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.

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

A306889 Brazilian primes that are also the greater of a pair of twin primes.

Original entry on oeis.org

7, 13, 31, 43, 73, 241, 421, 463, 601, 1093, 1483, 1723, 2551, 2971, 3541, 4423, 8011, 10303, 17293, 19183, 20023, 22621, 23563, 24181, 27061, 31153, 35533, 41413, 42643, 43891, 46441, 47743, 53593, 55933, 60763, 83233, 84391, 95791, 98911, 123553, 143263, 156421, 164431
Offset: 1

Views

Author

Michel Marcus, Mar 15 2019

Keywords

Links

Crossrefs

Intersection of A006512 and A085104.
Cf. A306845, A306849.

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 15 2019

A306407 Brazilian primes p such that p+2 and 2p+1 are also prime.

Original entry on oeis.org

78914411, 7294932341, 119637719305001, 937391863673981, 16737518900352251, 54773061508358111, 417560366367249821, 1103799812221103741, 1515990022247085221, 2748614000294776541, 2805758307714748481, 16359900662260777211, 19024521721109192201, 126048913814465881331, 138996334987487396981
Offset: 1

Views

Author

Bernard Schott, Apr 05 2019

Keywords

Comments

The initial terms of this sequence are of the form (11111)_b. The successive bases b are 94, 292, 3307, 5533, 11374, ...
The first term which is not of this form has 43 digits: it is 1137259672818014782224246589454763146442851 = 1 + 16054 + ... + 16054^9 + 16054^10 = (11111111111)_16054 with a string of eleven 1's.
Sophie Germain primes and lesser twins which are Brazilian both have the same property: if p = (b^q - 1)/(b - 1) is a term, necessarily q (prime) == 5 (mod 6) and b == 1 (mod 3). The smallest terms for the first pairs (q,b) are (5,94), (11,16054), (17,3247).
Intersection of A306845 and A306849.
Intersection of A045536 and A085104.

Examples

			The prime 78914411 is a term, because 78914411 = 1 + 94 + 94^2 + 94^3 + 94^4 is a Brazilian prime, 78914411 + 2 = 78914413 is prime and 2 * 78914411 + 1 = 157828823 is prime. The prime 78914411 is Brazilian, the lesser of a pair of twin primes and also a Sophie Germain prime.

Crossrefs

Cf. A001359 (lesser of twin primes), A005384 (Sophie Germain primes).
Cf. A045536 (intersection of A001359 and A005384).
Cf. A085104 (Brazilian primes).
Cf. A306845 (Sophie Germain Brazilian primes), A306849 (lesser of twin primes which is Brazilian).

Programs

  • PARI
    brazilp(N)=forprime(K=5, #binary(N+1)-1, for(n=4, sqrtnint(N-1, K-1), if((K%6==5)&&(n%3==1),if(isprime((n^K-1)/(n-1))&&isprime((n^K-1)/(n-1)+2)&&isprime(2*(n^K-1)/(n-1)+1), print1((n^K-1)/(n-1), ", "))))) \\ Davis Smith, Apr 06 2019
Showing 1-3 of 3 results.

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