Sum-product number¶
Sum-product number denotes a number equal to the product of the sum and product of its digits within recreational number theory.
Core Idea¶
A sum-product number in base b is a natural number equal to the product of the sum of its base-b digits and the product of those digits. If Sb(n) is the digit sum and Pb(n) the digit product, the defining fixed-point equation is n = Sb(n)Pb(n). The property is representation-dependent: changing the base changes the digits, the function, and generally the membership of n. In decimal notation, 144 is a nontrivial example because (1+4+4)(1×4×4)=9×16=144.
Scope of Application¶
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Fixed-point enumeration. Candidates are generated and checked exactly under one base and numeral convention.
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Finite-state dynamics. Growth bounds force trajectories into a bounded region where cycles and basins can be exhaustively studied.
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Sociable cycles. Longer periodic orbits extend the fixed-point question under the same map.
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Harshad relations. Divisibility by the digit sum is necessary for positive fixed points but not sufficient.
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Base comparison. The same integer can qualify in one base and fail in another because its digits change.
Clarity¶
Sum-product number denotes a fixed point of the base-dependent map that multiplies a number's digit sum by its digit product. It is distinct from numbers equal only to a digit sum or product and from longer periodic orbits under iteration. Because zero digits force the product to zero and changing base changes every digit, the base and treatment of zero are structural, not incidental.
Manages Complexity¶
Sum-product numbers compress a search over integers into the digit sum, digit product, base, and a fixed-point equation. Zero digits eliminate most candidates immediately; bounds on the maximum digit product and sum eventually dominate positional growth, limiting the number of digits that must be checked. Fixed points, transients, and longer cycles form branches of the same arithmetic dynamical system.
Abstract Reasoning¶
Representation move. From a number's digits in a fixed base, compute both digit sum and digit product and test the defining equality with the number. Constraint move. Use digit bounds, zero behavior, and place-value growth to restrict possible digit lengths and compositions. Construction move. Search by digit multisets or recurrences rather than blindly enumerating all integers, while restoring positional distinctions where they matter. Base-change move. Re-evaluate the property after changing radix because both operations are representation-dependent. Boundary move.
Knowledge Transfer¶
Within the home domain. Sum-product numbers transfer across recreational mathematics, digit-function dynamics, and radix-dependent enumeration when an integer equals the sum plus product of its digits under a specified base and convention. Digit length, zeros, place value, and search bounds retain formal force. Beyond the home domain (C — formal class). The definition applies literally to any radix, but its members generally change with representation. Its boundary is semantic: the term does not describe a number closed under sum and product, an intrinsic algebraic property, or a model of combined processes. Numerical pattern alone carries no cross-domain mechanism.
Relationships to Other Abstractions¶
Current abstraction Sum-product number Domain-specific
Parents (1) — more general patterns this builds on
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Sum-product number is a kind of Classification Prime
Sum-product number is a domain-specific kind of Classification: Sum-product number denotes a number equal to the product of the sum and product of its digits within recreational number theory.
Hierarchy path (1) — routes to 1 parentless root
- Sum-product number → Classification
Neighborhood in Abstraction Space¶
Sum-product number sits in a moderately populated region (48th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Numeral Systems & Digit Algorithms (11 abstractions)
Nearest neighbors
- Perfect digit-to-digit invariant — 0.94
- Multiplicative Digital Root — 0.88
- Cyclic Number — 0.86
- Average Order of an Arithmetic Function — 0.85
- Signedness — 0.85
Computed from structural-signature embeddings · 2026-10-08