A354596 -id:A354596 - cp's OEIS frontend

A327263 Array T(n,k) in which the i-th row consists of numbers > 1 not in array U(i;n,k) = (ink - (i-2)*A319929(n,k))/2 where i >= 1, n >= 1 and k >= 1, read by antidiagonals.

3, 5, 2, 9, 3, 2, 13, 5, 3, 2, 21, 7, 4, 3, 2, 25, 11, 5, 4, 3, 2, 33, 13, 7, 5, 4, 3, 2, 37, 17, 9, 6, 5, 4, 3, 2, 45, 19, 10, 7, 6, 5, 4, 3, 2, 57, 23, 13, 9, 7, 6, 5, 4, 3, 2, 61, 29, 15, 11, 8, 7, 6, 5, 4, 3, 2, 73, 31, 17, 12, 9, 8, 7, 6, 5, 4, 3, 2

Offset: 1

David Lovler, Oct 15 2019

All the U(i;n,k) mimic the ordinary multiplication table in that they are commutative, associative, have identity element 1 and have 0. However (except when i=2) they are partially distributive, meaning that distributivity works except if an even number is partitioned into a sum of two odd numbers. Only when i=2, the odd-even-dependent A319929 term disappears and normal distributivity holds.
U(0;n,k) = A319929(n,k);
U(1;n,k) = A322630(n,k);
U(2;n,k) = n*k;
U(3;n,k) = A322744(n,k);
U(4;n,k) = A327259(n,k);
U(i;n,k) = 2i*floor(n/2)*floor(k/2) + A319929(n,k).
Row 1 is 2p-1 where p is a prime number (A076274 without 1).
Row 2 is the prime numbers.
Row 3 is A307002.
Row 4 is A327261.
The i-th row of T(n,k) consists of numbers that sieve out of the array U(i;n,k) = (i*n*k - (i-2)*A319929(n,k))/2, in numerical order.
From David Lovler, Sep 02 2020: (Start)
Row 1 has no even numbers. Row 2 has one even number. Generally, the even numbers of the i-th row start with i-1 consecutive even numbers (from 2). This is because U(i;2,2) = 2*i gives the first even number not in row i.
Row 3 seems to have even numbers that, after 2, coincide with A112774 which has an infinite number of terms. For i > 3, as i increases, row i has a denser presence of even numbers, thus each row has an infinite number of even terms.
Generalization of the twin prime conjecture: Since row 2 is the prime numbers, we can observe the twin prime conjecture that after the first three odd primes, the sprinkling of pairs of consecutive prime numbers never ends. Concerning just odd terms, a similar conjecture can be stated for rows i >= 3. Row 3 starts with four odd numbers then the sprinkling of three consecutive odd number never ends. Row 4 starts with five odd numbers then the sprinkling of four consecutive odd numbers never ends. The pattern continues as row i starts with i+1 odd numbers then the sprinkling of i consecutive odd numbers never ends. We can take this back to row 1 which starts with two odd numbers then continues with isolated odd numbers.
Studying the even terms, there is an analog to the above generalization of the twin prime conjecture. Row 3 starts with two even numbers then continues with isolated even numbers. Row 4 starts with three even numbers then the sprinkling of pairs of consecutive even numbers never ends. Row 5 starts with four even numbers then the sprinkling of three consecutive even numbers never ends. The pattern continues as row i starts with i-1 even numbers then the sprinkling of i-2 consecutive even numbers never ends.
(End)

			3  5  9  13  21  25  33  37  45  57  61  73  81  85  93 105 117 121 133 141 145 ...
2  3  5   7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71  73 ...
2  3  4   5   7   9  10  13  15  17  21  22  23  25  29  31  34  37  39  41  45 ...
2  3  4   5   6   7   9  11  12  14  15  17  19  21  22  25  27  28  29  31  35 ...
2  3  4   5   6   7   8   9  11  13  14  16  17  18  19  21  23  25  26  28  29 ...
2  3  4   5   6   7   8   9  10  11  13  15  16  18  19  20  21  22  23  25  27 ...
2  3  4   5   6   7   8   9  10  11  12  13  15  17  18  20  21  22  23  24  25 ...
2  3  4   5   6   7   8   9  10  11  12  13  14  15  17  19  20  22  23  24  25 ...
2  3  4   5   6   7   8   9  10  11  12  13  14  15  16  17  19  21  22  24  25 ...
2  3  4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19  21  23  24 ...
...

Cf. A000040, A003991, A076274, A112774.
Cf. A319929, A322630, A322744, A327259, A345474.
Cf. A354596.

row=12;max=200;U[i_,n_,k_]:=(i*n*k-(i-2)If[OddQ@n,If[OddQ@k,n+k-1,k],If[OddQ@k,n,0]])/2;t=Table[c=Union@Flatten@Table[U[i,n,k],{n,2,max},{k,2,max}];Complement[Range[2,max],c][[;;row]],{i,row}];Flatten@Table[t[[m,k-m+1]],{k,row},{m,k}] (* Giorgos Kalogeropoulos, Jun 08 2021 *)

With one exception there are likely no formulas for the rows of T(n,k) since their creation is based on a sieving process like the familiar prime number sieve. The exception is T(1,k) = 2*T(2,k)-1.

A354594 a(n) = n^2 + 2*floor(n/2)^2.

Original entry on oeis.org

0, 1, 6, 11, 24, 33, 54, 67, 96, 113, 150, 171, 216, 241, 294, 323, 384, 417, 486, 523, 600, 641, 726, 771, 864, 913, 1014, 1067, 1176, 1233, 1350, 1411, 1536, 1601, 1734, 1803, 1944, 2017, 2166, 2243, 2400, 2481, 2646, 2731, 2904

Offset: 0

David Lovler, Jun 01 2022

The first bisection is A033581, the second bisection is A080859. - Bernard Schott, Jun 07 2022

David Lovler, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000290, A008794, A033581, A080859, A247375.

Cf. A213037, A322744 (diagonal), A354595, A354596.

a[n_] := n^2 + 2 Floor[n/2]^2
Table[a[n], {n, 0, 90}]    (* A354594 *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 6, 11, 24}, 60]

PARI
```
a(n) = n^2 + 2*(n\2)^2;
```

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), n >= 5.
a(n) = A000290(n) + 2*A008794(n).
G.f.: x*(1 + 5*x + 3*x^2 + 3*x^3)/((1 - x)^3*(1 + x)^2).
E.g.f.: (x*(1 + 3*x)*cosh(x) + (1 + 3*x + 3*x^2)*sinh(x))/2. - Stefano Spezia, Jun 07 2022

A354595 a(n) = n^2 + 4*floor(n/2)^2.

Original entry on oeis.org

0, 1, 8, 13, 32, 41, 72, 85, 128, 145, 200, 221, 288, 313, 392, 421, 512, 545, 648, 685, 800, 841, 968, 1013, 1152, 1201, 1352, 1405, 1568, 1625, 1800, 1861, 2048, 2113, 2312, 2381, 2592, 2665, 2888, 2965, 3200, 3281, 3528, 3613, 3872

Offset: 0

David Lovler, Jun 01 2022

The first bisection is A139098, the second bisection is A102083.

David Lovler, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000290, A008794, A102083, A139098, A247375.

Cf. A213037, A327259 (diagonal), A354594, A354596.

a[n_] := n^2 + 4 Floor[n/2]^2
Table[a[n], {n, 0, 90}]    (* A354595 *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 8, 13, 32}, 60]

PARI
```
a(n) = n^2 + 4*(n\2)^2;
```

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), n >= 5.
a(n) = A000290(n) + 4*A008794(n).
G.f.: x*(1 + 7*x + 3*x^2 + 5*x^3)/((1 - x)^3*(1 + x)^2).
E.g.f.: 2*x^2*cosh(x) + (1 + 2*x + 2*x^2)*sinh(x). - Stefano Spezia, Jun 07 2022

cp's OEIS Frontend

A327263 Array T(n,k) in which the i-th row consists of numbers > 1 not in array U(i;n,k) = (ink - (i-2)*A319929(n,k))/2 where i >= 1, n >= 1 and k >= 1, read by antidiagonals.

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A354594 a(n) = n^2 + 2*floor(n/2)^2.

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A354595 a(n) = n^2 + 4*floor(n/2)^2.

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cp's OEIS Frontend

A327263 Array T(n,k) in which the i-th row consists of numbers > 1 not in array U(i;n,k) = (i*n*k - (i-2)*A319929(n,k))/2 where i >= 1, n >= 1 and k >= 1, read by antidiagonals.

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Formula

A354594 a(n) = n^2 + 2*floor(n/2)^2.

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Author

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Programs

Mathematica

PARI

Formula

A354595 a(n) = n^2 + 4*floor(n/2)^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A327263 Array T(n,k) in which the i-th row consists of numbers > 1 not in array U(i;n,k) = (ink - (i-2)*A319929(n,k))/2 where i >= 1, n >= 1 and k >= 1, read by antidiagonals.