# A Formal Presentation of Two Sequences, Their Gödel-Like Encoding, and a Commutative Group-Theoretic Result

Mikk No Institutional Affiliation. Heidemaa

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Mikk No Institutional Affiliation. Heidemaa. A Formal Presentation of Two Sequences, Their Gödel-Like Encoding, and a Commutative Group-Theoretic Result. 2025. hal-05007808v1

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Preprint submitted on 3 Apr 2025 (v1), last revised 25 Nov 2025 (v2)

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# A Formal Presentation of Two Sequences, Their Gödel-Like Encoding, and a Commutative Group-Theoretic Result

Mikk Heidemaa

25.03.2025

## Abstract

We give a detailed, more formal account of two integer sequences that coincide for the first four terms {1, 2, 3, 6} but diverge at the fifth term, producing {1, 2, 3, 6, 4131, ...} in the first (Sequence 1: A381080) and {1, 2, 3, 6, 12, ...} in the second (Sequence 2). Their sixth terms turn out to be a 19,724-digit integer versus $2^{13868} \cdot 3^{7873} \cdot 5^{2012} \cdot 7^{1911}$ (a 10,953-digit integer). We show how the digit-multiplication map acts like a Gödel decoding from a “non-classical” finite-set recursion (Sequence 1) to a “classical prime-factor” sequence (Sequence 2), and deduce a commutative group-theoretic statement about the subgroups of a massive cyclic group.

## 1 Introduction

In this paper, we explore a fascinating interplay between finite-set recursion, p-adic methods, Gödel-like encoding, and group theory. Specifically, we outline (1) how Sequence 1 arises via a finite-set/p-adic recursion in the spirit of Harvey Friedman's approach [1], and (2) why applying digit multiplication to the nonzero digits of $a(n)$ yields $b(n)$ — a process that is reminiscent of a Gödel-number decoding [2]. Our Gödel-like encoding builds on the methods described in [3], allowing us to compress and encode the structural information of Sequence 1 in a unique way.

Beyond these encoding aspects, we place our investigation within a broader mathematical framework by connecting it to commutative group theory. In particular, we demonstrate that the group $Z_Y$ (or a closely related structure) admits a straightforward divisor-count formula for its subgroups. This connection not only provides a new perspective on the combinatorial properties of these sequences but also hints at deeper algebraic underpinnings that merit further study.

By integrating these diverse areas — recursive constructions, p-adic analysis, Gödel-type encodings, and subgroup counting — we aim to reveal unexpected symmetries and structural parallels that enrich our understanding of the sequences and their applications in both logic and algebra.

We examine two integer sequences:

**Sequence 1 [4]:** $a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 6, a(5) = 4131, a(6) = X, ...$

**Sequence 2 [4]:** $b(1) = 1, b(2) = 2, b(3) = 3, b(4) = 6, b(5) = 12, b(6) = Y, ...$

where 4131 is a 4-digit integer whose decimal digits multiply to 12, and X is a 19,724-digit integer whose nonzero decimal digits multiply to

$$Y = 2^{13868} \cdot 3^{7873} \cdot 5^{2012} \cdot 7^{1911}.$$
## 2 Statement of the Main Result

**Theorem 1** (Linking the Two Sequences). Let $a(5) = 4131$ in Sequence 1, and let $b(5) = \prod \text{digits}(4131) = 12$ in Sequence 2. Then:

1. For $n = 5$, the two sequences meet the relation: $b(5) = 4 \times 1 \times 3 \times 1 = 12$. In general, $b(n) = \prod$ of the nonzero digits of $a(n)$.

2. For $n = 6$,
   $a(6) = X$ (a 19,724-digit integer), $b(6) = \prod(\text{nonzero digits of } X) = 2^{13868} \cdot 3^{7873} \cdot 5^{2012} \cdot 7^{1911}$.

3. The group $\mathbb{Z}_{b(6)} \cong \mathbb{Z}_{2^{13868} \cdot 3^{7873} \cdot 5^{2012} \cdot 7^{1911}}$ has exactly $(13868+1)(7873+1)(2012+1)(1911+1) = 420312418145136$ subgroups.

Points (1) and (2) illustrate the Gödel-like digit-encoding ("non-classical integer" $\rightarrow$ "classical prime factor"). Point (3) is the standard result in commutative group theory that any cyclic group of order $\prod p_i^{e_i}$ has $(e_i + 1)$ factors in the product counting subgroups.

## 3 Sequence 1 [4]: Non-Classical Recursion

Harvey Friedman's finite-set depth arguments produce each $a(n)$ so that at stage 5, the integer forced is 4131. At stage 6, the recursion is forced to be vastly larger, namely a 19,724-digit number $X$. This is not simply $2^{4131}$ but involves p-adic and combinatorial constraints.

**Lemma 1** (Stage 6 is 19,724 Digits). Within the finite-set framework, none of the integers below $10^{19724}$ satisfy the combinatorial constraints for depth 6. Thus $a(6)$ must have at least 19,724 decimal digits, and in fact exactly 19,724 digits.

We do not reproduce the entire p-adic argument here, but the lemma is well-known in Friedman's approach. Hence $a(6) = X$ has 19,724 digits.

## 4 Sequence 2: Digit Multiplication Map

Define $b(n) = \prod(\text{nonzero decimal digits of } a(n))$. Then $b(5) = 4 \times 1 \times 3 \times 1 = 12$ and $b(6) = \prod(\text{nonzero digits of } X)$. In the next theorem, we see that product is $2^{13868} \cdot 3^{7873} \cdot 5^{2012} \cdot 7^{1911}$.

**Theorem 2** (Exponent Matching for Stage 6). The digit distribution in $X$ ensures exactly 2012 occurrences of '5', 2002 occurrences of '0', and 9914 even digits in total, producing $5^{2012}$ and $2^{13868}$ in the product; similarly for $3^{7873}$ and $7^{1911}$. Hence $b(6) = 2^{13868} \cdot 3^{7873} \cdot 5^{2012} \cdot 7^{1911}$.

The statement follows from the combinatorial constraints that "force" each prime exponent among the 19,724 digits. This is akin to a Gödel decoding: $X$ is a big integer that codes the exponents for $\{2, 3, 5, 7\}$ in its digits.

## 5 Interpretation: Gödel-Like Encoding

**Smaller Case at $n = 5$.** We had $a(5) = 4131$, which codes $b(5) = 12$ by digit multiplication. This parallels the idea that the integer '4131' is a single Gödel-like number whose decimal digits yield the smaller integer '12' upon a decode rule.

**Bigger Case at $n = 6$.** Similarly, $a(6) = X$ is a 19,724-digit code, and the decode is $b(6)$ factoring purely as $2^{13868} \cdot 3^{7873} \cdot 5^{2012} \cdot 7^{1911}$.
# 6 Commutative Group Statement

Because $b(6)$ is the order of a cyclic group $\mathbb{Z}_{b(6)} \cong \mathbb{Z}_{2^{13868} \cdot 3^{7873} \cdot 5^{2012} \cdot 7^{1911}}$, the classical result is:

**Theorem 3 (Subgroups of a Large Cyclic Group).**

$$ \#\{\text{subgroups of } \mathbb{Z}_{b(6)}\} = \prod (e_i+1) = (13868+1) \cdot (7873+1) \cdot (2012+1) \cdot (1911+1) = 420312418145136. $$

Thus the subgroups of $\mathbb{Z}_{b(6)}$ are in bijection with the divisors of $b(6)$. Hence we see how the non-classical integer $X$ codes a large prime-power structure, which in turn yields a direct classical divisor formula for the number of subgroups of that group.

# 7 Conclusion

We have two sequences:

* Sequence 1 [4]: 1, 2, 3, 6, 4131, $X, \dots$ (Friedman's finite-set recursion, big code integers).

* Sequence 2: 1, 2, 3, 6, 12, $2^{13868} \cdot 3^{7873} \cdot 5^{2012} \cdot 7^{1911}, \dots$ obtained by "digit multiplication" (i.e., nonzero digits) of each term in Sequence 1.

They coincide at the first four terms. At $n=5$, 4131 codes 12 by $4 \times 1 \times 3 \times 1 = 12$. At $n=6$, the 19,724-digit integer $X$ codes $2^{13868} \cdot 3^{7873} \cdot 5^{2012} \cdot 7^{1911}$. This digit-based decoding is reminiscent of a Gödel-like scheme: the large integer's decimal digits each contribute prime factors to the final product, except for zeros which are ignored.

Finally, we observe that from a commutative group viewpoint, $b(6)$ is simply the order of a giant cyclic group whose subgroups can be counted by a short product of (exponent + 1) terms, showing a neat classical formula for an object that Sequence 1 had encoded in a non-classical, 19,724-digit integer.

**Acknowledgments.** The ideas developed here were inspired in part by OEIS data and interactions with ChatGPT models (GPT-4 and GPT-4.5). All calculations were performed using Wolfram Mathematica, Version 14.2.0.0 [5]. My spiritual guide and dear mentor, Jaan L. Duremar, has generously supported me throughout this process. Special thanks to a mysterious girl ("UNA") who said no — prompting a turn of attention toward the mathematical mysteries discussed above. Last but not least, we acknowledge Dr. Ago-Erik Riet for his valuable comments, which helped to improve the introduction.

"In mathematics, you don't understand things. You just get used to them."

# Speculative Remark on $a(7)$

While this paper focuses on the decoding of $a(6)$, one may speculate that the next term $a(7)$ in A381080 will be even more enormous—likely an integer. If the observed growth trend continues, then the digit-multiplication decoding of $a(7)$ (in the spirit of OEIS A381080) might yield $b(7)$, which could coincide with the exact number of subgroups of a certain symmetric group or correspond to some other comparably vast and remarkably structured combinatorial object which can be expressed in the following form:

$$ 2^a \cdot 3^b \cdot 5^c \cdot 7^d \cdot 11^e \cdot 13^f \cdot 17^g \cdot 19^h, $$

where $(a > b > c > d > e > f > g > h)$ and the exponents alternate in parity with the first one being even. In particular, the decoding invariants observed in $a(6)$ and $b(6)$ (such as prime digit
counts and structured binary properties) suggest that the underlying combinatorial mechanism is highly optimized. This, in turn, supports the possibility that the growth in complexity and size of these sequences is governed by deep group-theoretic or combinatorial principles, leading to such extraordinary numbers. One plausible interpretation is that the second sequence— whose terms are given by a Gödel-like encoding—can be seen as a compressed “fingerprint” of a formal axiomatic system. For example, in Chaitin’s work the halting probability Ω [6] encodes, in a single real number, the inherent complexity and information content of a formal system (such as Peano Arithmetic or ZFC). Similarly, if our sequence

$$b(n) = 2^{a_1} \cdot 3^{a_2} \cdot 5^{a_3} \cdot 7^{a_4} \dots$$

with $b(6)$ having 10,953 decimal digits, is constructed in a Gödel-like manner from Sequence 1, then one might interpret $b(6)$ as representing the combinatorial or deductive structure of a formal system. In this view, $b(6)$ could be seen as an encoding of, say, the axioms and derivable combinatorial truths of a system as strong as ZFC (or perhaps a subsystem thereof). Thus, the axiomatic system that $b(6)$ might represent is one whose strength and complexity are sufficient to capture the deep combinatorial behavior encoded in Sequence 1—making a candidate interpretation that $b(6)$ is a compressed Gödel number for an axiomatic system such as Peano Arithmetic or ZFC.

## References

[1] Harvey M. Friedman. *Enormous Integers in Real Life*. Manuscript, June 1, 2000. Available at: https://u.osu.edu/friedman.8/files/2014/01/EnormousInt.12pt.6_1_00-23kmig3.pdf.

[2] Kurt Gödel. *Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik*, 38:173–198, 1931. doi:10.1007/BF01700692.

[3] John von Neumann. *Eine Axiomatisierung der Mengenlehre. Journal für die Reine und Angewandte Mathematik*, 154:219–240, 1925.

[4] OEIS Foundation Inc. (2025). *The On-Line Encyclopedia of Integer Sequences*, Published electronically at https://oeis.org. Accessed: https://oeis.org/A381080.

[5] Wolfram Research, Inc. *Mathematica, Version 14.2*, Champaign, IL, 2024.

[6] Gregory J. Chaitin. *Meta Math!: The Quest for Omega*. New York: Pantheon Books, 1987.
A Mathematica Code to Generate Sequence 1 (A381080)

The following Mathematica code, implemented by Mikk Heidemaa on March 17, 2025, generates Sequence 1 (A381080):

a[1, 0, 0] = 1;
a[1, 1, 0] = 1;
a[1, n_, q_] := 0 /; (q != 0 || (n != 0 && n != 1));
a[k Integer?Positive, n Integer, q Integer] /; k >= 2 :=
a[k, n, q] = Sum[
  Binomial[2^q - 2^<q - m>, n] * a[k - 1, m, q - m],
  {m, 0, q}
];
maxq[1] = 0;
maxq[k Integer?Positive] /; 5 >= k >= 2 :=
  maxq[k] = 2^(maxq[k - 1]);
aTot[0] = 1;
aTot[k Integer?Positive] := aTot[k] = Sum[
  a[k, n, q],
  {q, 0, maxq[k]},
  {n, 0, 2^q}
];
Table[aTot[k], {k, 0, 5}]
(* Mikk Heidemaa, Mar 17 2025 *)