Prime signature

<h2 id="properties">Properties</h2>
<p>The <a href="/facts/Divisor_function/rzvm1jw2">divisor function</a> τ(<i>n</i>), the <a href="/facts/M%25C3%25B6bius_function/geuvVfqS">Möbius function</a> <i>μ</i>(<i>n</i>), the number of distinct prime divisors ω(<i>n</i>) of <i>n</i>, the number of prime divisors Ω(<i>n</i>) of <i>n</i>, the <a href="/facts/Indicator_function/QEuh04NM">indicator function</a> of the <a href="/facts/Squarefree_integer/R3qR9hP1">squarefree integers</a>, and many other important functions in number theory, are functions of the prime signature of <i>n</i>.
</p><p>In particular, τ(<i>n</i>) equals the product of the incremented by 1 exponents from the prime signature of <i>n</i>. 
For example, 20 has prime signature {2,1} and so the number of divisors is (2+1) × (1+1) = 6. Indeed, there are six divisors: 1, 2, 4, 5, 10 and 20.
</p><p>The smallest number of each prime signature is a product of <a href="/facts/Primorial/NqlQBLDp">primorials</a>. The first few are:
</p>
1, 2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, 180, 192, 210, 216, ... (sequence A025487 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>).
<p>A number cannot divide another unless its prime signature is included in the other numbers prime signature in the <a href="/facts/Young%2527s_lattice/X90jpt0r">Young's lattice</a>.
</p>
<h2 id="numbers-with-same-prime-signature">Numbers with same prime signature</h2>
<table><tbody><tr><th>Signature</th><th>Numbers</th><th><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a> ID</th><th>Description</th></tr><tr><td>∅</td><td>1</td><td></td><td>The number 1, as an <a href="/facts/Empty_product/mj7nhGeD">empty product</a> of primes</td></tr><tr><td>{1}</td><td>2, 3, 5, 7, 11, ...</td><td>A000040</td><td>prime numbers</td></tr><tr><td>{2}</td><td>4, 9, 25, 49, 121, ...</td><td>A001248</td><td><a href="/facts/Square_number/jfakASPK">squares</a> of prime numbers</td></tr><tr><td>{1, 1}</td><td>6, 10, 14, 15, 21, ...</td><td>A006881</td><td>two distinct prime divisors (<a href="/facts/Square-free_integer/R3qR9hP1">square-free</a> <a href="/facts/Semiprime/J5cNFs7T">semiprimes</a>)</td></tr><tr><td>{3}</td><td>8, 27, 125, 343, ...</td><td>A030078</td><td><a href="/facts/Cube_(algebra)/SqmlEEog">cubes</a> of prime numbers</td></tr><tr><td>{2, 1}</td><td>12, 18, 20, 28, ...</td><td>A054753</td><td>squares of primes times another prime</td></tr><tr><td>{4}</td><td>16, 81, 625, 2401, ...</td><td>A030514</td><td>fourth powers of prime numbers</td></tr><tr><td>{3, 1}</td><td>24, 40, 54, 56, ...</td><td>A065036</td><td>cubes of primes times another prime</td></tr><tr><td>{1, 1, 1}</td><td>30, 42, 66, 70, ...</td><td>A007304</td><td>three distinct prime divisors (<a href="/facts/Sphenic_number/XsNyg0Jy">sphenic numbers</a>)</td></tr><tr><td>{5}</td><td>32, 243, 3125, ...</td><td>A050997</td><td>fifth powers of primes</td></tr><tr><td>{2, 2}</td><td>36, 100, 196, 225, ...</td><td>A085986</td><td>squares of square-free semiprimes</td></tr></tbody></table>
<h2 id="sequences-defined-by-their-prime-signature">Sequences defined by their prime signature</h2>
<p>Given a number with prime signature <i>S</i>, it is
</p>
<ul><li>A <a href="/facts/Prime_number/L20FLkgn">prime number</a> if <i>S</i> = {1},</li>
<li>A <a href="/facts/Square_(algebra)/Jzpq0R0z">square</a> if <a href="/facts/Greatest_common_divisor/kNFvBuKl">gcd</a>(<i>S</i>) is <a href="/facts/Parity_(mathematics)/7zq2UM4V">even</a>,</li>
<li>A <a href="/facts/Cube_(algebra)/SqmlEEog">cube</a> if gcd(<i>S</i>) is divisible by 3,</li>
<li>A <a href="/facts/Square-free_integer/R3qR9hP1">square-free integer</a> if max(<i>S</i>) = 1,</li>
<li>A <a href="/facts/Cube-free_integer/R3qR9hP1">cube-free integer</a> if max(<i>S</i>) ≤ 2,</li>
<li>A <a href="/facts/Powerful_number/uhIIrxcR">powerful number</a> if min(<i>S</i>) ≥ 2,</li>
<li>A <a href="/facts/Perfect_power/HSzSQBvb">perfect power</a> if gcd(<i>S</i>) > 1,</li>
<li>A <i>k</i>-<a href="/facts/Almost_prime/REQl0tuz">almost prime</a> if sum(<i>S</i>) = <i>k</i>, or</li>
<li>An <a href="/facts/Achilles_number/WIDTnft5">Achilles number</a> if min(<i>S</i>) ≥ 2 and gcd(<i>S</i>) = 1.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Canonical_representation_of_a_positive_integer/EzHUwlXX">Canonical representation of a positive integer</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="https://willnicholes.com/2010/06/06/list-of-prime-signatures/">List of the first 400 prime signatures</a></li>
<li><a href="https://willnicholes.com/2010/06/06/iterative-mapping-of-prime-signatures/">Iterative Mapping of Prime Signatures</a></li></ul>

Prime signature open-in-new

Prime signature