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 1-8 of 8 results.

A284594 Numbers whose square has a prime number of partitions.

Original entry on oeis.org

2, 6, 29, 36, 2480, 14881
Offset: 1

Views

Author

Serge Batalov, Mar 29 2017

Keywords

Comments

Because asymptotically A072213(n) = A000041(n^2) ~ exp(Pi*sqrt(2/3)*n) / (4*sqrt(3)*n^2), the sum of the prime probabilities ~ 1/log(A072213(n)) is diverging and there are no obvious restrictions on primality; therefore, this sequence may be conjectured to be infinite.
Curiously, both A000041(6^2) and A000041(6^4) are prime; in addition, A000041(6^3) and A000041(6^1) are prime, but for no other powers A000041(6^k) is known (or can be expected) to be prime.
a(7) > 649350.

Examples

			a(2) = 6 is in the sequence because A000041(6^2) = 17977 is a prime.

Links

Crossrefs

Cf. A000041, A046063, A072213, A285086, A285087, A285088.

Programs

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

A285086 Numbers n such that the number of partitions of n^2+1 (=A000041(n^2+1)) is prime.

Original entry on oeis.org

1, 2, 3914
Offset: 1

Views

Author

Serge Batalov, Apr 09 2017

Keywords

Comments

Because asymptotically A000041(n^2+1) ~ exp(Pi*sqrt(2/3*(n^2+1))) / (4*sqrt(3)*(n^2+1)), the sum of the prime probabilities ~ 1/log(A000041(n^2+1)) is diverging and there are no obvious restrictions on primality; therefore, this sequence may be conjectured to be infinite.
a(4) > 90000.

Examples

			a(2) = 2 is in the sequence because A000041(2^2+1) = 7 is a prime.

Links

Crossrefs

Cf. A000041, A046063, A072213, A284594, A285087, A285088.

Programs

  • PARI
    for(n=1,3920,if(ispseudoprime(numbpart(n^2+1)),print1(n,", ")))

A285087 Numbers n such that the number of partitions of n^2-1 is prime.

Original entry on oeis.org

2, 13, 21, 46909
Offset: 1

Views

Author

Serge Batalov, Apr 09 2017

Keywords

Comments

Because asymptotically A000041(n^2-1) ~ exp(Pi*sqrt(2/3*(n^2-1))) / (4*sqrt(3)*(n^2-1)), the sum of the prime probabilities ~1/log(A000041(n^2-1)) is diverging and there are no obvious restrictions on primality; therefore, this sequence may be conjectured to be infinite.
a(5) > 50000.

Examples

			13 is in the sequence because A000041(13^2-1) = 228204732751 is a prime.

Links

Crossrefs

Cf. A000041, A046063, A072213, A284594, A285086, A285088.

Programs

  • PARI
    for(n=1,2000,if(ispseudoprime(numbpart(n^2-1)),print1(n,", ")))
  • Python
    from itertools import count, islice
from sympy import isprime, npartitions
def A285087_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: isprime(npartitions(n**2-1)), count(max(startvalue,1)))
A285087_list = list(islice(A285087_gen(),3)) # Chai Wah Wu, Nov 20 2023

Formula

{n: A000041(n^2-1) in A000040}.

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, ", ")))

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 1-8 of 8 results.

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