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.

A173178 Numbers k such that 2*k+3 is a prime of the form 3*A024893(m) + 2.

Original entry on oeis.org

1, 4, 7, 10, 13, 19, 22, 25, 28, 34, 40, 43, 49, 52, 55, 64, 67, 73, 82, 85, 88, 94, 97, 112, 115, 118, 124, 127, 130, 133, 139, 145, 154, 157, 172, 175, 178, 190, 193, 199, 208, 214, 220, 223, 229, 232, 238, 244, 250, 253, 259, 277, 280, 283, 292, 295, 298, 307, 319
Offset: 1

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Author

Eric Desbiaux, Feb 11 2010

Keywords

Comments

With the Bachet-Bézout theorem implicating Gauss Lemma and the Fundamental Theorem of Arithmetic,
for k > 1, k = 2*a + 3*b (a and b integers)
first type
A001477 = (2*A080425) + (3*A008611)
A000040 = (2*A039701) + (3*A157966)
A024893 Numbers k such that 3*k + 2 is prime
A034936 Numbers k such that 3*k + 4 is prime
OR second type
A001477 = (2*A028242) + (3*A059841)
A000040 = (2*A067076) + (3*1)
A067076 Numbers k such that 2*k + 3 is prime
k a b OR a b
-- - - - -
0 0 0 0 0
1 - - - -
2 1 0 1 0
3 0 1 0 1
4 2 0 2 0
5 1 1 1 1
6 0 2 3 0
7 2 1 2 1
8 1 2 4 0
9 0 3 3 1
10 2 2 5 0
11 1 3 4 1
12 0 4 6 0
13 2 3 5 1
14 1 4 7 0
15 0 5 6 1
...
2* 1 + 3 OR 3* 1 + 2 = 5;
2* 4 + 3 OR 3* 3 + 2 = 11;
2* 7 + 3 OR 3* 5 + 2 = 17;
2*10 + 3 OR 3* 7 + 2 = 23;
2*13 + 3 OR 3* 9 + 2 = 29;
2*19 + 3 OR 3*13 + 2 = 41;
2*22 + 3 OR 3*15 + 2 = 47;
2*25 + 3 OR 3*17 + 2 = 53;
2*28 + 3 OR 3*19 + 2 = 59.
A024893 Numbers k such that 3k+2 is prime.
A007528 Primes of the form 6k-1.
A024898 Positive integers k such that 6k-1 is prime.
1, 4, 7, 10, 13, 19, ... = (3*(4*A024898 - A024893) - 7)/2 = (A112774 - 3*A024893 - 5)/2 = A003627 - (3*A024893 - 5)/2.

Links

Crossrefs

Cf. A067076, A024893, A007528, A024898, A059325.

Programs

  • Mathematica
    Select[Range[0, 320], PrimeQ[(p = 2*# + 3)] && Mod[p, 3] == 2 &] (* Amiram Eldar, Jul 30 2024 *)

Formula

a(n) = 3*A059325(n) + 1. - Amiram Eldar, Jul 30 2024

Extensions

Data corrected and extended by Amiram Eldar, Jul 30 2024

A157733 a(0)=2, a(1)=3. Then 2 or 22 followed by a string of 3's such that the sum of the digits of a(n) is equal to prime(n).

Original entry on oeis.org

2, 3, 23, 223, 2333, 22333, 233333, 2233333, 23333333, 2333333333, 22333333333, 2233333333333, 23333333333333, 223333333333333, 2333333333333333, 233333333333333333, 23333333333333333333, 223333333333333333333
Offset: 0

Views

Author

Paul Curtz, Mar 05 2009

Keywords

Comments

We search for w twos and t threes in prime(n) = 2*w + 3*t. If t = floor(prime(n)/3) would lead to w = 1/2, we decrease t by 1.
The number of twos is 3 - A039701(n) if n > 1.
If prime(n) is congruent to 1 mod 6, then a(n) starts with 22, but if prime(n) is congruent to 5 mod 6, then a(n) starts with 2. - Alonso del Arte, Dec 04 2013

Examples

			a(3) = 23 because the third prime is 5 and 2 + 3 = 5.
a(4) = 223 because the fourth prime is 7 and 2 + 2 + 3 = 7.
a(5) = 2333 because the fifth prime is 11 and 2 + 3 + 3 + 3 = 11.

Crossrefs

Cf. A139067, A157966.

Programs

  • Mathematica
    Module[{nn = 30, t1, t2}, t1 = FromDigits/@Select[Table[PadRight[{2}, n, 3], {n, 2, nn}], PrimeQ[Total[#]] &]; t2 = FromDigits/@ Select[ Table[ PadRight[{2, 2}, n, 3], {n, 2, nn}], PrimeQ[Total[#]] &]; Union[ Join[ {2, 3}, t1, t2]]] (* Harvey P. Dale, Mar 06 2013 *)

Extensions

Edited by R. J. Mathar, Mar 15 2009

A173177 Numbers k such that 2k+3 is a prime of the form 3*A034936(m) + 4.

Original entry on oeis.org

2, 5, 8, 14, 17, 20, 29, 32, 35, 38, 47, 50, 53, 62, 68, 74, 77, 80, 89, 95, 98, 104, 110, 113, 119, 134, 137, 140, 152, 155, 164, 167, 173, 182, 185, 188, 197, 203, 209, 215, 218, 227, 230, 242, 248, 260, 269, 272, 284, 287, 299
Offset: 1

Views

Author

Eric Desbiaux, Feb 11 2010

Keywords

Comments

With Bachet-Bézout theorem implicating Gauss Lemma and the Fundamental Theorem of Arithmetic,
for k > 1, k = 2*a + 3*b (a and b integers)
first type
A001477 = (2*A080425) + (3*A008611)
A000040 = (2*A039701) + (3*A157966)
A024893 Numbers k such that 3*k + 2 is prime
A034936 Numbers k such that 3*k + 4 is prime
OR
second type
A001477 = (2*A028242) + (3*A059841)
A000040 = (2*A067076) + (3*1)
A067076 Numbers k such that 2*k + 3 is prime
k a b OR a b
-- - - - -
0 0 0 0 0
1 - - - -
2 1 0 1 0
3 0 1 0 1
4 2 0 2 0
5 1 1 1 1
6 0 2 3 0
7 2 1 2 1
8 1 2 4 0
9 0 3 3 1
10 2 2 5 0
11 1 3 4 1
12 0 4 6 0
13 2 3 5 1
14 1 4 7 0
15 0 5 6 1
...
2* 2 + 3 OR 3* 1 + 4 = 7;
2* 5 + 3 OR 3* 3 + 4 = 13;
2* 8 + 3 OR 3* 5 + 4 = 19;
2*14 + 3 OR 3* 9 + 4 = 31;
2*17 + 3 OR 3*11 + 4 = 37;
2*20 + 3 OR 3*13 + 4 = 43;
2*29 + 3 OR 3*19 + 4 = 61;
2*32 + 3 OR 3*21 + 4 = 67;
2*35 + 3 OR 3*23 + 4 = 73.
A034936 Numbers k such that 3k+4 is prime.
A002476 Primes of the form 6k+1.
A024899 Nonnegative integers k such that 6k+1 is prime.
2, 5, 8, 14, 17, 20, ... = (3*(4*A024899 - A034936) - 5)/2.

Links

Crossrefs

Cf. A067076, A034936, A002476, A024899.

Programs

  • Mathematica
    Select[Range[300],PrimeQ[2#+3]&&Divisible[2#-1,3]&] (* Harvey P. Dale, Aug 25 2016 *)

Extensions

More terms from Harvey P. Dale, Aug 25 2016
Showing 1-3 of 3 results.

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