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The abstract prime factorization (APF) or prime signature of a positive integer is the list of exponents in its prime factorization, conventionally written in descending order.  The specific prime factors are ignored.

Examples Edit

  • The number *88 (decimal 104) has the prime factorization 23 × *111, and so an APF of [3, 1].
  • The number *90 (decimal 108) has the prime factorization 22 × 33, and so an APF of [3, 2].

Numbers grouped listed by APF Edit

All figures in dozenal. There are an infinite number of APF equivalence classes; the list below shows only those that include numbers less than or equal to *50.

APF Numbers OEIS ID Description
[] 1 N/A a finite class consisting only of the number 1
[1] 2, 3, 5, 7, E, 11, 15, ... A000040 prime numbers
[2] 4, 9, 21, 41, X1, 121, ... A001248 squares of primes
[1, 1] 6, X, 12, 13, 19, 1X, ... A006881 square-free semiprimes
[3] 8, 23, X5, 247, 92E, ... A030078 cubes of primes
[2, 1] 10, 16, 18, 24, 38, 39, ... A054753 squares of primes times a different prime
[4] 14, 69, 441, 1481, ... A030514 fourth powers of primes
[3, 1] 20, 34, 46, 48, 74, 88, ... A065036 cubes of primes times a different prime
[1, 1, 1] 26, 36, 56, 5X, 66, 86, ... A007304 products of three distinct primes (sphenic numbers)
[5] 28, 183, 1985, 9887, ... A050997 fifth powers of primes
[2, 2] 30, 84, 144, 169, 309, ... A085986 squares of squarefree semiprimes
[4, 1] 40, 68, 94, 116, 128, ... TODO fourth powers of primes times a different prime
[2, 1, 1] 50, 70, 76, X6, E0, E8, ... TODO square of a prime times two other distinct primes

OEIS Sequences Edit

  • A046523 = Smallest number with same prime signature as n.
  • A046523 = Least integer of each prime signature. This list includes all highly composite numbers, and also all primorials.

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