Baeldung Pro – NPI EA (cat = Baeldung)
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Baeldung Pro comes with both absolutely No-Ads as well as finally with Dark Mode, for a clean learning experience:

>> Explore a clean Baeldung

Once the early-adopter seats are all used, the price will go up and stay at $33/year.

Partner – Microsoft – NPI EA (cat = Baeldung)
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Azure Container Apps is a fully managed serverless container service that enables you to build and deploy modern, cloud-native Java applications and microservices at scale. It offers a simplified developer experience while providing the flexibility and portability of containers.

Of course, Azure Container Apps has really solid support for our ecosystem, from a number of build options, managed Java components, native metrics, dynamic logger, and quite a bit more.

To learn more about Java features on Azure Container Apps, visit the documentation page.

You can also ask questions and leave feedback on the Azure Container Apps GitHub page.

Partner – Microsoft – NPI EA (cat= Spring Boot)
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Azure Container Apps is a fully managed serverless container service that enables you to build and deploy modern, cloud-native Java applications and microservices at scale. It offers a simplified developer experience while providing the flexibility and portability of containers.

Of course, Azure Container Apps has really solid support for our ecosystem, from a number of build options, managed Java components, native metrics, dynamic logger, and quite a bit more.

To learn more about Java features on Azure Container Apps, you can get started over on the documentation page.

And, you can also ask questions and leave feedback on the Azure Container Apps GitHub page.

Partner – Orkes – NPI EA (cat=Spring)
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Modern software architecture is often broken. Slow delivery leads to missed opportunities, innovation is stalled due to architectural complexities, and engineering resources are exceedingly expensive.

Orkes is the leading workflow orchestration platform built to enable teams to transform the way they develop, connect, and deploy applications, microservices, AI agents, and more.

With Orkes Conductor managed through Orkes Cloud, developers can focus on building mission critical applications without worrying about infrastructure maintenance to meet goals and, simply put, taking new products live faster and reducing total cost of ownership.

Try a 14-Day Free Trial of Orkes Conductor today.

Partner – Orkes – NPI EA (tag=Microservices)
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Modern software architecture is often broken. Slow delivery leads to missed opportunities, innovation is stalled due to architectural complexities, and engineering resources are exceedingly expensive.

Orkes is the leading workflow orchestration platform built to enable teams to transform the way they develop, connect, and deploy applications, microservices, AI agents, and more.

With Orkes Conductor managed through Orkes Cloud, developers can focus on building mission critical applications without worrying about infrastructure maintenance to meet goals and, simply put, taking new products live faster and reducing total cost of ownership.

Try a 14-Day Free Trial of Orkes Conductor today.

eBook – Guide Spring Cloud – NPI EA (cat=Spring Cloud)
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Let's get started with a Microservice Architecture with Spring Cloud:

>> Join Pro and download the eBook

eBook – Mockito – NPI EA (tag = Mockito)
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Mocking is an essential part of unit testing, and the Mockito library makes it easy to write clean and intuitive unit tests for your Java code.

Get started with mocking and improve your application tests using our Mockito guide:

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eBook – Java Concurrency – NPI EA (cat=Java Concurrency)
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Handling concurrency in an application can be a tricky process with many potential pitfalls. A solid grasp of the fundamentals will go a long way to help minimize these issues.

Get started with understanding multi-threaded applications with our Java Concurrency guide:

>> Download the eBook

eBook – Reactive – NPI EA (cat=Reactive)
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Spring 5 added support for reactive programming with the Spring WebFlux module, which has been improved upon ever since. Get started with the Reactor project basics and reactive programming in Spring Boot:

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eBook – Java Streams – NPI EA (cat=Java Streams)
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Since its introduction in Java 8, the Stream API has become a staple of Java development. The basic operations like iterating, filtering, mapping sequences of elements are deceptively simple to use.

But these can also be overused and fall into some common pitfalls.

To get a better understanding on how Streams work and how to combine them with other language features, check out our guide to Java Streams:

>> Join Pro and download the eBook

eBook – Jackson – NPI EA (cat=Jackson)
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Do JSON right with Jackson

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eBook – HTTP Client – NPI EA (cat=Http Client-Side)
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Get the most out of the Apache HTTP Client

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eBook – Maven – NPI EA (cat = Maven)
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Get Started with Apache Maven:

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eBook – Persistence – NPI EA (cat=Persistence)
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Working on getting your persistence layer right with Spring?

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eBook – RwS – NPI EA (cat=Spring MVC)
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Building a REST API with Spring?

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Course – LS – NPI EA (cat=Jackson)
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Get started with Spring and Spring Boot, through the Learn Spring course:

>> LEARN SPRING
Course – RWSB – NPI EA (cat=REST)
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Explore Spring Boot 3 and Spring 6 in-depth through building a full REST API with the framework:

>> The New “REST With Spring Boot”

Course – LSS – NPI EA (cat=Spring Security)
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Yes, Spring Security can be complex, from the more advanced functionality within the Core to the deep OAuth support in the framework.

I built the security material as two full courses - Core and OAuth, to get practical with these more complex scenarios. We explore when and how to use each feature and code through it on the backing project.

You can explore the course here:

>> Learn Spring Security

Course – All Access – NPI EA (cat= Spring)
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All Access is finally out, with all of my Spring courses. Learn JUnit is out as well, and Learn Maven is coming fast. And, of course, quite a bit more affordable. Finally.

>> GET THE COURSE
Course – LSD – NPI EA (tag=Spring Data JPA)
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Spring Data JPA is a great way to handle the complexity of JPA with the powerful simplicity of Spring Boot.

Get started with Spring Data JPA through the guided reference course:

>> CHECK OUT THE COURSE

Partner – LambdaTest – NPI EA (cat=Testing)
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End-to-end testing is a very useful method to make sure that your application works as intended. This highlights issues in the overall functionality of the software, that the unit and integration test stages may miss.

Playwright is an easy-to-use, but powerful tool that automates end-to-end testing, and supports all modern browsers and platforms.

When coupled with LambdaTest (an AI-powered cloud-based test execution platform) it can be further scaled to run the Playwright scripts in parallel across 3000+ browser and device combinations:

>> Automated End-to-End Testing With Playwright

Course – Spring Sale 2025 – NPI EA (cat= Baeldung)
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Yes, we're now running our Spring Sale. All Courses are 25% off until 26th May, 2025:

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Course – Spring Sale 2025 – NPI (cat=Baeldung)
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Yes, we're now running our Spring Sale. All Courses are 25% off until 26th May, 2025:

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1. Overview

In this tutorial, we’ll explore how to determine whether a given number can be expressed as the sum of two or more consecutive integers.

We’ll begin by understanding the mathematical principles behind consecutive integer sequences and how their properties can simplify our problem-solving process. Then, we’ll implement two approaches: a brute-force method and an optimized approach.

2. Mathematical Background

To see how these decompositions work in practice, let’s consider a few examples:

  • 9 = 2 + 3 + 4 or 4 + 5

  • 15 = 4 + 5 + 6

  • 10 = 1 + 2 + 3 + 4

  • 7 = 3 + 4

By contrast, we don’t find two or more consecutive positive integers that add up to 8.

Let’s begin by revisiting a result that directly leads to the criterion for identifying powers of two.

Every positive integer a can be written as a sum of two or more consecutive positive integers precisely when there exist integers k ≥ 1 and n ≥ 1 such that:

a = (n+1)*k + (n*(n+1))/2

The formula comes from the standard “sum of the first n integers” shifted to start at k:

n*(n+1)/2

Multiplying by 2 on both sides, we get:

2a  =  (n+1) *2k + n*(n+1) = (n+1) * (2k + n)

With this equation in hand, we can now prove why the powers of two fail to decompose and why every other positive integer succeeds.

2.1. Powers of Two Fail the Test

Let’s suppose a is a power of two. Then 2a is also a power of two, so its only positive divisors are powers of two. But in any factorization:

2a  =  (n+1) * (2k+n)

Regardless of whether n is even or odd, the two factors n+1 and 2k+n always have opposite parity, so they can’t be powers of two.

Moreover, to avoid the trivial cases (using only one summand), we require n ≥ 1 and k > 0. The only way two powers of two multiply to another power of two with opposite parity would force either n+1=1 (so n=0) or 2k+n=1 (so k=0), both of which violate our constraints. Hence, no power of two can be written as a sum of two or more consecutive positive integers.

2.2. Every Other Integer Works

Conversely, any integer a that isn’t a power of two has at least one odd divisor d > 1:

d=2m+1, a=d*q

For some integer q, the sequence:

(a/d - m), (a/d - m + 1), ..., (a/d + m)

Consists of d consecutive integers whose sum is exactly a. Even if a/d − m ≤ 0, one can shift the sequence by dropping negative terms and adding positive ones symmetrically to maintain the sum (or choose a different odd divisor), guaranteeing a valid representation.

Thus, all positive integers except powers of two can be expressed as the sum of two or more consecutive positive integers.

3. Brute-Force Approach

We’ll start with a straightforward method: try every possible sequence length until we either find a valid sum or exhaust our options. This approach is easy to understand and works well for moderate values of n:

@Test
void whenIsSumOfConsecutiveUsingBruteForce_thenItsTrue() {
    int n = 15;

    boolean isSumOfConsecutive = false;
    for (int k = 2; (k * (k - 1)) / 2 < n; k++) {
        int diff = n - k * (k - 1) / 2;
        if (diff % k == 0 && diff / k > 0) {
            isSumOfConsecutive = true;
            break;
        }
    }

    assertTrue(isSumOfConsecutive);
}

We loop over each possible length k, compute whether the remaining amount divides evenly, and check that the starting number stays positive. We return true if we find a match; otherwise, we return false after all lengths fail.

4. Optimized Approach

We can reduce the check to a single bitwise operation. If a positive integer n isn’t a power of two, it can be written as the sum of two or more consecutive integers. Otherwise, it can’t:

@Test
void whenIsSumOfConsecutiveUsingBitwise_thenReturnsTrue() {
    int n = 15;
    boolean result = (n > 0) && ((n & (n - 1)) != 0);
    assertTrue(result);
}

We return true when n is positive and fails the power of two tests. This runs in constant time and space, making it extremely efficient even for very large n.

5. Conclusion

In this article, we demonstrated that any positive integer can be expressed as the sum of two or more consecutive positive integers precisely when it isn’t a power of two.

Our brute-force approach methodically tested each possible sequence length, making it straightforward and reliable for moderate values of n. By contrast, the optimized bitwise check gives an O(1) solution that instantly identifies non-powers of two, making it ideal for very large inputs.

As always, the source code is available over on GitHub.

Baeldung Pro – NPI EA (cat = Baeldung)
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Baeldung Pro comes with both absolutely No-Ads as well as finally with Dark Mode, for a clean learning experience:

>> Explore a clean Baeldung

Once the early-adopter seats are all used, the price will go up and stay at $33/year.

Partner – Microsoft – NPI EA (cat = Spring Boot)
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Azure Container Apps is a fully managed serverless container service that enables you to build and deploy modern, cloud-native Java applications and microservices at scale. It offers a simplified developer experience while providing the flexibility and portability of containers.

Of course, Azure Container Apps has really solid support for our ecosystem, from a number of build options, managed Java components, native metrics, dynamic logger, and quite a bit more.

To learn more about Java features on Azure Container Apps, visit the documentation page.

You can also ask questions and leave feedback on the Azure Container Apps GitHub page.

Partner – Orkes – NPI EA (cat = Spring)
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Modern software architecture is often broken. Slow delivery leads to missed opportunities, innovation is stalled due to architectural complexities, and engineering resources are exceedingly expensive.

Orkes is the leading workflow orchestration platform built to enable teams to transform the way they develop, connect, and deploy applications, microservices, AI agents, and more.

With Orkes Conductor managed through Orkes Cloud, developers can focus on building mission critical applications without worrying about infrastructure maintenance to meet goals and, simply put, taking new products live faster and reducing total cost of ownership.

Try a 14-Day Free Trial of Orkes Conductor today.

Partner – Orkes – NPI EA (tag = Microservices)
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Modern software architecture is often broken. Slow delivery leads to missed opportunities, innovation is stalled due to architectural complexities, and engineering resources are exceedingly expensive.

Orkes is the leading workflow orchestration platform built to enable teams to transform the way they develop, connect, and deploy applications, microservices, AI agents, and more.

With Orkes Conductor managed through Orkes Cloud, developers can focus on building mission critical applications without worrying about infrastructure maintenance to meet goals and, simply put, taking new products live faster and reducing total cost of ownership.

Try a 14-Day Free Trial of Orkes Conductor today.

eBook – HTTP Client – NPI EA (cat=HTTP Client-Side)
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The Apache HTTP Client is a very robust library, suitable for both simple and advanced use cases when testing HTTP endpoints. Check out our guide covering basic request and response handling, as well as security, cookies, timeouts, and more:

>> Download the eBook

eBook – Java Concurrency – NPI EA (cat=Java Concurrency)
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Handling concurrency in an application can be a tricky process with many potential pitfalls. A solid grasp of the fundamentals will go a long way to help minimize these issues.

Get started with understanding multi-threaded applications with our Java Concurrency guide:

>> Download the eBook

eBook – Java Streams – NPI EA (cat=Java Streams)
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Since its introduction in Java 8, the Stream API has become a staple of Java development. The basic operations like iterating, filtering, mapping sequences of elements are deceptively simple to use.

But these can also be overused and fall into some common pitfalls.

To get a better understanding on how Streams work and how to combine them with other language features, check out our guide to Java Streams:

>> Join Pro and download the eBook

eBook – Persistence – NPI EA (cat=Persistence)
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Working on getting your persistence layer right with Spring?

Explore the eBook

Course – LS – NPI EA (cat=REST)

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Get started with Spring Boot and with core Spring, through the Learn Spring course:

>> CHECK OUT THE COURSE

Course – Spring Sale 2025 – NPI EA (cat= Baeldung)
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Yes, we're now running our Spring Sale. All Courses are 25% off until 26th May, 2025:

>> EXPLORE ACCESS NOW

Course – Spring Sale 2025 – NPI (All)
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Yes, we're now running our Spring Sale. All Courses are 25% off until 26th May, 2025:

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eBook Jackson – NPI EA – 3 (cat = Jackson)
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