| Displaying 1-8 of 8 results found.
page
1
Numbers n such that the number of partitions of n^2+1 (= A000041(n^2+1)) is prime.
+10
8
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.
EXAMPLE
a(2) = 2 is in the sequence because A000041(2^2+1) = 7 is a prime.
PROG
(PARI) for(n=1, 3920, if(ispseudoprime(numbpart(n^2+1)), print1(n, ", ")))
Numbers n such that the number of partitions of n^2-1 is prime.
+10
8
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.
EXAMPLE
13 is in the sequence because A000041(13^2-1) = 228204732751 is a prime.
PROG
(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)))
Numbers n such that the number of partitions of n(n+1)/2 (= A000041( A000217(n))) is prime.
+10
8
2, 3, 8, 3947, 43968, 61681
COMMENTS
Because asymptotically A000041(n*(n+1)/2) ~ exp(Pi*sqrt(2/3*(n*(n+1)/2))) / (4*sqrt(3)*(n*(n+1)/2)), the sum of the prime probabilities ~1/log( A000041(n*(n+1)/2)) is diverging and there are no obvious restrictions on primality; therefore, this sequence may be conjectured to be infinite.
EXAMPLE
a(3) = 8 is in the sequence because A000041(8*9/2) = 17977 is a prime.
PROG
(PARI) for(n=1, 2000, if(ispseudoprime(numbpart(n*(n+1)/2)), print1(n, ", ")))
Indices k of partition function p where p(k) and p(k) + 2 are twin primes.
+10
4
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.
EXAMPLE
13 is a term because A000041(13) = 101 is prime and 103 is prime.
PROG
(PARI) for(n=1, 2500, if(ispseudoprime(p=numbpart(n))&&ispseudoprime(p+2), print1(n, ", ")))
Indices k of partition function p where p(k) is a twin prime.
+10
4
3, 4, 5, 6, 13, 186, 3542, 2335166
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.
EXAMPLE
13 is a term because A000041(13) = 101 is prime and 101 and 103 are twin primes.
PROG
(PARI) for(n=1, 3600, if(ispseudoprime(p=numbpart(n))&&(ispseudoprime(p-2)||ispseudoprime(p+2)), print1(n, ", ")))
Indices k of partition function p where p(k) and p(k) - 2 are twin primes.
+10
3
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.
EXAMPLE
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.
PROG
(PARI) for(n=1, 3600, if(ispseudoprime(p=numbpart(n))&&ispseudoprime(p-2), print1(n, ", ")))
Indices k of partition function where consecutive p(k) and p(k+1) are prime.
+10
0
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.
EXAMPLE
5 is in the sequence because A000041(5) = 7 and A000041(6) = 11 are prime.
PROG
(PARI) for(k=1, 5000, if(ispseudoprime(numbpart(k))&&ispseudoprime(numbpart(k+1)), print1(k, ", ")))
Index k of partition function p such that p(k) is a member of a cousin prime pair.
+10
0
3, 5, 6, 13, 36, 157, 302, 546, 2502, 2732, 19439060
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.
EXAMPLE
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.
PROG
(PARI) for(n=1, 10000, if(ispseudoprime(p=numbpart(n))&&(ispseudoprime(p-4)||ispseudoprime(p+4)), print1(n, ", ")))
Search completed in 0.007 seconds
| 