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Perfect Totient Number

An integer greater than two whose successive Euler totients through one add up exactly to the starting integer.

Version
v1 · 2026-10-25 · History
Domain-specific #
14428
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomain
Number Theory → Mathematics

Core Idea

A perfect totient number is a whole number greater than \(2\) with a special equality.[1] Apply Euler's totient function \(\varphi\) to the number, apply it again to each result, and continue until the value is \(1\).[1] Add the values after the starting number, including that final \(1\).[1] If the sum equals the starting number, it belongs to this class.[1]

For example, \(15\) gives \(8,4,2,1\); their sum is \(15\).[2] Loomis and colleagues found this chain by first dividing \(1,\ldots,15\) into groups according to each number's greatest common divisor with \(15\).[2] That partition helps explain the example, but it is not part of the test for every perfect totient number.

Scope of Application

This is a class of positive integers in number theory, defined using the ordinary Euler totient and an exact sum.[1] It is useful both when checking one candidate and when proving that an entire family belongs. The research papers state and prove that every positive power of \(3\) is a perfect totient number.[1][2]

Clarity

The starting number itself is not one of the terms added. The terminal \(1\) is added. “Perfect” means that this precise total comes back to the start; it does not mean that the integer is a perfect number in the proper-divisor sense. A normal totient chain can miss the target: \(21\) gives \(12,4,2,1\), whose sum is \(19\), not \(21\). This \(21\) calculation follows the standard totient rules rather than a printed example in the papers.[2]

Manages Complexity

For one integer, the task reduces to four steps: generate the complete totient chain, omit the starting value, sum the remaining values through \(1\), and compare with the start. A successful calculation proves membership for that integer. A theorem about all numbers of a form needs a separate argument; checking several values is not enough. The published theorem for powers of \(3\) provides that broader warrant.[2]

Abstract Reasoning

At \(27=3^3\), the calculated chain after the start is \(18,6,2,1\). Since \(18+6+2+1=27\), the exact-return test succeeds. The papers prove the wider power-of-three family; this particular full orbit is our independently reproduced calculation from the standard totient rules.[1][2]

For a candidate such as \(21\), inequality at the final comparison proves nonmembership. Stopping at an intermediate value would leave the calculation incomplete. This distinction matters when a large candidate is known only through a partial computation.

Knowledge Transfer

The same test works whether a candidate was discovered through a gcd partition, a theorem about prime powers, or direct computation. What transfers within number theory is the orbit and exact-return rule, not the special way the number was found. Replacing Euler's totient with another recurrence would define another class, even if it had a similar looking equality.

Examples

Gcd-class discovery at \(15\). The source works through four disjoint groups of \(1,\ldots,15\) with sizes \(8,4,2,1\), identifies those sizes with the successive totients from \(15\), and shows that they sum to \(15\).[2] Mapped back: input \(15\) → totient orbit \(8,4,2,1\) → sum \(15\) → membership.

Prime-power family at \(27\). A source-proved result covers all positive powers of \(3\); \(27\) illustrates it with the independently calculated orbit \(18,6,2,1\) and sum \(27\).[1][2] Mapped back: input \(27\) → the same totient rule → sum \(27\) → membership. The two settings use the same mathematical test, but one source works through a gcd partition while the other proves an algebraic family.

Relationships to Other Abstractions

Local relationship map for Perfect Totient NumberParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.PerfectTotient NumberDOMAINDomain-specific abstraction: Euler's totient function — presupposesEuler's totientfunctionDOMAIN

Current abstraction Perfect Totient Number Domain-specific

Parents (1) — outgoing typed relations

  • Perfect Totient Number presupposes Euler's totient function Domain-specific

    Perfect-totient membership requires Euler's totient to generate every post-start value; the function can exist without this number class.

Hierarchy paths (5) — routes to 3 parentless roots

Neighborhood in Abstraction Space

Perfect Totient Number sits in a sparse region of the domain-specific corpus (81st percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2769 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25

Not to Be Confused With

A perfect number is selected by a sum of proper divisors. A highly totient number counts preimages of a totient value and compares that count with earlier values. A sparsely totient number compares a chosen input's totient with those of every larger input. The direct graph relation here is a strict presupposition of Euler's Totient Function: without that function there is no named orbit test, while the function can be applied to integers that fail the test.

References

[1] Douglas E. Iannucci, Deng Moujie, and Graeme L. Cohen, “On Perfect Totient Numbers,” Journal of Integer Sequences 6 (2003), Article 03.4.5, abstract, §1 and Table 1 (PDF pp. 1–2). registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h

[2] Paul Loomis, Michael Plytage, and John Polhill, “Summing Up the Euler Phi Function,” The College Mathematics Journal 39, no. 1 (2008), 34–42, especially pp. 34–37, 39. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h