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.

A306845 Sophie Germain primes which are Brazilian.

Original entry on oeis.org

28792661, 78914411, 943280801, 7294932341, 30601685951, 919548423641, 2275869057821, 4172851565741, 4801096143881, 27947620155401, 29586967653101, 43573806645461, 119637719305001, 124484682222941, 148908227169101, 172723673300501
Offset: 1

Views

Author

Bernard Schott, Mar 13 2019

Keywords

Comments

These terms point out that the conjecture proposed in Quadrature "No Sophie Germain prime is Brazilian (prime)" (see link) was false.
Giovanni Resta has found the first counterexample of Sophie Germain prime which is Brazilian. It's the 141385th Sophie Germain prime 28792661 = 1 + 73 + 73^2 + 73^3 + 73^4 = (11111)73. The other counterexamples have been found by _Michel Marcus.
These numbers are relatively rare: only 25 terms < 10^15.
The 47278 initial terms of this sequence are of the form (11111)_b. The successive bases b are 73, 94, 175, 292, 418, 979, 1228, 1429, ...
The first term which is not of this form has 32 digits, it is 14781835607449391161742645225951 = 1 + 1309 + ... + 1309^9 + 1309^10 = (11111111111)_1309 with a string of eleven 1's. In this case, the successive bases b are 1309, 1348, 2215, 2323, 2461, ...
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: (5,73), (11,1309), (17,1945), (23,20413), (29,5023), (41,9565), (47,2764) (See link Jon Grantham, Hester Graves).
Other smallest pairs (q, b) are: (53, 139492), (59, 154501), (71, 7039), (83, 9325), (89, 78028), (101, 8869), (107, 86503), (113, 89986), (131, 429226), (137, 929620), (149, 1954), (167, 175), (173, 1368025). - David A. Corneth, Mar 13 2019

Examples

			78914411 is a term because 2 * 78914411 + 1 = 157828823 is prime, so 78914411 is Sophie Germain prime, then, 78914411 = 1 + 94 + 94^2 + 94^3 + 94^4 = (11111)_94 and 78914411 is also a Brazilian prime.

Links

Crossrefs

Intersection of A005384 and A085104.
Cf. A007528, 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(2*t+1), listput(v, t)))); v = vecsort(Vec(v), , 8); \\ 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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