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 1-4 of 4 results.

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

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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, ", ")))

A355705 Indices k of partition function p where p(k) and p(k) - 2 are twin primes.

Original entry on oeis.org

4, 5, 186, 3542
Offset: 1

Views

Author

Serge Batalov, Jul 15 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(5) > 10^7.

Examples

			4 is a term because A000041(4) = 5, and 3 and 5 are twin primes.
5 is a term because A000041(5) = 7, and 5 and 7 are twin primes.

Crossrefs

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

Programs

  • PARI
    for(n=1, 3600, if(ispseudoprime(p=numbpart(n))&&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, ", ")))

A355956 Index k of partition function p such that p(k) is a member of a cousin prime pair.

Original entry on oeis.org

3, 5, 6, 13, 36, 157, 302, 546, 2502, 2732, 19439060
Offset: 1

Views

Author

Serge Batalov, Jul 21 2022

Keywords

Comments

Because asymptotically the size of the partition number function p(n) ~ O(exp(sqrt(n))), and the probability of primality of p(n) ~ O(1/sqrt(n)) and the combined probability of primality of p(n) and p(n)+-4 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(12) > 4*10^7.

Examples

			5 is in the sequence because A000041(5) = 7 and 7 + 4 = 11 are cousin primes.
13 is in the sequence because A000041(13) = 101 and 101 - 4 = 97 are cousin primes.

Crossrefs

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

Programs

  • PARI
    for(n=1, 10000, if(ispseudoprime(p=numbpart(n))&&(ispseudoprime(p-4)||ispseudoprime(p+4)), print1(n, ", ")))
Showing 1-4 of 4 results.

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