A378384 Contribution to the OEIS

 A378384

Digital root of the sum of the previous 3 terms; a(0) = a(1) = a(2) = 1.

0

1, 1, 1, 3, 5, 9, 8, 4, 3, 6, 4, 4, 5, 4, 4, 4, 3, 2, 9, 5, 7, 3, 6, 7, 7, 2, 7, 7, 7, 3, 8, 9, 2, 1, 3, 6, 1, 1, 8, 1, 1, 1, 3, 5, 9, 8, 4, 3, 6, 4, 4, 5, 4, 4, 4, 3, 2, 9, 5, 7, 3, 6, 7, 7, 2, 7, 7, 7, 3, 8, 9, 2, 1, 3, 6, 1, 1, 8, 1, 1, 1, 3, 5, 9, 8, 4, 3, 6

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OFFSET

0,4

COMMENTS

This differs from A112661 which is sum of digits of sum of previous 3 terms.

Digital root of A000213 (tribonacci numbers beginning {1,1,1}).

This has a period of 39 beginning with the first term.

Decimal expansion of 12373315960504936995263080863765792902/111111111111111111111111111111111111111 = 0.[111359843644544432957367727773892136118] (periodic).

LINKS

Table of n, a(n) for n=0..87.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

FORMULA

a(n) = A010888(A000213(n)).

MATHEMATICA

Nest[Append[#, ResourceFunction["AdditiveDigitalRoot"][Total[Take[#, -3]]]]&, {1, 1, 1}, 85]

CROSSREFS

Cf. A000213, A010888, A112661.

Sequence in context: A221967 A079428 A094548 * A112661 A230725 A254689

Adjacent sequences: A378381 A378382 A378383 * A378386 A378388 A378389

KEYWORD

nonn,base,easy,new

AUTHOR

James C. McMahon, Nov 24 2024

STATUS

approved

Primes that can be expressed in the form p² + 4q², where both p and q are prime numbers

In a significant advancement for number theory, mathematicians Ben Green of the University of Oxford and Mehtaab Sawhney of Columbia University have introduced a novel method for identifying specific types of prime numbers. Their work, detailed in a recent Quanta Magazine article, focuses on primes that can be expressed in the form p² + 4q², where both p and q are prime numbers.

Quanta Magazine

Prime numbers, defined as numbers greater than 1 that have no positive divisors other than 1 and themselves, are fundamental to mathematics. Understanding their distribution has been a longstanding challenge. While the infinitude of primes was established by Euclid around 300 BCE, identifying primes that satisfy additional constraints has proven difficult. Green and Sawhney's achievement in proving the existence of infinitely many primes of the form p² + 4q² represents a significant breakthrough in this area.

Their approach diverged from traditional methods by incorporating tools from other mathematical disciplines, demonstrating the potential for interdisciplinary techniques to address complex problems in number theory. This innovative strategy not only resolved a specific conjecture but also opened avenues for applying similar methods to other mathematical challenges.

The implications of this discovery extend beyond the immediate result. By enhancing our understanding of prime distribution, it contributes to the broader field of analytic number theory and may influence related areas such as cryptography, where prime numbers play a crucial role.

For a more comprehensive exploration of Green and Sawhney's work and its significance, the full article is available on Quanta Magazine's website.

Quanta Magazine

Unveiling the Mysteries of Sequences: A Dive into A048720, A065621, and A379129/A379130

 Unveiling the Mysteries of Sequences: A Dive into A048720, A065621, and A379129/A379130

In the realm of mathematics, fascinating sequences emerge, each with unique properties and applications. Today, we'll delve into three such sequences: A048720, A065621, and the intriguing pair A379129 and A379130.

This exploration is inspired by a recent article (link: https://news.google.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?hl=en-US&gl=US&ceid=US%3Aen), which piqued our curiosity about these specific sequences.

A Glimpse into the Sequences:

• A048720: This sequence involves converting binary representations of two numbers, multiplying them as polynomials (considering the digits as coefficients), and taking the modulo 2 result.
• A065621: Here, we perform a bitwise XOR operation between n-1 and 2n-1.
• A379129 and A379130: These sequences delve deeper, utilizing concepts like sum of divisors, greatest common divisor (GCD), and potentially building upon A048720 and A065621. However, the details of their calculations differ slightly (refer to the provided link for the original PARI/GP code).

Why are these sequences interesting?

While the specific applications of these sequences might not be readily apparent, their existence and properties contribute to the vast tapestry of mathematical knowledge. Studying them can lead to new discoveries, connections between seemingly disparate areas of mathematics, and even potential applications in cryptography or computer science.

Further Exploration:

The provided link offers the original PARI/GP code for calculating these sequences. We've also included equivalent Mathematica code within the comments of this blog post (accessible if you have access to edit the post).

Feel free to delve deeper into these sequences, explore their properties, and potentially discover fascinating relationships or applications. The world of mathematics is full of surprises waiting to be unveiled!