Recreational Mathematics · Two-Way Number Relationships

Amicable Numbers Generator
Divisor Sums · Powers · Products

Created by Inder J. Taneja

Compare the classical proper-divisor sum with two digit-split constructions using powers or products. In every mode, check whether one number leads to another and back.

Three ways to form a two-way relationship

Classical: N → M = s(N) → N = s(M)

s(N) is the sum of positive proper divisors. For example, s(220) = 284 and s(284) = 220.

Power: N = a|b → M = a^p ± b^p → N = ±d^p ± e^p

Example: 240 = 2|40 → 2² + 40² = 1604 → 16² − 04² = 240.

Product: N = a|b → M = k(a ± b) → N = k(d ± e)

Example: 150 = 1|50 → 5(1 + 50) = 255 → 5(25 + 5) = 150.

Complete original 300625 example

300625 = 300|625 → 300² − 625² = −300625

−300625 = (−300)|625 → −(−300)² + 625² = 300625

Here 300² = 90,000 and 625² = 390,625, so both steps are exact. The parentheses around −300 make the second step unambiguous.

Here a|b means a cut between digits, not division. The second cut is M = d|e. Classical amicable numbers use proper-divisor sums; the Power and Product modes apply the two-way idea to digit blocks. The intermediate M may be negative; its minus sign belongs to its first block.

Choose a search

Modes

Classical divisor sums Power Product Include N → N self-cycles

From NTo NMinimum exponent p / factor kMaximum exponent p / factor kForward operationReturn operationMaximum solutions

The exponent/factor and operation controls apply only to Power and Product. Searches are limited to 10,000 starting numbers per run. All found identities are retained for download; the screen shows the first 200. The Stop button ends after the current starting number.

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References

This work is based on the author's research:

Inder J. Taneja, Amicable Numbers with Patterns in Products and Powers, RGMIA Research Report Collection, vol. 20, art. 156, 2017, pp. 1–25, rgmia.org/papers/v20/v20a156.pdf.
Inder J. Taneja, Amicable Numbers with Patterns in Products and Powers, Zenodo, March 05, 2019, pp. 1–25, doi.org/10.5281/zenodo.2583306.
Inder J. Taneja, Product and Power Type Amicable Numbers, Numbers Magic, December 19, 2017, inderjtaneja.wordpress.com/2017/12/19/product-and-power-type-amicable-numbers/.