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.

A355704 Indices k of partition function p where p(k) and p(k) + 2 are twin primes.

Original entry on oeis.org

3, 4, 6, 13, 2335166
Offset: 1

Views

Author

Serge Batalov, Jul 14 2022

Keywords

Comments

Because asymptotically size of partitions number function p(n) ~ O(exp(sqrt(n))), and probability of primality of p(n) ~ O(1/sqrt(n)) and combined probability of primality of p(n) and p(n)+-2 is ~ O(1/n), the sum of the prime probabilities is diverging and there are no obvious restrictions on primality; therefore, this sequence may be conjectured to be infinite.
a(6) > 10^7.

Examples

			13 is a term because A000041(13) = 101 is prime and 103 is prime.

Crossrefs

Cf. A000041, A046063, A072213, A285086, A285087, A285088, A284594, A355705, A355706.

Programs

  • PARI
    for(n=1, 2500, if(ispseudoprime(p=numbpart(n))&&ispseudoprime(p+2), print1(n,", ")))

A355706 Indices k of partition function p where p(k) is a twin prime.

Original entry on oeis.org

3, 4, 5, 6, 13, 186, 3542, 2335166
Offset: 1

Views

Author

Serge Batalov, Jul 15 2022

Keywords

Comments

Because asymptotically the size of the partition number function p(n) is ~ O(exp(sqrt(n))), and the probability of primality of p(n) is ~ O(1/sqrt(n)) and the combined probability of primality of p(n) and p(n)+-2 is ~ O(1/n), the sum of the prime probabilities is diverging and there are no obvious restrictions on primality; therefore, this sequence may be conjectured to be infinite.
p(2335166), a 1696-digit number, was known to be prime and proven prime by F. Morain using his software (ca. April 2001), but the primality of p(2335166)+2 was found by targeted search (for this sequence) in July 2022.
a(9) > 10^7.

Examples

			13 is a term because A000041(13) = 101 is prime and 101 and 103 are twin primes.

Crossrefs

Subsequence of A046063.
Union of A355704 and A355705.
Cf. A000041, A072213, A285086, A285087, A285088, A284594.

Programs

  • PARI
    for(n=1, 3600, if(ispseudoprime(p=numbpart(n))&&(ispseudoprime(p-2)||ispseudoprime(p+2)), print1(n, ", ")))

A355728 Indices k of partition function where consecutive p(k) and p(k+1) are prime.

Original entry on oeis.org

2, 3, 4, 5, 1085
Offset: 1

Views

Author

Serge Batalov, Jul 15 2022

Keywords

Comments

Because asymptotically the size of the partition number function p(n) is ~ O(exp(sqrt(n))), and the probability of primality of p(n) is ~ O(1/sqrt(n)) and the combined probability of primality of p(n) and p(n+1) is ~ O(1/n), the sum of the prime probabilities is diverging and there are no obvious restrictions on primality; therefore, this sequence may be conjectured to be infinite.
a(6) > 10^8.

Examples

			5 is in the sequence because A000041(5) = 7 and A000041(6) = 11 are prime.

Crossrefs

Cf. A000041, A046063, A072213, A284594, A285086, A285087, A285088, A355704, A355705, A355706.

Programs

  • PARI
    for(k=1, 5000, if(ispseudoprime(numbpart(k))&&ispseudoprime(numbpart(k+1)), print1(k, ", ")))
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.