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.

Showing 1-5 of 5 results.

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

Views

Author

Gus Wiseman, Oct 17 2024

Keywords

Comments

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

Examples

			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

Crossrefs

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.

Programs

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

Formula

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

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

Views

Author

Gus Wiseman, Oct 17 2024

Keywords

Examples

			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.

Links

Crossrefs

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.

Programs

  • Mathematica
    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]]]}]

Extensions

Offset 2 from Michel Marcus, Oct 18 2024
a(17)-a(54) from Alois P. Heinz, Oct 18 2024

A350001 Iterated differences of lucky numbers. Array read by antidiagonals, n >= 0, k >= 1: T(0,k) = A000959(k), T(n,k) = T(n-1,k+1) - T(n-1,k) for n > 0.

Original entry on oeis.org

1, 3, 2, 7, 4, 2, 9, 2, -2, -4, 13, 4, 2, 4, 8, 15, 2, -2, -4, -8, -16, 21, 6, 4, 6, 10, 18, 34, 25, 4, -2, -6, -12, -22, -40, -74, 31, 6, 2, 4, 10, 22, 44, 84, 158, 33, 2, -4, -6, -10, -20, -42, -86, -170, -328, 37, 4, 2, 6, 12, 22, 42, 84, 170, 340, 668
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   11   12
  ---+--------------------------------------------------------------
   0 |    1    3    7    9   13   15   21    25   31    33   37   43
   1 |    2    4    2    4    2    6    4     6    2     4    6    6
   2 |    2   -2    2   -2    4   -2    2    -4    2     2    0   -4
   3 |   -4    4   -4    6   -6    4   -6     6    0    -2   -4   14
   4 |    8   -8   10  -12   10  -10   12    -6   -2    -2   18  -32
   5 |  -16   18  -22   22  -20   22  -18     4    0    20  -50   56
   6 |   34  -40   44  -42   42  -40   22    -4   20   -70  106  -82
   7 |  -74   84  -86   84  -82   62  -26    24  -90   176 -188  102
   8 |  158 -170  170 -166  144  -88   50  -114  266  -364  290 -100
   9 | -328  340 -336  310 -232  138 -164   380 -630   654 -390   50
  10 |  668 -676  646 -542  370 -302  544 -1010 1284 -1044  440   78

Links

Crossrefs

Cf. A000959 (row n = 0), A031883 (row n = 1), A123593 (column k = 1).
Cf. A254967 (absolute differences), A095195 (iterated differences of primes), A350004 (iterated differences of ludic numbers).

Formula

T(n,k) = Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*A000959(k+j).

A350007 Array read by antidiagonals, n >= 2, m >= 0: T(n,m) is the smallest ludic number L(k) such that all n-th differences of (L(k), ..., L(k+n+m)) are zero, where L is A003309; T(n,m) = 0 if no such number exists.

Original entry on oeis.org

1, 71, 11, 6392047, 41, 41
Offset: 2

Views

Author

Pontus von Brömssen, Dec 08 2021

Keywords

Comments

Equivalently, T(n,m) is the smallest ludic number L(k) such that there is a polynomial f of degree at most n-1 such that f(j) = L(j) for k <= j <= k+n+m.
T(n,m) = A003309(k), where k is the smallest positive integer such that A350004(n,k+j) = 0 for 0 <= j <= m.

Examples

			Array begins:
  n\m|     0        1        2        3        4        5
  ---+---------------------------------------------------
   2 |     1       71  6392047        ?        ?        ?
   3 |    11       41     1111  2176387 61077491 93320837
   4 |    41     1111   545977 27244691 93320837        ?
   5 |    47       91 27244691 93320837        ?        ?
   6 |    91    23309 93320837        ?        ?        ?
   7 |  1361  9899189        ?        ?        ?        ?
   8 |  4261    26233        ?        ?        ?        ?
   9 |   481  7110347        ?        ?        ?        ?
  10 | 46067 79241951        ?        ?        ?        ?
For n = 5 and m = 1, the first seven (n+m+1) consecutive ludic numbers for which all fifth (n-th) differences are 0 are (91, 97, 107, 115, 119, 121, 127), so T(5,1) = 91. The successive differences are (6, 10, 8, 4, 2, 6), (4, -2, -4, -2, 4), (-6, -2, 2, 6), (4, 4, 4), and (0, 0).

Crossrefs

Cf. A350005 (row n = 2), A350006 (column m = 0).
Cf. A003309, A349644 (counterpart for primes), A350003 (counterpart for lucky numbers), A350004.

Formula

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

A350006 a(n) is the smallest ludic number L(k) such that the n-th difference of (L(k), ..., L(k+n)) is zero, where L is A003309; a(n) = 0 if no such number exists.

Original entry on oeis.org

1, 11, 41, 47, 91, 1361, 4261, 481, 46067, 5027, 31499, 888893, 126205, 36191, 7676353, 26794127, 206527, 2560375, 7716073
Offset: 2

Views

Author

Pontus von Brömssen, Dec 08 2021

Keywords

Comments

Equivalently, a(n) is the smallest ludic number L(k) such that there is a polynomial f of degree at most n-1 such that f(j) = L(j) for k <= j <= k+n.
a(n) = A003309(k), where k is the smallest positive integer such that A350004(n,k) = 0.
a(21) > 10^8 (unless a(21) = 0).

Examples

			The first six consecutive ludic numbers for which the fifth difference is 0 are (47, 53, 61, 67, 71, 77), so a(5) = 47. The successive differences are (6, 8, 6, 4, 6), (2, -2, -2, 2), (-4, 0, 4), (4, 4), and (0).

Crossrefs

First column of A350007.
Cf. A003309, A349643, A350002, A350004.

Formula

Sum_{j=0..n} (-1)^j*binomial(n,j)*A003309(k+j) = 0, where A003309(k) = a(n).
Showing 1-5 of 5 results.

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