In this Deep Dive we unpack OEIS A000338. We explore its generating function, explain what the offset (offset 3, 1) means, and show how the infinite power-series expansion yields the sequence beginning 5, 18, 42, 75, 117. We derive the linear recurrence and connect the terms to a combinatorial story about discordant permutations, as discussed in J. Reordan's 1954 work and traced back to N. J. Sloan’s 1973 Handbook of Integer Sequences. The episode illustrates how a compact algebraic form hide...

Number of the Day - "353"This podcast is produced by Klassic Studios. Check out our other podcasts there! Learn more about your ad choices. Visit megaphone.fm/adchoices

Anton Korinek's NBER working paper argues that AI is evolving from responsive tools to autonomous agents that can plan, run multi-step analyses, and collaborate across tools to advance economic research. We trace the arc from System 1 to System 2 reasoning to agentic AI, explore vibe coding, private-data protocols, and cost/governance issues, and discuss why human judgment remains essential even as AI reshapes the questions we can tackle. Source: https://www.nber.org/papers/w34202 Note: Thi...

In this Deep Dive, we explore A000337, a small-seeming sequence that threads through binary arithmetic, geometry, and number theory. We'll trace how its simple definition links to counting zeros and bits in binary lists, to directed column-convex polyominoes, and to the genus of cube graphs. Along the way we’ll encounter prime and semiprime patterns noted by researchers, a neat generating function and a linear recurrence, and a combinatorial interpretation as sums of subset maxima. Join us as...

Number of the Day - "352" This podcast is produced by Klassic Studios. Check out our other podcasts there! Learn more about your ad choices. Visit megaphone.fm/adchoices

Number of the Day - "351" This podcast is produced by Klassic Studios. Check out our other podcasts there! Learn more about your ad choices. Visit megaphone.fm/adchoices

Number of the Day - "350" This podcast is produced by Klassic Studios. Check out our other podcasts there! Learn more about your ad choices. Visit megaphone.fm/adchoices

Number of the Day - "349" This podcast is produced by Klassic Studios. Check out our other podcasts there! Learn more about your ad choices. Visit megaphone.fm/adchoices

Number of the Day - "348" This podcast is produced by Klassic Studios. Check out our other podcasts there! Learn more about your ad choices. Visit megaphone.fm/adchoices

From its 2005 discovery by Mike Brown's team to its high-inclination orbit that kept it hidden in dense star fields, Makemake is a bright but enigmatic dwarf planet beyond Neptune. We explore how its Easter Island name origin became Makemake, its red, methane- and ethane-ice surface at about 40 K, and the surprising hints that it may host geothermal activity and a subsurface ocean. We also examine the 2011 occultation showing no substantial atmosphere, the single moon MK2, and what Makemake t...

In this Deep Dive we zoom in on OEIS A000336, the classic product-recurrence sequence. Starting with a1=1, a2=2, a3=3, a4=4 and, for n≥5, an = an−1 · an−2 · an−3 · an−4, the seeds explode into astonishing growth: a5=24, a6=576, a7=165,888, a8=9,172,942,848, and far beyond. We’ll unpack why such a simple rule yields such rapid, almost astronomical expansion and how later terms acquire hundreds of digits (a12, for example, reaches 139 digits). A standout twist is Hasler’s elegant alternative fo...

What is a spin glass? A disordered magnetic state with random couplings that freezes into many metastable configurations. We explore frustration, non-ergodic dynamics, and slow, non-exponential relaxation that can persist for days. The rugged energy landscape isn’t just about magnets—it’s a framework for thinking about learning in neural networks, optimization challenges, and even protein folding, culminating in Giorgio Parisi’s Nobel-winning insights into complex systems. Note: This podcast ...

¿Las fracciones te parecen complicadas? En este episodio del Calculadora Alicia Podcast: Aprende y Resuelve, un profesor y un estudiante conversan paso a paso sobre cómo dividir fracciones de manera sencilla y sin errores. Descubrirás el truco del inverso, aprenderás a simplificar los resultados y verás ejemplos prácticos que podrás aplicar de inmediato.Un recurso ideal para estudiantes, padres y profesores que buscan aprender matemáticas de forma clara, rápida y entretenida.

Number of the Day - "347" This podcast is produced by Klassic Studios. Check out our other podcasts there! Learn more about your ad choices. Visit megaphone.fm/adchoices

We explore A00331, the OEIS entry listing the coefficient of nu in the Rayleigh polynomial of even index 2n. From the numbers 5, 14, 1026, 4324 onward, we see a growth that encodes a deep link to the zeros of the Bessel function. We outline how these coefficients are constructed via standard formulas (psi_on and phi_pn) and why this sequence helps connect pure number theory to applied physics through Bessel zeros. We’ll also touch the historical trail—from Sloan’s Handbook to modern reference...

This week’s episode isn’t our usual deep dive—it’s a behind-the-scenes update. On Monday, Breaking Math vanished from Spotify, Apple, YouTube, and more without warning. After digging in, we discovered a strange RSS glitch that merged our feed with another podcast.In this quick update, Autumn shares: What happened behind the scenes when the show disappeared Which platforms are already fixed (and which we’re still waiting on) How you can make sure you never lose track of Breaking Math again What

Number of the Day - "346" This podcast is produced by Klassic Studios. Check out our other podcasts there! Learn more about your ad choices. Visit megaphone.fm/adchoices

Number of the Day - "345" This podcast is produced by Klassic Studios. Check out our other podcasts there! Learn more about your ad choices. Visit megaphone.fm/adchoices

We explore A000329, the tangent-iteration sequence defined by b(0) = 1 and the nearest integer to b(n), where b(n) = tan(b(n-1)). The interplay between the continuous, blow-up behavior of tan near odd multiples of π/2 and the discrete rounding step yields a surprisingly erratic sequence (with terms wandering through 1, 2, 75, -1, -1, -2 … and beyond). We unpack how the sensitivity of tan to its input, combined with rounding, acts as a powerful nonlinearity that can send the next term in a com...

Number of the Day - "344" This podcast is produced by Klassic Studios. Check out our other podcasts there! Learn more about your ad choices. Visit megaphone.fm/adchoices