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.

User: Serge Batalov

Serge Batalov's wiki page.

Serge Batalov has authored 27 sequences. Here are the ten most recent ones:

A360994 Numbers k such that (2^k + 1)^3 - 2 is a semiprime.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 13, 14, 18, 27, 43, 45, 63, 76, 85, 108, 115, 119, 123, 187, 211, 215, 283, 312
Offset: 1

Views

Author

Serge Batalov, Feb 27 2023

Keywords

Comments

a(25) >= 355.
623, 674, 711, 766, 767 are also in this sequence, but their position cannot be established before finding any factor for the values corresponding to the following "blockers": 355, 511, 587, 707, 731.
1424, 1470, 1580, 1946, 2117, 2693, 3000, 3540, 4164, 7043, 9475, 10632, 15018, 19064, 27130, 28266, 28532, 46434, 58768, 103536 are some larger members of this sequence, but their position cannot be established. These produce "trivial" semiprimes where one prime is small (e.g., 3 or 5).

Links

Crossrefs

Cf. A001358, A091513, A091514, A093069, A099359, A100899, A100900, A269264, A360993.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [1..70]| IsSemiprime(s) where s is (2^n+1)^3-2];
  • Mathematica
    Select[Range[70], PrimeOmega[(2^# + 1)^3 - 2] == 2 &]
  • PARI
    isok(n) = bigomega((2^n+1)^3-2) == 2;

Formula

{ k >= 0 : A099359(k) in { A001358 } }.

A360993 Numbers k such that (2^k - 1)^3 + 2 is a semiprime.

Original entry on oeis.org

4, 5, 8, 12, 13, 18, 20, 29, 38, 56, 60, 62, 76, 82, 101, 118, 202, 210, 230, 276, 328, 332, 336, 338, 368
Offset: 1

Views

Author

Serge Batalov, Feb 27 2023

Keywords

Comments

a(26) >= 406.
438, 500, 526, 604, 648, 696 are also in this sequence, but their positions cannot be established before finding any factor for the values corresponding to the following "blockers": 406, 496, 528.
2382, 2733, 2910, 3368, 3508, 5338, 7705, 11185, 19905, 23814, 38545, 179294 are larger terms of this sequence, but their positions cannot be established. These produce "trivial" semiprimes where one prime is small (e.g., 3 or 11).

Examples

			a(1) = 4 because 15^3 + 2 = 3377 = 11 * 307, which is semiprime.
a(2) = 5 because 31^3 + 2 = 29793 = 3 * 9931, which is semiprime.

Links

Crossrefs

Cf. A091515, A091516, A100899, A100900, A268574, A269264, A360994.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [2..70]| IsSemiprime(s) where s is (2^n-1)^3+2];
  • Mathematica
    Select[Range[70], PrimeOmega[(2^# - 1)^3 - 2] == 2 &]
  • PARI
    isok(n) = bigomega((2^n-1)^3+2) == 2;

Extensions

a(20)-a(26) from Serge Batalov, Mar 03 2023

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, ", ")))

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, ", ")))

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, ", ")))

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,", ")))

A334993 Numbers k such that 2*3^k + 1 is prime and divides Phi(3^m, 2).

Original entry on oeis.org

1, 5, 9, 17, 57, 65, 897, 4217, 6225, 152529, 3648969, 5570081
Offset: 1

Views

Author

Serge Batalov, May 18 2020

Keywords

Comments

A subset of odd values from A003306.
If p = 2*3^k + 1 is prime then p divides 2^(3^k) + (-1)^k, due to Euler's criterion.
Only odd terms of sequence A003306 can divide the cyclotomic expression Phi(3^m, 2); none of the even terms of sequence A003306 can divide 2^3^k-1 and therefore cannot divide Phi(3^m, 2).

Links

Crossrefs

Cf. A003306.

Programs

  • PARI
    dp(n)=Mod(2,2*3^n+1)^3^n==1;
forstep(n=1,6225,2,if(dp(n),print1(n,", ")))

A298206 a(n) is the smallest b >= 2 such that b^(6*2^n) - b^(3*2^n) + 1 is prime.

Original entry on oeis.org

6, 3, 3, 6, 5, 106, 207, 569, 224, 736, 2854, 21234, 14837, 165394, 24743, 62721, 237804, 143332
Offset: 0

Views

Author

Serge Batalov, Jan 14 2018

Keywords

Comments

a(13) = 165394 is a significant outlier from the generally expected trend, which can be conjectured to be 6*2^n*gamma, where gamma is the Euler-Mascheroni constant A001620. Additionally, the next b > a(13) such that b^(6*2^n) - b^(3*2^n) + 1 is prime is 165836, which is remarkably close to a(13). - Serge Batalov, Jan 24 2018

Examples

			2^12 - 2^6 + 1 = 4033 is composite and 3^12 - 3^6 + 1 = 530713 is prime, so a(1) = 3.

Links

Crossrefs

Subsequence of A205506.
Cf. A001620, A085398, A153438, A246119.

Programs

  • PARI
    for(n=0,9,for(b=2,1000,x=b^(3*2^n); if(isprime(x*(x-1)+1), print1(b,", "); break)))

Formula

a(n) = A085398(18*2^n). - Jinyuan Wang, Dec 21 2022

Extensions

a(13) from Serge Batalov, Jan 24 2018

A288451 Numbers n such that !n + 7 is prime.

Original entry on oeis.org

0, 3, 4, 5, 7, 10, 12, 20, 37, 52, 73, 149, 304, 540, 2135, 7112, 7436, 9357
Offset: 1

Views

Author

Serge Batalov, Jul 14 2017

Keywords

Comments

At present the terms >= 2135 are only probable primes.
Expected to be finite, similar to Živković (1999).

Examples

			4 is a term, because 0! + 1! + 2! + 3! + 7 = 17 is prime.

Links

Crossrefs

Cf. A001272, A063833, A100614.

Programs

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