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 1-3 of 3 results.

A096447 Odd primes p such that the number of primes less than p that are congruent to 1 (mod 4) is equal to the number of primes less than p that are congruent to 3 (mod 4).

Original entry on oeis.org

3, 7, 19, 43, 463, 26839, 26861, 26879, 26891, 26903, 26927, 616783, 616799, 616841, 616849, 616877, 617039, 617269, 617369, 617401, 617429, 617453, 617471, 617479, 617521, 617537, 617587, 617689, 617717, 617723, 618439, 618547, 618619, 618643
Offset: 1

Views

Author

Yasutoshi Kohmoto, Aug 12 2004

Keywords

Comments

Assign the odd prime numbers to the rows of an array as follows:
Assign the first odd prime, prime(2) = 3, to row 0 (the top row).
For m > 2, assign prime(m) to the row immediately above or below the row to which prime(m-1) was assigned: above if prime(m-1) == 1 (mod 4), below otherwise.
The following array results:
row 0 (this sequence): 3, 7, 19, 43, 463, 26839, ...
row 1 (A096448): 5, 11, 17, 23, 31, 41, 47, 59, 67, 103, 127, ...
row 2 (A096451): 13, 29, 37, 53, 61, 71, 79, 101, 107, 113 ...
row 3: 73, 83, 97, 109, ...
row 4: 89, ...

Links

Crossrefs

Cf. A007351, A151800.
Cf. A096448, A096449, A096450, A096451, A096452, A096453, A096454, A096455.

Programs

  • Mathematica
    lim = 10^5; k1 = 0; k3 = 0; p = 2; t = {}; Do[p = NextPrime[p]; If[k1 == k3, AppendTo[t, p]]; If[Mod[p, 4] == 1, k1++, k3++], {lim}]; t (* T. D. Noe, Sep 07 2011 *)

Formula

a(n) = A151800(A007351(n)), the next prime after A007351(n). - Joshua Zucker, May 03 2006

Extensions

More terms from Joshua Zucker, May 03 2006
"odd" added to definition by N. J. A. Sloane, Sep 09 2015

A096448 Primes p such that the number of primes less than p equal to 1 mod 4 is one less than the number of primes less than p equal to 3 mod 4.

Original entry on oeis.org

5, 11, 17, 23, 31, 41, 47, 59, 67, 103, 127, 419, 431, 439, 461, 467, 1259, 1279, 1303, 26833, 26849, 26881, 26893, 26921, 26947, 615883, 616769, 616787, 616793, 616829, 617051, 617059, 617087, 617257, 617473, 617509, 617647, 617681, 617731, 617819, 617879
Offset: 1

Views

Author

Yasutoshi Kohmoto, Aug 12 2004

Keywords

Examples

			First term prime(2) = 3 is placed on 0th row.
If prime(n-1) = +1 mod 4 is on k-th row then we put prime(n) on (k-1)-st row.
If prime(n-1) = -1 mod 4 is on k-th row then we put prime(n) on (k+1)-st row.
This process makes an array of prime numbers:
  0th row:  3,  7, 19,  43, ...
  1st row:  5, 11, 17,  23, 31, 41, 47, 59, 67, 103, 127, ...
  2nd row: 13, 29, 37,  53, 61, 71, 79, 101, 107, 113 ...
  3rd row: 73, 83, 97, 109, ...
  4th row: 89, ...

Links

Crossrefs

Cf. A096447, A096449, A096450, A096451, A096452, A096453, A096454, A096455.

Programs

  • Mathematica
    Prime[#]&/@(Flatten[Position[Accumulate[If[Mod[#,4]==1,1,-1]&/@ Prime[ Range[ 2,51000]]],-1]]+2) (* Harvey P. Dale, Mar 08 2015 *)
  • PARI
    lista(nn) = my(vp=primes(nn), nb1=0, nb3=0); for (i=2, #vp, my(p = vp[i]); if (nb1 == nb3-1, print1(p, ", ")); if ((p % 4) == 1, nb1++, nb3++);); \\ Michel Marcus, May 30 2024

Extensions

More terms from Joshua Zucker, May 03 2006

A096449 Primes p such that the number of primes q, 5 <= q < p, congruent to 1 mod 3, is equal to the number of such primes congruent to 2 mod 3.

Original entry on oeis.org

5, 11, 17, 23, 41, 47, 83, 167, 227, 233, 608981812919, 608981812961, 608981813017, 608981813569, 608981813677, 608981813833, 608981813851, 608981813927, 608981813939, 608981813963, 608981814043, 608981814149, 608981814251, 608981814827
Offset: 1

Views

Author

Yasutoshi Kohmoto, Aug 12 2004

Keywords

Comments

First term prime(3) = 5 is placed on 0th row.
If prime(n-1) = +1 mod 3 is on k-th row then we put prime(n) on (k-1)-st row.
If prime(n-1) = -1 mod 3 is on k-th row then we put prime(n) on (k+1)-st row.
This process makes an array of prime numbers:
5, 11, 17, 23, 41, 47, 83, ... (this sequence)
7, 13, 19, 29, 37, 43, 53, 71, 79, 89, 101, .. (A096452).
31, 59, 67, 73, 97, ... (A096453)
61, ...

Crossrefs

Cf. A096448-A096455.

Programs

  • Mathematica
    lst = {5}; p = 0; q = 0; r = 5; While[r < 10^9, If[ Mod[r, 3] == 2, p++, q++ ]; r = NextPrime@r; If[p == q, AppendTo[lst, r]; Print@r]]; lst (* Robert G. Wilson v, Sep 20 2009 *)

Formula

For n>1, a(n) = prime(A096629(n-1)+1) = A000040(A096629(n-1)+1). - Max Alekseyev, Sep 19 2009
a(n) = A151800(A098044(n)) = A007918(A098044(n)+1).

Extensions

More terms and better definition from Joshua Zucker, May 21 2006
Terms a(11) onward from Max Alekseyev, Feb 10 2011
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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