682.
Row n = 0 is A281425.
Row n = 1 is A320590 except first term.
For composites we have A377033.
For squarefree numbers we have A377038.
For nonsquarefree numbers we have A377046.
For prime powers we have A377051.
Antidiagonal sums are A377056, absolute value version A378621.
The version for strict partitions is A378622, first column A293467.
A000009 counts strict integer partitions, differences A087897, A378972.
Cf. A008284, A047966, A098859, A116608, A225486, A325242, A225485 or A325280, A377285, A378970.

A41:= combinat[numbpart]:
DD:= proc(p) proc(n) option remember; p(n+1) -p(n) end end:
A:= (n,k)-> (DD@@k)(A41)(n):
seq(seq(A(n, d-n), n=0..d), d=0..11);

max = 11; a41 = Array[PartitionsP, max+1, 0]; a[n_, k_] := Differences[a41, k][[n+1]]; Table[a[n, k-n], {k, 0, max}, {n, 0, k}] // Flatten (* Jean-François Alcover, Aug 29 2014 *)
nn=5;Table[Table[Sum[(-1)^(k-i)*Binomial[k,i]*PartitionsP[n+i],{i,0,k}],{k,0,nn}],{n,0,nn}] (* Gus Wiseman, Dec 15 2024 *)

A(n,k) = (Delta^(k) A000041)(n).

A(n,k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k,i) * A000041(n+i). In words, row x is the inverse zero-based binomial transform of A000041 shifted left x times. - Gus Wiseman, Dec 15 2024

A378033 Greatest nonsquarefree number <= n, or 1 if there is none (the case n <= 3).

Original entry on oeis.org

1, 1, 1, 4, 4, 4, 4, 8, 9, 9, 9, 12, 12, 12, 12, 16, 16, 18, 18, 20, 20, 20, 20, 24, 25, 25, 27, 28, 28, 28, 28, 32, 32, 32, 32, 36, 36, 36, 36, 40, 40, 40, 40, 44, 45, 45, 45, 48, 49, 50, 50, 52, 52, 54, 54, 56, 56, 56, 56, 60, 60, 60, 63, 64, 64, 64, 64, 68

Offset: 1

Gus Wiseman, Nov 18 2024

nonn

			The nonsquarefree numbers <= 10 are {4, 8, 9}, so a(10) = 9.

For prime-powers we have A031218, differences A377782.
Greatest of the nonsquarefree numbers counted by A057627.
The opposite for squarefree is A067535, differences A378087.
For squarefree we have A070321, differences A378085.
The opposite is A120327 (union A162966), differences A378039.
The restriction to the primes is A378032, opposite A377783 (union A378040).
First-differences are A378036, restriction A378034.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers, differences A076259, seconds A376590.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A061398 counts squarefree numbers between primes (sums A337030), zeros A068360.
A061399 counts nonsquarefree numbers between primes (sums A378086), zeros A068361.
A112925 gives the greatest squarefree number < prime(n), differences A378038.
A112926 gives the least squarefree number > prime(n), differences A378037.
A377046 encodes k-differences of nonsquarefree numbers, zeros A377050.
Cf. A053797, A053806, A377047, A377048, A377049, A377784, A378082, A378083, A378084.

Table[NestWhile[#-1&,n,#>1&&SquareFreeQ[#]&],{n,100}]

a(n) = my(k=n); while (issquarefree(k), k--); if(!k, 1, k); \\ Michel Marcus, Jul 26 2025

a(prime(n)) = A378032(n).

a(n) = A013929(A057627(n)), for n > 3. - Ridouane Oudra, Jul 26 2025

A378032 a(1) = a(2) = 1; a(n>2) is the greatest nonsquarefree number < prime(n).

Original entry on oeis.org

1, 1, 4, 4, 9, 12, 16, 18, 20, 28, 28, 36, 40, 40, 45, 52, 56, 60, 64, 68, 72, 76, 81, 88, 96, 100, 100, 104, 108, 112, 126, 128, 136, 136, 148, 150, 156, 162, 164, 172, 176, 180, 189, 192, 196, 198, 208, 220, 225, 228, 232, 236, 240, 250, 256, 261, 268, 270

Offset: 1

Gus Wiseman, Nov 16 2024

nonn

			The terms together with their prime indices begin:
    1: {}
    1: {}
    4: {1,1}
    4: {1,1}
    9: {2,2}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   28: {1,1,4}
   28: {1,1,4}
   36: {1,1,2,2}
   40: {1,1,1,3}
   40: {1,1,1,3}
   45: {2,2,3}
   52: {1,1,6}
   56: {1,1,1,4}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}
   68: {1,1,7}
   72: {1,1,1,2,2}

Terms appearing twice are A061351 + 1.
For prime-powers we have A065514 (diffs A377781), opposite A345531 (diffs A377703).
For squarefree we have A112925 (differences A378038).
The opposite for squarefree is A112926 (differences A378037).
The opposite is A377783 (union A378040), restriction of A120327 (differences A378039).
Restriction of A378033, which has differences A378036.
The first-differences are A378034, opposite A377784.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A061398 counts squarefree numbers between primes (sums A337030), zeros A068360.
A061399 counts nonsquarefree numbers between primes (sums A378086), zeros A068361.
A070321 gives the greatest squarefree number up to n.
A377046 encodes k-differences of nonsquarefree numbers, zeros A377050.
Cf. A053797, A053806, A072284, A065890, A224363, A366833, A377431.
Cf. A377047, A377048, A377049, A377430, A378082, A378083, A378084.

Table[NestWhile[#-1&,Prime[n],#>1&&SquareFreeQ[#]&],{n,100}]

a(n) = A378033(prime(n)).

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

Original entry on oeis.org

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

Gus Wiseman, Oct 18 2024

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

			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

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.

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}]

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

A377033 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the composite numbers (A002808).

Original entry on oeis.org

4, 6, 2, 8, 2, 0, 9, 1, -1, -1, 10, 1, 0, 1, 2, 12, 2, 1, 1, 0, -2, 14, 2, 0, -1, -2, -2, 0, 15, 1, -1, -1, 0, 2, 4, 4, 16, 1, 0, 1, 2, 2, 0, -4, -8, 18, 2, 1, 1, 0, -2, -4, -4, 0, 8, 20, 2, 0, -1, -2, -2, 0, 4, 8, 8, 0, 21, 1, -1, -1, 0, 2, 4, 4, 0, -8, -16, -16

Offset: 0

Gus Wiseman, Oct 17 2024

Row n is the k-th differences of A002808 = the composite numbers.

			Array begins:
        n=1:  n=2:  n=3:  n=4:  n=5:  n=6:  n=7:  n=8:  n=9:
  ----------------------------------------------------------
  k=0:   4     6     8     9    10    12    14    15    16
  k=1:   2     2     1     1     2     2     1     1     2
  k=2:   0    -1     0     1     0    -1     0     1     0
  k=3:  -1     1     1    -1    -1     1     1    -1    -1
  k=4:   2     0    -2     0     2     0    -2     0     2
  k=5:  -2    -2     2     2    -2    -2     2     2    -2
  k=6:   0     4     0    -4     0     4     0    -4    -1
  k=7:   4    -4    -4     4     4    -4    -4     3    10
  k=8:  -8     0     8     0    -8     0     7     7   -29
  k=9:   8     8    -8    -8     8     7     0   -36    63
Triangle begins:
    4
    6    2
    8    2    0
    9    1   -1   -1
   10    1    0    1    2
   12    2    1    1    0   -2
   14    2    0   -1   -2   -2    0
   15    1   -1   -1    0    2    4    4
   16    1    0    1    2    2    0   -4   -8
   18    2    1    1    0   -2   -4   -4    0    8
   20    2    0   -1   -2   -2    0    4    8    8    0
   21    1   -1   -1    0    2    4    4    0   -8  -16  -16

Initial rows: A002808, A073783, A073445.
The version for primes is A095195 or A376682.
A version for partitions is A175804, cf. A053445, A281425, A320590.
Triangle row-sums are A377034, absolute version A377035.
Column n = 1 is A377036, for primes A007442 or A030016.
First position of 0 in each row is A377037.
Other arrays of differences: A095195 (prime), A376682 (noncomposite), A377033 (composite), A377038 (squarefree), A377046 (nonsquarefree), A377051 (prime-power).
A000040 lists the primes, differences A001223, seconds A036263.
A008578 lists the noncomposites, differences A075526.
Cf. A065310, A065890, A084758, A173390, A350004, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative), A376680.

nn=9;
t=Table[Take[Differences[NestList[NestWhile[#+1&, #+1,PrimeQ]&,4,2*nn],k],nn],{k,0,nn}]

A(i,j) = Sum_{k=0..j} (-1)^(j-k) binomial(j,k) A002808(i+k).

A378036 First differences of A378033 (greatest positive integer < n that is 1 or nonsquarefree).

Original entry on oeis.org

0, 0, 3, 0, 0, 0, 4, 1, 0, 0, 3, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 4, 1, 0, 2, 1, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 4, 1, 0, 0, 3, 1, 1, 0, 2, 0, 2, 0, 2, 0, 0, 0, 4, 0, 0, 3, 1, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 3, 1, 0, 0, 0, 4, 1, 0, 0, 3, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 4, 0, 2, 1, 1, 0, 0, 0, 4, 0, 0, 0, 4

Offset: 1

Gus Wiseman, Nov 18 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65539

Positions of 0 are A005117 - 1, complement A013929 - 1.
Sums for squarefree numbers are A070321 (restriction A112925).
The restricted opposite is A377784, differences of A377783 (union A378040).
First-differences of A378033.
The restriction is A378034, differences of A378032.
The restricted opposite for squarefree is A378037, differences of A112926.
The opposite is A378039, differences of A120327 (union A162966).
For squarefree numbers we have A378085, restriction A378038.
The opposite for squarefree is A378087, differences of A067535.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers, differences A076259, seconds A376590.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A061398 counts squarefree numbers between primes (sums A337030), zeros A068360.
A061399 counts nonsquarefree numbers between primes (sums A378086), zeros A068361.
A377046 encodes k-differences of nonsquarefree numbers, zeros A377050.
Cf. A053797, A053806, A065514, A377047, A377048, A377049, A377703, A378082, A378083, A378084.

Differences[Table[NestWhile[#-1&,n,#>1&&SquareFreeQ[#]&],{n,100}]]

A378033(n) = if(n<=3, 1, forstep(k=n, 0, -1, if(!issquarefree(k), return(k))));
A378036(n) = (A378033(1+n)-A378033(n)); \\ Antti Karttunen, Jan 28 2025

a(prime(n)) = A378034(n).

Data section extended to a(107) by Antti Karttunen, Jan 28 2025

cp's OEIS Frontend

A377048 Antidiagonal-sums of the absolute value of the array A377046(n,k) = n-th term of k-th differences of nonsquarefree numbers (A013929).

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A377047 Antidiagonal-sums of the array A377046(n,k) = n-th term of k-th differences of nonsquarefree numbers (A013929).

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A377051 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the powers of primes.

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A095195 T(n,0) = prime(n), T(n,k) = T(n,k-1)-T(n-1,k-1), 0<=k

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A175804 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the n-th term of the k-th differences of partition numbers A000041.

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A378033 Greatest nonsquarefree number <= n, or 1 if there is none (the case n <= 3).

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A378032 a(1) = a(2) = 1; a(n>2) is the greatest nonsquarefree number < prime(n).

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

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