A118914 Table of the prime signatures (sorted lists of exponents of distinct prime factors) of the positive integers.

1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 4, 2, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 6, 1, 1

Offset: 2

Author

Eric W. Weisstein, May 05 2006

Keywords

• nonn
• tabf

Comments

Since the prime factorization of 1 is the empty product (i.e., the multiplicative identity, 1), it follows that the prime signature of 1 is the empty multiset { }. (Cf. http://oeis.org/wiki/Prime_signature)

MathWorld wrongly defines the prime signature of 1 as {1}, which is actually the prime signature of primes.

The sequences A025487, A036035, A046523 consider the prime signatures of 1 and 2 to be distinct, implying { } for 1 and {1} for 2.

Since the prime signature of n is a partition of Omega(n), also true for Omega(1) = 0, the order of exponents is only a matter of convention (using reverse sorted lists of exponents would create a different sequence).

Here the multisets of nonzero exponents are sorted in increasing order; it is slightly more common to order them, as the parts of partitions, in decreasing order. This yields A212171. - M. F. Hasler, Oct 12 2018

Examples

			The table starts:
  n : prime signature of n  (factorization of n)
  1 : {},                   (empty product)
  2 : {1},                  (2^1)
  3 : {1},                  (3^1)
  4 : {2},                  (2^2)
  5 : {1},                  (5^1)
  6 : {1, 1},               (2^1 * 3^1)
  7 : {1},                  (5^1)
  8 : {3},                  (2^3)
  9 : {2},                  (3^2)
  10 : {1, 1},              (2^1 * 5^1)
  11 : {1},                 (11^1)
  12 : {1, 2},              (2^2 * 3^1, but exponents are sorted increasingly)
  etc.
		

Links

• Reinhard Zumkeller, Rows n = 2..1000 of table, flattened
• Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 73.
• Eric Weisstein's World of Mathematics, Prime Signature
• OEIS Wiki, Prime signatures
• OEIS Wiki, Ordered prime signatures

Crossrefs

Cf. A025487, A036035, A046523, A095904.

Cf. A124010.

Cf. A001221 (row lengths), A001222 (row sums).

Programs

Haskell

import Data.List (sort)
a118914 n k = a118914_tabf !! (n-2) !! (k-1)
a118914_row n = a118914_tabf !! (n-2)
a118914_tabf = map sort $ tail a124010_tabf
-- Reinhard Zumkeller, Mar 23 2014

Mathematica

primeSignature[n_] := Sort[ FactorInteger[n] , #1[[2]] < #2[[2]]&][[All, 2]]; Flatten[ Table[ primeSignature[n], {n, 2, 65}]](* Jean-François Alcover, Nov 16 2011 *)

PARI

A118914_row(n)=vecsort(factor(n)[,2]~) \\ M. F. Hasler, Oct 12 2018

Extensions

Corrected and edited by Daniel Forgues, Dec 22 2010