A377038 -id:A377038 - cp's OEIS frontend

A376683 Antidiagonal-sums of the array A376682(n,k) = n-th term of the k-th differences of the noncomposite numbers (A008578).

1, 3, 4, 9, 6, 27, -20, 109, -182, 471, -868, 1737, -2872, 4345, -4700, 1133, 14060, -55275, 150462, -346093, 717040, -1369351, 2432872, -4002905, 5964846, -7524917, 6123130, 4900199, -40900410, 134309057, -348584552, 798958881, -1678213106, 3277459119

Offset: 0

Gus Wiseman, Oct 15 2024

sign

			The fourth anti-diagonal of A376682 is: (7, 2, 0, -1, -2), so a(4) = 6.

The modern version (for A000040 instead of A008578) is A140119.
The absolute version is A376681.
Antidiagonal-sums of A376682 (modern version A095195).
For composite instead of noncomposite we have A377033.
For squarefree instead of noncomposite we have A377038, nonsquarefree A377046.
A000040 lists the modern primes, differences A001223, second A036263.
A008578 lists the noncomposites, first differences A075526.
Cf. A007442, A030016, A064113, A065890, A002808, A173390, A233671, A333214, A333254, A376678, A376684, A376855.

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

A377034 Antidiagonal-sums of the array A377033(n,k) = n-th term of the k-th differences of the composite numbers (A002808).

Original entry on oeis.org

4, 8, 10, 8, 14, 14, 11, 24, 10, 20, 37, -10, 56, 26, -52, 260, -659, 2393, -8128, 25703, -72318, 184486, -430901, 933125, -1888651, 3597261, -6479654, 11086964, -18096083, 28307672, -42644743, 62031050, -86466235, 110902085, -110907437, 52379, 483682985

Offset: 1

Gus Wiseman, Oct 17 2024

sign

Row-sums of the triangle version of A377033.

			The fourth antidiagonal of A377033 is (9, 1, -1, -1), so a(4) = 8.

The version for prime instead of composite is A140119, noncomposite A376683.
This is the antidiagonal-sums of the array A377033, absolute version A377035.
For squarefree instead of composite we have A377039, absolute version A377040.
For nonsquarefree instead of composite we have A377047, absolute version A377048.
For prime-power instead of composite we have A377052, absolute version A377053.
Other arrays of differences: A095195 (prime), A376682 (noncomposite), A377033 (composite), A377038 (squarefree), A377046 (nonsquarefree), A377051 (prime-power).
A000040 lists the primes, differences A001223, second A036263.
A002808 lists the composite numbers, differences A073783, second A073445.
A008578 lists the noncomposites, differences A075526.
Cf. A018252, A065310, A065890, A333254, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative), A376680, A377036.

q=Select[Range[100],CompositeQ];
t=Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[i+k]],{k,0,j}],{j,0,Length[q]/2},{i,Length[q]/2}];
Total/@Table[t[[j,i-j+1]],{i,Length[q]/2},{j,i}]

A377037 Position of first zero in the n-th differences of the composite numbers (A002808), or 0 if it does not appear.

Original entry on oeis.org

1, 14, 2, 65, 1, 83, 2, 7, 1, 83, 2, 424, 12, 32, 11, 733, 10, 940, 9, 1110, 8, 1110, 7, 1110, 6, 1110, 112, 1110, 111, 1110, 110, 2192, 109, 13852, 108, 13852, 107, 13852, 106, 13852, 105, 17384, 104, 17384, 103, 17384, 102, 17384, 101, 27144, 552, 28012, 551

Offset: 2

Gus Wiseman, Oct 17 2024

nonn

			The third differences of the composite numbers are:
  -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -2, 1, 0, 0, 1, -1, -1, ...
so a(3) = 14.

Alois P. Heinz, Table of n, a(n) for n = 2..100

The version for prime instead of composite is A376678.
For noncomposite numbers we have A376855.
This is the first position of 0 in row n of the array A377033.
For squarefree instead of composite we have A377042, nonsquarefree A377050.
For prime-power instead of composite we have A377055.
Other arrays of differences: A095195 (prime), A376682 (noncomposite), A377033 (composite), A377038 (squarefree), A377046 (nonsquarefree), A377051 (prime-power).
A000040 lists the primes, differences A001223, second A036263.
A002808 lists the composite numbers, differences A073783, second A073445.
A008578 lists the noncomposites, differences A075526.
A377036 gives first term of the n-th differences of the composite numbers, for primes A007442 or A030016.
Cf. A018252, A064113, A065310, A065890, A140119, A173390, A233671, A258025, A258026, A350004, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative), A376680, A377034, A377035.

nn=10000;
u=Table[Differences[Select[Range[nn],CompositeQ],k],{k,2,16}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
m=Table[Position[u[[k]],0][[1,1]],{k,mnrm[Union[First/@Position[u,0]]]}]

Offset 2 from Michel Marcus, Oct 18 2024

a(17)-a(54) from Alois P. Heinz, Oct 18 2024

A379308 Number of integer partitions of n with a unique squarefree part.

Original entry on oeis.org

0, 1, 1, 1, 0, 2, 2, 2, 0, 3, 5, 5, 1, 6, 9, 9, 2, 10, 14, 18, 6, 18, 24, 30, 11, 28, 39, 47, 24, 48, 63, 76, 41, 74, 95, 118, 65, 120, 149, 181, 107, 181, 221, 266, 169, 266, 335, 398, 262, 394, 487, 578, 391, 578, 697, 844, 592, 834, 997, 1198, 867

Offset: 0

Gus Wiseman, Dec 26 2024

nonn

			The a(1) = 1 through a(11) = 5 partitions:
  (1)  (2)  (3)  .  (5)    (6)    (7)    .  (5,4)    (10)     (11)
                    (4,1)  (4,2)  (4,3)     (8,1)    (6,4)    (7,4)
                                            (4,4,1)  (8,2)    (8,3)
                                                     (9,1)    (9,2)
                                                     (4,4,2)  (4,4,3)

If all parts are squarefree we have A073576 (strict A087188), ranks A302478.
If no parts are squarefree we have A114374 (strict A256012), ranks A379307.
For composite instead of squarefree we have A379302 (strict A379303), ranks A379301.
For prime instead of squarefree we have A379304, (strict A379305), ranks A331915.
The strict case is A379309.
For old prime instead of squarefree we have A379314, (strict A379315), ranks A379312.
Ranked by A379316, positions of 1 in A379306.
A000041 counts integer partitions, strict A000009.
A005117 lists the squarefree numbers, differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147.
A377038 gives k-th differences of squarefree numbers.
A379310 counts nonsquarefree prime indices.
Cf. A000586, A000607, A002095, A013928, A023895, A034891, A072284, A073247, A120327, A175804, A376657, A377430.

Table[Length[Select[IntegerPartitions[n],Count[#,_?SquareFreeQ]==1&]],{n,0,30}]

A379309 Number of strict integer partitions of n with a unique squarefree part.

Original entry on oeis.org

0, 1, 1, 1, 0, 2, 2, 2, 0, 2, 4, 4, 1, 4, 7, 7, 2, 6, 8, 11, 4, 9, 13, 17, 7, 13, 20, 22, 13, 20, 29, 33, 21, 29, 40, 47, 27, 41, 56, 64, 42, 59, 77, 85, 60, 74, 104, 115, 83, 101, 141, 155, 113, 138, 179, 206, 156, 183, 236, 272, 212, 239, 309, 343, 282, 315

Offset: 0

Gus Wiseman, Dec 27 2024

nonn

			The a(9) = 2 through a(15) = 7 partitions:
  (5,4)  (10)   (11)   (9,3)  (13)     (14)     (15)
  (8,1)  (6,4)  (7,4)         (8,5)    (8,6)    (8,7)
         (8,2)  (8,3)         (12,1)   (9,5)    (9,6)
         (9,1)  (9,2)         (8,4,1)  (10,4)   (11,4)
                                       (12,2)   (12,3)
                                       (8,4,2)  (8,4,3)
                                       (9,4,1)  (9,4,2)

If all parts are squarefree we have A087188, non-strict A073576 (ranks A302478).
If no parts are squarefree we have A256012, non-strict A114374 (ranks A379307).
For composite instead of squarefree we have A379303, non-strict A379302 (ranks A379301).
For prime instead of squarefree we have A379305, non-strict A379304 (ranks A331915).
The non-strict version is A379308, ranks A379316.
For old prime instead of squarefree we have A379315, non-strict A379314 (ranks A379312).
Ranked by A379316 /\ A005117 = squarefree positions of 1 in A379306.
A000041 counts integer partitions, strict A000009.
A005117 lists the squarefree numbers, differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147.
A377038 gives k-th differences of squarefree numbers.
A379310 counts nonsquarefree prime indices.
Cf. A000586, A000607, A002095, A023895, A034891, A036497, A072284, A073247, A096258, A204389, A377430.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,_?SquareFreeQ]==1&]],{n,0,30}]

lista(nn) = my(r=1, s=0); for(k=1, nn, if(issquarefree(k), s+=x^k, r*=1+x^k)); concat(0, Vec(r*s+O(x^(1+nn)))); \\ Jinyuan Wang, Feb 21 2025

More terms from Jinyuan Wang, Feb 21 2025

A377035 Antidiagonal-sums of the absolute value of the array A377033(n,k) = n-th term of the k-th differences of the composite numbers (A002808).

Original entry on oeis.org

4, 8, 10, 12, 14, 18, 21, 28, 34, 40, 47, 74, 96, 110, 138, 286, 715, 2393, 8200, 25731, 72468, 184716, 431575, 934511, 1892267, 3605315, 6494464, 11116110, 18134549, 28348908, 42701927, 62290660, 88313069, 120999433, 159769475, 221775851, 483797879

Offset: 1

Gus Wiseman, Oct 18 2024

nonn

			The fourth antidiagonal of A377033 is (9, 1, -1, -1), so a(4) = 12.

The version for prime instead of composite is A376681, absolute version of A140119.
The version for noncomposite is A376684, absolute version of A376683.
This is the antidiagonal-sums of absolute value of the array A377033.
For squarefree instead of composite we have A377040, absolute version of A377039.
For nonsquarefree instead of composite we have A377048, absolute version of A377047.
For prime-power instead of composite we have A377053, absolute version of A377052.
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.
A002808 lists the composite numbers, differences A073783, seconds A073445.
A008578 lists the noncomposites, differences A075526.
Cf. A018252, A065310, A065890, A333254, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative), A376680, A377036.

q=Select[Range[120],CompositeQ];
t=Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[i+k]],{k,0,j}],{j,0,Length[q]/2},{i,Length[q]/2}];
Total/@Table[Abs[t[[j,i-j+1]]],{i,Length[q]/2},{j,i}]

A377042 Position of first zero in the n-th differences of the squarefree numbers (A005117), or 0 if it does not appear.

Original entry on oeis.org

0, 0, 1, 11, 8, 57, 14, 11, 13, 1019, 44, 1250, 43, 2721, 42, 249522, 2840, 1989839, 2839, 3373774, 4933, 142715511, 42793, 435650856, 5266, 30119361, 104063, 454172978707, 100285, 434562125244, 2755089, 2409925829164, 2485612

Offset: 0

Gus Wiseman, Oct 18 2024

a(n) for n even appear to be smaller than a(n) for n odd. - Chai Wah Wu, Oct 19 2024

a(33) > 10^13, unless it is 0. - Lucas A. Brown, Nov 15 2024

			The fourth differences begin: -3, 3, 1, -6, 7, -5, 3, 0, -2, ... so a(4) = 8

Lucas A. Brown, Python program.

The version for primes is A376678, noncomposites A376855, composites A377037.
This is the first position of 0 in each row of A377038.
For nonsquarefree numbers we have A377050.
For prime-powers we have A377055.
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.
A377039 gives antidiagonal-sums of A377038, absolute version A377040.
A377041 gives first column of A377038, for primes A007442 or A030016.
Cf. A007674, A053797, A053806, A061398, A072284, A076259, A112925, A120992, A376311, A376590, A376591, A377046.

nn=10000;
u=Table[Differences[Select[Range[nn],SquareFreeQ],k],{k,2,16}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
m=Table[Position[u[[k]],0][[1,1]],{k,mnrm[Union[First/@Position[u,0]]]}]

a(15)-a(20) from Chai Wah Wu, Oct 19 2024

a(21)-a(32) from Lucas A. Brown, Nov 15 2024

A378622 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the strict partition numbers A000009.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 2, 1, 1, 1, 2, 0, -1, -2, -3, 3, 1, 1, 2, 4, 7, 4, 1, 0, -1, -3, -7, -14, 5, 1, 0, 0, 1, 4, 11, 25, 6, 1, 0, 0, 0, -1, -5, -16, -41, 8, 2, 1, 1, 1, 1, 2, 7, 23, 64, 10, 2, 0, -1, -2, -3, -4, -6, -13, -36, -100, 12, 2, 0, 0, 1, 3, 6, 10, 16, 29, 65, 165

Offset: 0

Gus Wiseman, Dec 13 2024

			As a table (read by antidiagonals downward):
        n=0:  n=1:  n=2:  n=3:  n=4:  n=5:  n=6:  n=7:  n=8:
  ----------------------------------------------------------
  k=0:   1     1     1     2     2     3     4     5     6
  k=1:   0     0     1     0     1     1     1     1     2
  k=2:   0     1    -1     1     0     0     0     1     0
  k=3:   1    -2     2    -1     0     0     1    -1     0
  k=4:  -3     4    -3     1     0     1    -2     1     1
  k=5:   7    -7     4    -1     1    -3     3     0    -3
  k=6: -14    11    -5     2    -4     6    -3    -3     7
  k=7:  25   -16     7    -6    10    -9     0    10   -14
  k=8: -41    23   -13    16   -19     9    10   -24    24
  k=9:  64   -36    29   -35    28     1   -34    48   -34
As a triangle (read by rows):
   1
   1   0
   1   0   0
   2   1   1   1
   2   0  -1  -2  -3
   3   1   1   2   4   7
   4   1   0  -1  -3  -7 -14
   5   1   0   0   1   4  11  25
   6   1   0   0   0  -1  -5 -16 -41
   8   2   1   1   1   1   2   7  23  64

Rows are: A000009 (k=0), A087897 (k=1, without first term), A378972 (k=2).
For primes we have A095195 or A376682.
For partitions we have A175804.
First column is A293467 (up to sign).
For composites we have A377033.
For squarefree numbers we have A377038.
For nonsquarefree numbers we have A377046.
For prime powers we have A377051.
Position of first zero in each row is A377285.
Triangle's row-sums are A378970, absolute A378971.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A047966, A098859, A225486, A325244, A325282.
Cf. A008284, A116608, A325242, A225485 or A325280.

nn=20;
t=Table[Take[Differences[PartitionsQ/@Range[0,2nn],k],nn],{k,0,nn}];
Table[t[[j,i-j+1]],{i,nn/2},{j,i}]

A379307 Positive integers whose prime indices include no squarefree numbers.

Original entry on oeis.org

1, 7, 19, 23, 37, 49, 53, 61, 71, 89, 97, 103, 107, 131, 133, 151, 161, 173, 193, 197, 223, 227, 229, 239, 251, 259, 263, 281, 307, 311, 337, 343, 359, 361, 371, 379, 383, 409, 419, 427, 433, 437, 457, 463, 479, 497, 503, 521, 523, 529, 541, 569, 593, 613, 623

Offset: 1

Gus Wiseman, Dec 27 2024

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices begin:
    1: {}
    7: {4}
   19: {8}
   23: {9}
   37: {12}
   49: {4,4}
   53: {16}
   61: {18}
   71: {20}
   89: {24}
   97: {25}
  103: {27}
  107: {28}
  131: {32}
  133: {4,8}
  151: {36}
  161: {4,9}
  173: {40}

Partitions of this type are counted by A114374, strict A256012.
Positions of zero in A379306.
For a unique squarefree part we have A379316, counted by A379308 (strict A379309).
A000040 lists the primes, differences A001223.
A005117 lists the squarefree numbers, differences A076259.
A008966 is the characteristic function for the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A061398 counts squarefree numbers between primes, zeros A068360.
A377038 gives k-th differences of squarefree numbers.
Other counts of prime indices:
- A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A330944 nonprime, see A000586, A000607, A076610, A330945.
- A379300 composite, see A023895, A034891, A036497, A302540, A379301.
- A379310 nonsquarefree, see A302478.
- A379311 old prime, see A204389, A320629, A379312-A379315.
Cf. A000720, A013928, A038550, A057627, A068361, A070321, A071403, A072284, A087436, A112929, A377430, A378086.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Length[Select[prix[#],SquareFreeQ]]==0&]

A379310 Number of nonsquarefree prime indices of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 1

Gus Wiseman, Dec 27 2024

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 39 are {2,6}, so a(39) = 0.
The prime indices of 70 are {1,3,4}, so a(70) = 1.
The prime indices of 98 are {1,4,4}, so a(98) = 2.
The prime indices of 294 are {1,2,4,4}, a(294) = 2.
The prime indices of 1911 are {2,4,4,6}, so a(1911) = 2.
The prime indices of 2548 are {1,1,4,4,6}, so a(2548) = 2.

Positions of first appearances are A000420.
Positions of zero are A302478, counted by A073576 (strict A087188).
No squarefree parts: A379307, counted by A114374 (strict A256012).
One squarefree part: A379316, counted by A379308 (strict A379309).
A000040 lists the primes, differences A001223.
A005117 lists the squarefree numbers, differences A076259.
A008966 is the characteristic function for the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A061398 counts squarefree numbers between primes, zeros A068360.
A377038 gives k-th differences of squarefree numbers.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.
- A330944 nonprime, see A000586, A000607, A076610, A330945.
- A379300 composite, see A023895, A034891, A036497, A302540, A379301.
- A379311 old prime, see A204389, A320629, A379312-A379315.
Cf. A000720, A013928, A038550, A057627, A068361, A070321, A071403, A087436, A112929, A120327, A378086.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[prix[n],Not@*SquareFreeQ]],{n,100}]

Totally additive with a(prime(k)) = A107078(k) = 1 - A008966(k).

cp's OEIS Frontend

A376683 Antidiagonal-sums of the array A376682(n,k) = n-th term of the k-th differences of the noncomposite numbers (A008578).

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A377034 Antidiagonal-sums of the array A377033(n,k) = n-th term of the k-th differences of the composite numbers (A002808).

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A377037 Position of first zero in the n-th differences of the composite numbers (A002808), or 0 if it does not appear.

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A379308 Number of integer partitions of n with a unique squarefree part.

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A379309 Number of strict integer partitions of n with a unique squarefree part.

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A377035 Antidiagonal-sums of the absolute value of the array A377033(n,k) = n-th term of the k-th differences of the composite numbers (A002808).

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A377042 Position of first zero in the n-th differences of the squarefree numbers (A005117), or 0 if it does not appear.

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A378622 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the strict partition numbers A000009.

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A379307 Positive integers whose prime indices include no squarefree numbers.

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