Bönisch, Duhr and Maggio introduced C_6b and conjectured that it is magnetic of depth 2. Shvets proved the stronger denominator-one statement a(n) in Z for every n >= 1.
Up to scaling, C_6b is the canonical CM form F_{-4,2} of discriminant -4 attached to the level-2 K3 packet: F_{-4,2} = 32*C_6b. Here F_{-4,2} is meromorphic of weight 6 for Gamma_0(2), vanishing at the cusps, with poles at the associated CM points; it is the specialization f_{3,-4,2,1,1} of the higher-level forms of Löbrich and Schwagenscheidt. In particular C_6b is not holomorphic, and its coefficients grow exponentially.
If n = 2^r*m with m odd, then 2^(3*r) divides a(n). In fact, the stronger estimate v_2(c_6b(2^r*m)) >= 5*r holds.
This is the (e,d) = (3,2) member of the equianharmonic symmetric-power family. It is the self-convolution of A276018, hence all terms are integers.
The exponent 3 is uniformly optimal in both prime classes, and the inert orbit length 2 is minimal: v_7(a(7)-a(1))=3; also a(5) != a(1) (mod 5), while v_5(a(25)-a(1))=3.
n^3*a(n) = 6*(3n-2)*(3n^2-4n+2)*a(n-1) - 27*(3n-4)^3*a(n-2), n >= 2, with a(0)=1 and a(1)=6.
(theta^3 - 6*x*(3*theta+1)*(3*theta^2+2*theta+1) + 27*x^2*(3*theta+2)^3)*G(x) = 0 where theta=x*d/dx and G(x) is the g.f.
Sum_{n>=0} (-1)^n*a(n)*t_3(tau)^n = eta(tau)^6/eta(3*tau)^2 where t_3(tau) = (eta(3*tau)/eta(tau))^12.
a(m*p^(f*r)) == a(m*p^(f*(r-1))) (mod p^(3*(r+floor(v_p(m)/f)))) for every prime p >= 5, and m,r >= 1, where f = ord_3(p), so f=1 if p == 1 (mod 3) and f=2 if p == 2 (mod 3).
The geode G is here the 5-dimensional array arising from operator inversion of the hyper-Catalan generating series in variables t_2, t_3, t_4, t_5, t_6: if S denotes that series, then S - 1 = (t_2 + t_3 + t_4 + t_5 + t_6) * G. This sequence records the main diagonal a(n) = G(n,n,n,n,n).
The value G(1000,1000,1000,1000,1000) is an 8367-digit integer, answering a challenge of Zeilberger (see the Cgeode link, where a donation to the OEIS Foundation was offered for its computation). It was computed in August 2026 by two independent implementations (an alternating-sum/generating-function collapse in exact arithmetic, and an independently written second implementation), each run modulo disjoint sets of more than 390 word-sized primes and reconstructed by CRT with agreement on every prime.
T. Amdeberhan, M. Kauers, and D. Zeilberger, The Challenge of Computing Geode Numbers, arXiv:2508.10245 [math.CO], 2025; Palestine J. Math. 14(4) (2025), 709-713.
The results of the division are: 2,3,5,5,7,17,11... (is this related to A338141?) which are equivalent to the tau values of the numbers plus one (does this hold true for larger numbers?).
p^(p-2) (A130614) is a term for all primes p since tau(p^(p-2)) * sigma(p^(p-2)) + 1 = p^(p-1). - Amiram Eldar, Aug 13 2026
Put simply, the rows represent the states when the task of generating all permutations of [1,...,n] is halfway complete.
REFERENCES
Donald E. Knuth, The Art of Computer Programming, Volume 4A, Combinatorial Algorithms, Part 1, Pearson Education, 2011. Chapter 7.2.1.2 Generating all permutations, Exercise 2, page 344 with answer on page 703.
Consider a prime number prime(i) with k digits (d_1, d_2, ..., d_k), where d_k is the units digit. Consider the k preceding prime numbers prime(i-1), prime(i-2), ..., prime(i-k) and the numbers obtained by concatenating [d_1, prime(i-1)], [d_2, prime(i-2)], ..., [d_k, prime(i-k)]. If the latter are all prime, then prime(i) belongs to the sequence.
Carlos Rivera, Puzzle 1271. 91997, The Prime Puzzles & Problems Connection.
EXAMPLE
5 is a term: 5 has one digit; the preceding prime is 3 -> [5,3] = 53 is prime.
367 is a term: 367 has three digits; the three preceding primes are 359, 353 and 349 -> [3, 359] = 3359, [6, 353] = 6353 and [7, 349] = 7349 are all primes.
These are the even terms of A398316. A398316 is the disjoint union of A398569 and this sequence.
Since all the terms are even, the prime 2 appears with an exponent of 1 or 2: either 2 is the only prime appearing with an exponent of 2, or the unique prime appearing with an exponent of 2 is an odd prime.
The asymptotic density of this sequence is (2/Pi^2) * (Sum_{p prime} 1/(p*(p+1)) + 1/3) = A185197 * (A179119 + 1/3) = 0.1344660297679805099613... . - Amiram Eldar, Aug 10 2026
