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 1-4 of 4 results.

A220519 Permutation of prime numbers in the order of sequential reading the antidiagonals of A220508.

Original entry on oeis.org

3, 2, 5, 7, 17, 11, 13, 23, 37, 19, 47, 29, 31, 61, 79, 101, 83, 67, 53, 41, 43, 59, 97, 103, 167, 197, 71, 73, 139, 193, 223, 257, 227, 199, 173, 149, 127, 107, 89, 113, 137, 163, 191, 229, 151, 109, 251, 359, 401, 293, 131, 283, 397, 439, 443, 263, 233, 179, 157, 281, 317, 577, 487, 367, 331, 181, 353
Offset: 1

Views

Author

Omar E. Pol, Feb 09 2013

Keywords

Crossrefs

Cf. A000040, A220508.

Extensions

More terms from R. J. Mathar, May 05 2013

A220516 Nonnegative integers in square maze arrangement T(n,k), read by antidiagonals, n>=0, k>=0.

Original entry on oeis.org

0, 1, 3, 8, 2, 4, 9, 7, 5, 15, 24, 10, 6, 14, 16, 25, 23, 11, 13, 17, 35, 48, 26, 22, 12, 18, 34, 36, 49, 47, 27, 21, 19, 33, 37, 63, 80, 50, 46, 28, 20, 32, 38, 62, 64, 81, 79, 51, 45, 29, 31, 39, 61, 65, 99, 120, 82, 78, 52, 44, 30, 40, 60, 66, 98, 100
Offset: 0

Views

Author

Omar E. Pol, Feb 09 2013

Keywords

Comments

This sequence consists of 0 together with a permutation of the natural numbers. The structure is the same as A081344 but starting with 0, not 1.
It appears that in the n-th layer there is at least a prime number <= g and also there is at least a prime number > g, where g is the number on the main diagonal, the n-th oblong number A002378(n), if n >= 1.

Examples

			The first layer is [1, 2, 3] which looks like this:
.  3,
1, 2,
The second layer is [4, 5, 6, 7, 8] which looks like this:
.  .  4
.  .  5,
8, 7, 6,
Square array T(0,0)..T(10,10) begins:
0,     3,   4,  15,  16,  35,  36,  63,  64,  99, 100,...
1,     2,   5,  14,  17,  34,  37,  62,  65,  98, 101,...
8,     7,   6,  13,  18,  33,  38,  61,  66,  97, 102,...
9,    10,  11,  12,  19,  32,  39,  60,  67,  96, 103,...
24,   23,  22,  21,  20,  31,  40,  59,  68,  95, 104,...
25,   26,  27,  28,  29,  30,  41,  58,  69,  94, 105,...
48,   47,  46,  45,  44,  43,  42,  57,  70,  93, 106,...
49,   50,  51,  52,  53,  54,  55,  56,  71,  92, 107,...
80,   79,  78,  77,  76,  75,  74,  73,  72,  91, 108,...
81,   82,  83,  84,  85,  86,  87,  88,  89,  90, 109,...
120, 119, 118, 117, 116, 115, 114, 113, 112, 111, 110,...
...

Links

Crossrefs

Main diagonal is A002378.
Cf. A000040, A000290, A002620, A081344, A220508.

Formula

a(n) = A081344(n+1) - 1.
T(n,k) = n^2 + k , if n is odd and k<=n.
T(n,k) = n(n + 2) - k, if n is even and k<=n.
T(n,k) = k(k + 2) - n, if n is odd and n
T(n,k) = k^2 + n , if n is even and n

A222030 Primes and quarter-squares.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 11, 12, 13, 16, 17, 19, 20, 23, 25, 29, 30, 31, 36, 37, 41, 42, 43, 47, 49, 53, 56, 59, 61, 64, 67, 71, 72, 73, 79, 81, 83, 89, 90, 97, 100, 101, 103, 107, 109, 110, 113, 121, 127, 131, 132, 137, 139, 144, 149, 151, 156, 157, 163, 167, 169
Offset: 0

Views

Author

Omar E. Pol, Feb 05 2013

Keywords

Comments

Union of A002620 and A000040.
It appears that there is always a prime between two consecutive quarter squares, if n >= 2. Therefore in a square spiral, or zig-zag, whose vertices are the quarter-squares, it appears that there is always a prime between two consecutive vertices, if n >= 2.
Apparently the above comment is equivalent to the Oppermann's conjecture. - Omar E. Pol, Oct 26 2013
Union of A000040 and A000290 and A002378. - Omar E. Pol, Oct 28 2013

Links

Crossrefs

Cf. A000040, A002620, A000290, A014085, A220492, A220506, A220508, A220516.

Programs

  • Mathematica
    mx = 13; Union[Prime[Range[PrimePi[mx^2]]], Floor[Range[2*mx]^2/4]] (* Alonso del Arte, Mar 03 2013 *)

Formula

a(n) ~ n log n. - Charles R Greathouse IV, Mar 04 2013

A342397 Expansion of the o.g.f. (2*x^2 + 3*x + 2)*x/((x + 1)^2*(x - 1)^4).

Original entry on oeis.org

0, 2, 7, 18, 35, 62, 98, 148, 210, 290, 385, 502, 637, 798, 980, 1192, 1428, 1698, 1995, 2330, 2695, 3102, 3542, 4028, 4550, 5122, 5733, 6398, 7105, 7870, 8680, 9552, 10472, 11458, 12495, 13602, 14763, 15998, 17290, 18660, 20090, 21602, 23177, 24838, 26565, 28382, 30268, 32248, 34300, 36450
Offset: 0

Views

Author

Petros Hadjicostas, Mar 10 2021

Keywords

Comments

One-half of the antidiagonal sums of array A220508.

Links

Crossrefs

Cf. A004126, A169607, A220508, A342462.

Programs

  • Mathematica
    CoefficientList[Series[(2x^2+3x+2) x/((x+1)^2 (x-1)^4),{x,0,70}],x] (* or *) LinearRecurrence[{2,1,-4,1,2,-1},{0,2,7,18,35,62},70] (* Harvey P. Dale, Jul 08 2023 *)
  • PARI
    /* First program */
seq1(n)={my(x='x+O('x^n)); Vec((2*x^2 + 3*x + 2)*x/((x + 1)^2*(x - 1)^4), -n)}
/* Second program (array M is A220508) */
seq2(nn) = {my(M=matrix(nn+1, nn+1)); my(a=vector(nn+1)); for(n=1, nn+1, for(k=1, nn+1, M[n, k]=if(k == n, n^2-n, if(k < n, n^2-2*n+k, k^2-n)))); for(n=1, nn+1, a[n] = sum(k=1, n, M[n-k+1,k])/2); a}

Formula

a(n) = (n+1)*(1 - (-1)^n)/16 + (7/4)*(binomial(n+3, 3) - binomial(n+2, 2)).
a(n) = (A342362(n) - (n + 1))/4.
a(2*n) = A169607(n) and a(2*n + 1) = 2*A004126(n + 1).
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n > 5. - Chai Wah Wu, Mar 11 2021
Showing 1-4 of 4 results.

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