Autonomous Integer Sequence Research System

Bias Rules

Prefer candidates that are likely to be multiplicative — they are richer, easier to characterize at prime powers, and more likely to be novel. Structural signals to pursue:

• φ compositions over divisors
• Products involving unitary divisors
• Dirichlet convolutions (μ ★ f, id ★ φ, etc.)
• Functions of the prime signature only (depend on exponents, not primes)

Avoid: trivial linear combinations of known sequences, constant multiples of φ or σ, anything obviously reducible by inspection.

Example Constructions Worth Exploring

φ(∏_{d|n} φ(d))          — §2 shows this is non-multiplicative, interesting growth
∑_{d|n} d·φ(d)            — multiplicative, near-quadratic growth, strong candidate
∏_{d|n, gcd(d,n/d)=1} φ(d)  — unitary variant; check against dual impl
φ(n) · 2^{ω(n)}           — multiplicative; geometric prime-power profile
(μ ★ φ)(n)                 — Dirichlet convolution; vanishes on many n
∑_{d|n} gcd(d, n/d)        — multiplicative; sub-linear growth
∏_{p^a ∥ n} p^{φ(a)}      — depends only on prime signature

§2 · Multi-Implementation Evaluation

Rule: every sequence must have ≥2 independent implementations before conjecture work begins.

Implementation Types

A. Definition-based — iterate over divisors(n) directly.

B. Factorization-based — exploit the prime factorization n = ∏ pᵢ^aᵢ. For multiplicative functions this gives a cleaner formula: f(n) = ∏ f(pᵢ^aᵢ), computable from factorint(n) alone.

C. Hybrid — cached + partial symbolic where helpful for large n.

Differential Validation

Run both implementations on n = 1…30 plus the weak zones (§4). If they disagree at any n: the sequence definition is ambiguous or one implementation is wrong. Fix before proceeding. Do not conjecture on a sequence with unresolved implementation disagreements.

§3 · Multiplicativity Detection

Test f(ab) = f(a)·f(b) for gcd(a,b) = 1 across:

• Pairs of distinct primes: (2,3), (2,5), (3,7), (5,11), …
• Prime × prime square: (2, 9), (4, 3), (4, 25), …
• Larger coprime products: (4, 9), (8, 25), (16, 27), …

If multiplicative: proceed to prime-power profiling (§3.1) and enforce multiplicative modeling in conjecture work (§5).

If not multiplicative: compute the interaction residual f(ab)/(f(a)·f(b)) for coprime pairs and check whether the residual has structure. If it does, the function may be "almost multiplicative" in a useful sense. If it doesn't, the function is unlikely to have a clean closed form.

3.1 · Prime-Power Profiling

For any multiplicative (or candidate-multiplicative) function, compute:

f(p^k) for p ∈ {2, 3, 5, 7, 11}, k = 1…8

Look for:

Once the prime-power formula is identified, the full multiplicative function follows immediately from f(n) = ∏ f(p^a) over the factorization.

§4 · Adversarial Falsification (Most Important Stage)

The goal is to find the smallest n where the conjecture fails before claiming success. Always run this before reporting any formula.

4.1 · Weak Zones (Test These First)

These inputs break naive formulas most often:

WEAK_ZONES = (
    # Prime powers
    [p**k for p in [2,3,5,7,11,13] for k in range(1,6)] +
    # Products of two primes and their powers
    [p**a * q**b for p,q in [(2,3),(2,5),(3,5)] for a,b in [(1,1),(2,1),(1,2)]] +
    # Highly composite: 12, 24, 60, 120, 360, 720, 840, 2520
    [12, 24, 60, 120, 360, 720, 840, 2520] +
    # Powers of 2 and near-powers
    [2**k for k in range(1,20)] + [2**k - 1 for k in range(2,20)] +
    # Three-prime products
    [2*3*5, 2*3*7, 2*5*7, 3*5*7, 2*3*5*7]
)

4.2 · Mutation Attack

Given a failing n₀, propagate:

n₀ → n₀ · p   (extend by small prime)
n₀ → n₀ / p   (reduce if p² | n₀)
n₀ → n₀²      (square)

This often reveals whether the failure is isolated or structural.

4.3 · Delta Debugging

If a counterexample is found at composite n, find the smallest n' where the formula fails. Prime powers and small semiprimes are the most informative.

4.4 · Red Team Checklist

Before accepting any formula, verify:

1. Does it work for n = 1? (edge case: φ(1)=1, σ(1)=1, ω(1)=0, μ(1)=1)
2. Does it work for prime n? For p²? For p³?
3. Does it work for pq (two distinct primes)?
4. Does it work for highly composite n (60, 120, 720)?
5. Is the formula actually a known sequence in disguise? (→ §6 OEIS check)
6. Does the conjectured formula match the factorization-based dual? (→ §2)
7. Is the growth rate consistent across the tested range? (→ §5)

§5 · Conjecture Engine

Only conjecture after multiplicativity testing (§3) and adversarial testing (§4) have passed.

5.1 · Template Hierarchy

Try formulas in this order (simpler first):

1. f(n) = n, f(n) = φ(n), f(n) = σ(n), f(n) = τ(n), f(n) = rad(n)
2. f(n) = φ(n)^a, f(n) = n^a · φ(n)^b
3. f(n) = 2^{g(ω(n))} · h(n)
4. For multiplicative f: derive from the prime-power formula directly

5.2 · Log-Space Regression

For growth estimation and exponent fitting:

alpha = cov(log n, log f(n)) / var(log n)

This gives the asymptotic exponent in f(n) ~ C · n^alpha. Use n > 10 to avoid distortion from small-n irregularities.

5.3 · Multiplicative Conjecture Construction

If f is multiplicative with prime-power formula f(p^k) = g(p,k), then:

f(n) = ∏_{p^a ∥ n}  g(p, a)

This is the canonical form. Test the fully reconstructed multiplicative function against the definition-based implementation on n = 1…100 and all weak zones.

§6 · OEIS Collision Filter

Compute the first 100 terms and check against:

• Standard arithmetic functions: φ, σ, τ, rad, ω, Ω, μ, id, id²
• Their Dirichlet convolutions
• Known transforms: partial sums, Euler transform, Möbius transform

If the first 30 terms match a known sequence exactly, the candidate is not novel — report the collision and move on.

If the first 30 terms are unique: compute 100 terms and search OEIS directly (format: 1, 2, 4, 4, 8, 8, 12, 8, 12, 16 — comma-separated, no spaces around commas).

§7 · Confidence Scoring

Each sequence receives a score 0–100:

Confidence levels:

§8 · Output Standard (OEIS-Grade)

A sequence report must include:

1. Definition — at least two equivalent formulations
2. First 100–500 terms — verified across ≥2 implementations
3. Multiplicativity — proved or disproved with witness if not
4. Prime-power characterization — f(p^k) formula with evidence
5. Closed form — if found, with falsification report (how many n tested, result)
6. Counterexample search — what was tested, what survived
7. Growth exponent — asymptotic α from log-regression
8. OEIS collision check — which known sequences were compared
9. Confidence score — with breakdown
10. Reference implementation — clean Python, importable

§9 · Known Failure Modes and Mitigations

§10 · Meta-Insight Tracking

After running multiple candidates, record which constructions tend to produce useful sequences and which are dead ends. Updated after Batch 1 and Batch 2.

Productive:

• φ(n)·c^{ω(n)} family — multiplicative for any constant c; f(p^k) = c·p^{k-1}(p-1); c=2-6 confirmed OEIS-READY
• n·2^{Ω(n)} — "totally doubled" variant of n; f(p^k) = (2p)^k; exact closed form
• Σ_{d|n} d²·φ(n/d) — Dirichlet id²★φ; f(p^k) = p^{2k} + p^{k-1}(p^k - 1); near-quadratic (α≈2)
• Σ_{d|n} φ(d)² — Dirichlet φ²★1; f(p^k) = 1+(p-1)²(p^{2k}-1)/(p²-1); near-quadratic (α≈2)
• Σ_{d|n} gcd(d, n/d) — multiplicative; piecewise formula for even/odd exponents; slow growth (α≈0.2)
• ∏_{p^a ∥ n} φ(p^a + 1) — product of φ of prime-power+1; α≈0.8; OEIS-READY
• Completely additive g(a) family (Σ_{p^a ∥ n} g(a)); prime-signature only; slow growth (α≈0.1) · g(a) = a(a+1)/2 — triangular of exponent; f(p^k) = k(k+1)/2 (same all p) · g(a) = 2^a — exponential exponent; f(p^k) = 2^k (same all p) · g(a) = a² — squared exponent; f(p^k) = k² (same all p)
• Dirichlet convolutions involving μ and φ (when convergent)
• Products over unitary divisors (when α < 3.0)

Dead ends:

• φ ∘ (∏_{d|n} φ(d)) — non-multiplicative, hard to characterize
• gcd(σ(n), φ(n)) — non-multiplicative, already in OEIS
• lcm(σ(n), φ(n)) — non-multiplicative, complex growth
• σ(rad(n)), rad(σ(n)), Σ rad(d) — sympy's rad is symbolic (radians!), not integer radical; implement manually: rad(n) = prod(p for p in factorint(n))
• φ(n)·σ(n)/n — only integer at specific n; not a well-defined integer sequence
• Σ_{d|n, d sqfree} φ(d) = rad(n) — A007947; see §11
• Σ_{d|n, d sqfree} d = σ(rad(n)) = ∏_{p|n}(1+p) — derivable from known; see §11
• ∏_{d unitary|n} (d+1) — α>3.0 (too fast, out of range)

Watch list (experimental, unresolved):

• ∑_{d|n} (d XOR n/d) — non-multiplicative but interesting XOR structure
• ω(n)^{φ(n)} mod n — slow growth, heavily prime-power-dominated
• φ(n)·ω(n) — not multiplicative, but residual f(ab)/(f(a)f(b)) is governed exactly by the harmonic mean of ω(a) and ω(b); might be worth characterizing as a type
• Σ_{d unitary} φ(d) = ∏(1 + φ(p^a)) — clean but α≈2.9, too fast growth
• (μ ★ φ)(n) = φ(n) — Möbius inversion, known, score 55 (weak zones problematic)
• ∏_{d unitary} φ(d) — non-multiplicative, α≈2.9, experimental

§11 · Confirmed Classical Identities (Do Not Resubmit)

These were rediscovered by the pipeline and confirmed as known. Do not submit.

§12 · Parameterized Families

When a sequence is confirmed as part of a parameterized family, note which parameter values are already in OEIS to guide novelty targeting.

Family: φ(n)·c^{ω(n)} for integer c ≥ 1

Multiplicative for any constant c. Prime-power formula: f(p^k) = c·p^{k-1}(p-1).

Family: Σ_{p^a ∥ n} g(a) for various g (completely additive, prime-signature functions)

Family: Σ_{d|n} φ(d)^k for k≥1

Family: Σ_{d|n} d^k·φ(n/d)

When testing a new member of a known family, reduce the novelty check burden by verifying only the first 30 terms against OEIS before full pipeline.

Appendix: Quick Reference

# Key imports
from sympy import (factorint, totient, divisors, divisor_sigma,
                   divisor_count, isprime, gcd, mobius,
                   primeomega, primenu)
import math

# Core functions (all return Python int, not sympy.Integer)
phi   = lambda n: int(totient(n))
tau   = lambda n: int(divisor_count(n))
sigma = lambda n: int(divisor_sigma(n))
omega = lambda n: int(primenu(n))       # distinct prime factors
Omega = lambda n: int(primeomega(n))    # total with multiplicity
mu    = lambda n: int(mobius(n))

# rad(n) - product of distinct prime factors (NOT sympy.rad which is radians!)
def rad(n):
    if n <= 1: return 1
    result = 1
    for p in factorint(n):
        result *= p
    return result

# Unitary divisors
def unitary_divisors(n):
    return [d for d in divisors(n) if gcd(d, n // d) == 1]

# Multiplicativity test
MULT_PAIRS = [(2,3),(2,5),(3,5),(4,9),(2,7),(5,9),(4,25),(8,27),
              (9,25),(16,9),(4,49),(8,125),(27,25),(2,15),(4,21)]

def is_multiplicative(fn):
    for a, b in MULT_PAIRS:
        if gcd(a, b) == 1:
            fa, fb, fab = fn(a), fn(b), fn(a*b)
            if fa * fb != fab:
                return False, (a, b, fa, fb, fab)
    return True, None

# Weak zone test (handles sympy.Integer)
WEAK_ZONES = sorted(set(
    [p**k for p in [2,3,5,7,11,13] for k in range(1,6)] +
    [p**a * q**b for p,q in [(2,3),(2,5),(3,5)] for a,b in [(1,1),(2,1),(1,2)]] +
    [12, 24, 60, 120, 360, 720, 840, 2520] +
    [2**k for k in range(1,20)] + [2**k - 1 for k in range(2,15)] +
    [2*3*5, 2*3*7, 2*5*7, 3*5*7, 2*3*5*7]
))

def test_weak_zones(fn):
    failures = []
    for n in WEAK_ZONES:
        try:
            val = fn(n)
            if hasattr(val, 'is_Integer'):  # sympy.Integer
                val = int(val)
            if not isinstance(val, (int, float)) or (isinstance(val, float) and math.isnan(val)):
                failures.append((n, 'non-numeric'))
        except Exception as e:
            failures.append((n, str(e)))
    return failures

# Dirichlet convolution
def dirichlet(f, g, n):
    return sum(f(d) * g(n // d) for d in divisors(n))

# Prime-power profile
def pp_profile(f, primes=(2,3,5,7,11), kmax=7):
    for p in primes:
        vals = [f(p**k) for k in range(1, kmax+1)]
        ratios = [round(vals[i+1]/vals[i], 5) if vals[i] != 0 else None
                  for i in range(len(vals)-1)]
        print(f"p={p}: {vals}")
        print(f"      ratios: {ratios}")

# Growth exponent (log-space regression)
def growth_alpha(f, lo=10, hi=200):
    pts = [(n, f(n)) for n in range(lo, hi+1) if f(n) > 0]
    if len(pts) < 10:
        return None
    xs = [math.log(n) for n, _ in pts]
    ys = [math.log(v) for _, v in pts]
    xm = sum(xs) / len(xs)
    ym = sum(ys) / len(ys)
    cov = sum((xs[i] - xm) * (ys[i] - ym) for i in range(len(xs)))
    var = sum((x - xm)**2 for x in xs)
    return round(cov / var, 4) if var > 0 else None

# OEIS string format
def oeis_str(f, count=30):
    return ", ".join(str(f(n)) for n in range(1, count + 1))

Appendix 2: Various analysis

Analyze how the sequence behaves for a(n) where n is odd, n is even, n is a perfect square, n=2^k, n=(2^k)-1, n=(2^k)+1 and a(prime(n)).