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.

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A306849 Brazilian primes that are also the lesser of a pair of twin primes.

Original entry on oeis.org

2801, 637421, 2625641, 78914411, 195534851, 7294932341, 19408913261, 57765899591, 133311428141, 212872312241, 1508520377381, 1960226457281, 5412080545901, 11543487851801, 19383356741711, 20748237948131, 24212632812551, 25413171899021, 28240486488581, 46922470889141
Offset: 1

Views

Author

Bernard Schott, Mar 13 2019

Keywords

Comments

As for Sophie Germain primes which are Brazilian (A306845), these terms are relatively rare (only 28 terms < 10^15).
The first 26051 terms of this sequence are of the form (11111)_b. The successive bases b are 7, 28, 40, 94, 118, 292, 373, 490, 604, 679, 1108, 1183, ... These 26051 terms end in 1: If base b ends in 1 or 6, (11111)_b ends in 5 and cannot be prime; if base b ends in another digit, then (11111)_b always ends in 1.
The first term which is not of this form has 31 digits; it's 1425663266336265377189900884061 = 1 + 1036 + ... + 1036^9 + 1036^10 = (11111111111)_1036 with a string of eleven 1's. In this case, the successive bases are 1036, 2089, 6961, 7894, 9775, ...
If (b^q - 1)/(b - 1) is a term, necessarily q (prime) == 5 (mod 6) and b == 1 (mod 3). The smallest term for each pair (q,b) is (5,7), (11,1036), (17,1603), (23,6697), (29,2779), (41,26719), (47,98506), (53,2110).

Examples

			2801 is a term because 2801 + 2 = 2803 is prime, so 2801 is a lesser of twin primes, then 2801 = 1 + 7 + 7^2 + 7^3 + 7^4 = (11111)_7 and 2801 is also a Brazilian prime.

Crossrefs

Intersection of A001359 and A085104.
Cf. A001097, A006512, A306845, A306889.

Programs

  • PARI
    lista(lim)=my(v=List(), t, k); for(n=2, sqrt(lim), t=1+n; k=1; while((t+=n^k++)<=lim, if(isprime(t) && isprime(t+2), listput(v, t)))); v = vecsort(Vec(v), , 8); \\ Michel Marcus, Mar 14 2019

Extensions

Terms computed by Giovanni Resta and Michel Marcus, Mar 13 2019

A265006 Twin prime pairs of the form (k^2 + k - 1, k^2 + k + 1).

Original entry on oeis.org

5, 7, 11, 13, 29, 31, 41, 43, 71, 73, 239, 241, 419, 421, 461, 463, 599, 601, 1481, 1483, 1721, 1723, 2549, 2551, 2969, 2971, 3539, 3541, 4421, 4423, 8009, 8011, 10301, 10303, 17291, 17293, 19181, 19183, 20021, 20023, 23561, 23563, 24179, 24181, 27059, 27061, 31151, 31153, 35531, 35533
Offset: 1

Views

Author

Bill McEachen, Nov 29 2015

Keywords

Comments

This is a subset of A002327 and A002383 taken together. Note that 3 is not a member, as the pairing (3, 5) is excluded as defined, as 3 and 5 associate to different centers.
The corresponding n are in A088485.
The average of each twin prime pair is an oblong number (A002378). - Michel Marcus, Feb 04 2017

Examples

			For k = 6, k^2 + k = 6^2 + 6 = 42, and (41,43) is a twin prime pair, so 41 and 43 are in the sequence.

Links

Crossrefs

Cf. A002327, A002378, A002383, A088485, A088486, A306889.

Programs

  • Magma
    &cat[[n^2+n-1, n^2+n+1]: n in [0..250]| IsPrime(n^2+n-1) and IsPrime(n^2+n+1)]; // Vincenzo Librandi, Feb 05 2017
  • Mathematica
    {#^2 + # - 1, #^2 + # + 1} & /@ Select[Range@ 200, PrimeQ[#^2 + # - 1] && PrimeQ[#^2 + # + 1] &] // Flatten (* Michael De Vlieger, Nov 30 2015 *)
Flatten[Select[Table[n^2 + n + {-1, 1}, {n, 0, 200}], And@@PrimeQ[#] &]] (* Vincenzo Librandi, Feb 05 2017 *)
  • PARI
    genit()={my(maxx=1000);n=0;while(n

Formula

a(2n-1) = A088486(n). a(2n)=2+a(2n-1).
Showing 1-2 of 2 results.

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