cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A376678 Position of first zero in the n-th differences of the primes, or 0 if it does not appear.

Original entry on oeis.org

0, 0, 2, 7, 69, 13, 47, 58, 9, 43, 3553, 100, 7019, 14082, 68097, 14526, 149677, 2697, 481054, 979719, 631894, 29811, 25340978, 50574254, 7510843, 210829337, 67248861, 224076286, 910615647, 931510269, 452499644, 2880203722, 396680865, 57954439970, 77572822440, 35394938648
Offset: 0

Views

Author

Gus Wiseman, Oct 14 2024

Keywords

Comments

Do the k-th differences of the primes contain a zero for all k > 1?

Examples

			The third differences of the primes begin:
  -1, 2, -4, 4, -4, 4, 0, -6, 8, ...
so a(3) = 7.

Links

Crossrefs

If 1 is considered prime (A008578) we get A376855.
The zeros of second differences are A064113, complement A333214.
This is the position at which 0 first appears in row n of A095195.
For composite instead of prime we have A377037.
For squarefree instead of prime we have A377042, nonsquarefree A377050.
For prime-power instead of prime we have A377055.
A000040 lists the primes, first differences A001223, second A036263.
Cf. A000720, A007442, A030016, A065890, A084758, A140119, A258025, A258026, A333254, A349643, A376681, A376682, A376683.

Programs

  • Mathematica
    nn=100000;
u=Table[Differences[Select[Range[nn],PrimeQ],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]]]}]

Formula

a(n) = A000720(A349643(n)) for n >= 2. - Pontus von Brömssen, Oct 17 2024

Extensions

a(17)-a(32) from Pontus von Brömssen, Oct 17 2024
a(33)-a(35) from Lucas A. Brown, Nov 03 2024

A376684 Antidiagonal-sums of the absolute value of the array A376682(n,k) = n-th term of the k-th differences of the noncomposite numbers (A008578).

Original entry on oeis.org

1, 3, 4, 9, 12, 27, 50, 109, 224, 471, 942, 1773, 3118, 4957, 7038, 9373, 16256, 55461, 150622, 346763, 718972, 1377101, 2462220, 4114987, 6387718, 9112455, 12051830, 17160117, 40946860, 134463917, 349105370, 800713921, 1684145408, 3297536923, 6040907554
Offset: 0

Views

Author

Gus Wiseman, Oct 15 2024

Keywords

Examples

			The fourth antidiagonal of A376682 is: (7, 2, 0, -1, -2), so a(4) = 12.

Crossrefs

For the modern primes (A000040) we have A376681, absolute version of A140119.
For firsts instead of row-sums we have A030016, modern A007442.
These are the antidiagonal-sums of the absolute value of A376682 (modern A095195).
This is the absolute version of A376683.
For first zero-positions we have A376855, modern A376678.
A000040 lists the modern primes, differences A001223, seconds A036263.
A008578 lists the noncomposites, first differences A075526.
Cf. A002808, A064113, A065890, A173390, A233671, A258025, A333214, A333254, A376680, A377033, A377037.

Programs

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

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

Views

Author

Gus Wiseman, Oct 18 2024

Keywords

Examples

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

Crossrefs

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.

Programs

  • Mathematica
    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}]

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

Views

Author

Gus Wiseman, Dec 13 2024

Keywords

Examples

			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

Crossrefs

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.

Programs

  • Mathematica
    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}]

A379314 Number of integer partitions of n with a unique 1 or prime part.

Original entry on oeis.org

0, 1, 1, 1, 0, 2, 1, 3, 1, 4, 3, 8, 3, 10, 6, 14, 8, 22, 12, 30, 18, 40, 26, 58, 33, 76, 53, 103, 69, 140, 94, 185, 132, 239, 176, 323, 232, 417, 320, 536, 414, 704, 544, 900, 721, 1145, 936, 1481, 1198, 1867, 1571, 2363, 2001, 3003, 2550, 3768, 3275, 4712
Offset: 0

Views

Author

Gus Wiseman, Dec 28 2024

Keywords

Examples

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

Links

Crossrefs

For all prime parts we have A000607 (strict A000586), ranks A076610.
For no prime parts we have A002095 (strict A096258), ranks A320628.
Ranked by A379312 = positions of 1 in A379311.
For a unique composite part we have A379302 (strict A379303), ranks A379301.
The strict case is A379315.
For squarefree instead of old prime we have A379308 (strict A379309), ranks A379316.
Considering 1 nonprime gives A379304 (strict A379305), ranks A331915.
A000040 lists the prime numbers, differences A001223.
A000041 counts integer partitions, strict A000009.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A376682 gives k-th differences of old primes.
Cf. A000070, A023895, A034891, A036497, A095195, A175804, A204389, A257994, A302540, A330944.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Count[#,_?(#==1||PrimeQ[#]&)]==1&]],{n,0,30}]
  • PARI
    seq(n)={Vec(sum(k=1, n, if(isprime(k) || k==1, x^k))/prod(k=4, n, 1 - if(!isprime(k), x^k), 1 + O(x^n)), -n-1)} \\ Andrew Howroyd, Dec 28 2024

A179278 Largest nonprime integer <= n.

Original entry on oeis.org

1, 1, 1, 4, 4, 6, 6, 8, 9, 10, 10, 12, 12, 14, 15, 16, 16, 18, 18, 20, 21, 22, 22, 24, 25, 26, 27, 28, 28, 30, 30, 32, 33, 34, 35, 36, 36, 38, 39, 40, 40, 42, 42, 44, 45, 46, 46, 48, 49, 50, 51, 52, 52, 54, 55, 56, 57, 58, 58, 60, 60, 62, 63, 64, 65, 66, 66, 68, 69, 70, 70, 72
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 08 2010

Keywords

Examples

			From _Gus Wiseman_, Dec 04 2024: (Start)
The nonprime integers <= n:
  1  1  1  4  4  6  6  8  9  10  10  12  12  14  15  16
           1  1  4  4  6  8  9   9   10  10  12  14  15
                 1  1  4  6  8   8   9   9   10  12  14
                       1  4  6   6   8   8   9   10  12
                          1  4   4   6   6   8   9   10
                             1   1   4   4   6   8   9
                                     1   1   4   6   8
                                             1   4   6
                                                 1   4
                                                     1
(End)

Crossrefs

Cf. A014684, A113523, A113638, A062298.
For prime we have A007917.
For nonprime we have A179278 (this).
For squarefree we have A070321.
For nonsquarefree we have A378033.
For prime power we have A031218.
For non prime power we have A378367.
For perfect power we have A081676.
For non perfect power we have A378363.
A000040 lists the primes, differences A001223.
A002808 lists the composite numbers, differences A073783.
A018252 lists the nonprimes, differences A065310.
A095195 has row n equal to the k-th differences of the prime numbers.
A113646 gives least nonprime >= n.
A151800 gives the least prime > n, weak version A007918.
A377033 has row n equal to the k-th differences of the composite numbers.

Programs

  • Mathematica
    Array[# - Boole[PrimeQ@ #] - Boole[# == 3] &, 72] (* Michael De Vlieger, Oct 13 2018 *)
Table[Max@@Select[Range[n],!PrimeQ[#]&],{n,30}] (* Gus Wiseman, Dec 04 2024 *)
  • PARI
    a(n) = if (isprime(n), if (n==3, 1, n-1), n); \\ Michel Marcus, Oct 13 2018

Formula

For n > 3: a(n) = A113523(n) = A014684(n);
For n > 0: a(n) = A113638(n). - Georg Fischer, Oct 12 2018
A005171(a(n)) = 1; A010051(a(n)) = 0.
a(n) = A018252(A062298(n)). - Ridouane Oudra, Aug 22 2025

Extensions

Inequality in the name reversed by Gus Wiseman, Dec 05 2024

A377036 First term of the n-th differences of the composite numbers. Inverse zero-based binomial transform of A002808.

Original entry on oeis.org

4, 2, 0, -1, 2, -2, 0, 4, -8, 8, 0, -16, 32, -32, -1, 78, -233, 687, -2363, 8160, -25670, 72352, -184451, 430937, -933087, 1888690, -3597221, 6479696, -11086920, 18096128, -28307626, 42644791, -62031001, 86466285, -110902034, 110907489, -52325, -483682930
Offset: 0

Views

Author

Gus Wiseman, Oct 18 2024

Keywords

Crossrefs

The version for prime instead of composite is A007442.
For noncomposite numbers we have A030016.
This is the first column (n=1) of A377033.
For row-sums we have A377034, absolute version A377035.
First zero positions are A377037, cf. A376678, A376855, A377042, A377050, A377055.
For squarefree instead of composite we have A377041, nonsquarefree A377049.
For prime-power instead of composite we have A377054.
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, A140119, A173390, A333214, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative), A376680.

Programs

  • Mathematica
    q=Select[Range[100],CompositeQ];
t=Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[1+k]],{k,0,j}],{j,0,Length[q]-1}]

Formula

The inverse zero-based binomial transform of a sequence (q(0), q(1), ..., q(m)) is the sequence p given by:
p(j) = sum_{k=0..j} (-1)^(j-k) binomial(j,k) q(k)

A320591 Expansion of Product_{k>=1} (1 + x^k/(1 + x)^k).

Original entry on oeis.org

1, 1, 0, 1, -2, 4, -7, 11, -16, 23, -36, 65, -129, 256, -473, 772, -1028, 835, 776, -5755, 17562, -41750, 86678, -165145, 299949, -541837, 1020029, -2068203, 4509512, -10252952, 23465297, -52762788, 115160832, -243018459, 496094524, -982431070, 1894710043, -3574095362
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 16 2018

Keywords

Comments

After the first term, this is the second term of the n-th differences of A000009, or column n=1 of A378622. - Gus Wiseman, Feb 03 2025

Links

Crossrefs

Cf. A000593, A320589, A307548.
The version for non-strict partitions is A320590, row n=1 of A175804.
Column n=1 (except first term) of A378622. See also A293467, A377285, A378970, A378971, A380412 (column n=0).
A000009 counts strict integer partitions, differences A087897, A378972.
A266232 gives zero-based binomial transform of strict partitions, differences A129519.
Cf. A000041, A007442, A002865, A008284, A095195, A103446, A116608, A218482, A281425, A320568.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1 + x^k/(1 + x)^k): k in [1..(m+2)]]) )); // G. C. Greubel, Oct 29 2018
  • Maple
    seq(coeff(series(mul((1+x^k/(1+x)^k),k=1..n),x,n+1), x, n), n = 0 .. 37); # Muniru A Asiru, Oct 16 2018
  • Mathematica
    nmax = 37; CoefficientList[Series[Product[(1 + x^k/(1 + x)^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 37; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k/d + 1) d, {d, Divisors[k]}] x^k/(k (1 + x)^k), {k, 1, nmax}]], {x, 0, nmax}], x]
Prepend[Table[Differences[PartitionsQ/@Range[0,k+1],k][[2]],{k,0,30}],1] (* Gus Wiseman, Jan 29 2025 *)
  • PARI
    m=50; x='x+O('x^m); Vec(prod(k=1, m+2, (1 + x^k/(1 + x)^k))) \\ G. C. Greubel, Oct 29 2018

Formula

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*((1 + x)^k - x^k))).
G.f.: exp(Sum_{k>=1} A000593(k)*x^k/(k*(1 + x)^k)).
From Peter Bala, Dec 22 2020: (Start)
O.g.f.: Sum_{n >= 0} x^(n*(n+1)/2)/Product_{k = 1..n} ((1 + x)^k - x^k). Cf. A307548.
Conjectural o.g.f.: (1/2) * Sum_{n >= 0} x^(n*(n-1)/2)*(1 + x)^n/( Product_{k = 1..n} ( (1 + x)^k - x^k ) ). (End)
a(n+1) = Sum_{k=0..n} (-1)^(n-k) binomial(n,k) A000009(k+1). - Gus Wiseman, Feb 03 2025

A349643 Smallest prime p = prime(k) such that the n-th difference of (prime(k), ..., prime(k+n)) is zero.

Original entry on oeis.org

3, 17, 347, 41, 211, 271, 23, 191, 33151, 541, 70891, 152681, 856637, 158047, 2010581, 24239, 7069423, 15149419, 9472693, 347957, 479691493, 994339579, 132480637, 4462552643, 1342424483, 4757283367, 20674291411, 21170786093, 9941224877, 68864319317, 8660066477
Offset: 2

Views

Author

Pontus von Brömssen, Nov 23 2021

Keywords

Comments

Equivalently, a(n) is the smallest prime p = prime(k) such that there is a polynomial f of degree at most n-1 such that f(j) = prime(j) for k <= j <= k+n.
a(n) = prime(k), where k is the smallest positive integer such that A095195(k+n,n) = 0.

Examples

			The first six consecutive primes for which the fifth difference is 0 are (41, 43, 47, 53, 59, 61), so a(5) = 41. The successive differences are (2, 4, 6, 6, 2), (2, 2, 0, -4), (0, -2, -4), (-2, -2), and (0).

Crossrefs

First column of A349644.
Cf. A095195.

Programs

  • Mathematica
    With[{prs=Prime[Range[10^6]]},Table[SelectFirst[Partition[prs,n+1,1],Differences[#,n]=={0}&][[1]],{n,2,21}]] (* The program generates the first 20 terms of the sequence. *) (* Harvey P. Dale, Aug 10 2024 *)
  • Python
    from math import comb
from sympy import nextprime
def A349643(n):
    plist, clist = [2], [1]
    for i in range(1,n+1):
        plist.append(nextprime(plist[-1]))
        clist.append((-1)**i*comb(n,i))
    while True:
        if sum(clist[i]*plist[i] for i in range(n+1)) == 0: return plist[0]
        plist = plist[1:]+[nextprime(plist[-1])] # Chai Wah Wu, Nov 25 2021

Formula

Sum_{j=0..n} (-1)^j*binomial(n,j)*prime(k+j) = 0, where prime(k) = a(n).

A350004 Iterated differences of ludic numbers. Array read by antidiagonals, n >= 0, k >= 1: T(0,k) = A003309(k), T(n,k) = T(n-1,k+1)-T(n-1,k) for n > 0.

Original entry on oeis.org

1, 2, 1, 3, 1, 0, 5, 2, 1, 1, 7, 2, 0, -1, -2, 11, 4, 2, 2, 3, 5, 13, 2, -2, -4, -6, -9, -14, 17, 4, 2, 4, 8, 14, 23, 37, 23, 6, 2, 0, -4, -12, -26, -49, -86, 25, 2, -4, -6, -6, -2, 10, 36, 85, 171, 29, 4, 2, 6, 12, 18, 20, 10, -26, -111, -282
Offset: 0

Views

Author

Pontus von Brömssen, Dec 08 2021

Keywords

Examples

			Array begins:
  n\k|    1    2    3   4    5   6    7    8   9   10
  ---+-----------------------------------------------
   0 |    1    2   3    5    7  11   13   17  23   25
   1 |    1    1   2    2    4   2    4    6   2    4
   2 |    0    1   0    2   -2   2    2   -4   2    4
   3 |    1   -1   2   -4    4   0   -6    6   2   -8
   4 |   -2    3  -6    8   -4  -6   12   -4 -10   10
   5 |    5   -9  14  -12   -2  18  -16   -6  20   -8
   6 |  -14   23 -26   10   20 -34   10   26 -28    2
   7 |   37  -49  36   10  -54  44   16  -54  30    8
   8 |  -86   85 -26  -64   98 -28  -70   84 -22  -26
   9 |  171 -111 -38  162 -126 -42  154 -106  -4   64
  10 | -282   73 200 -288   84 196 -260  102  68 -142

Links

Crossrefs

Cf. A003309 (row n = 0), A260723 (row n = 1).
Cf. A095195 (iterated differences of primes), A350001 (iterated differences of lucky numbers).

Formula

T(n,k) = Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*A003309(k+j).
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