OFFSET
2,3
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
LINKS
Reinhard Zumkeller, Rows n = 2..1000 of table, flattened
Eric Weisstein's World of Mathematics, Prime Signature
OEIS Wiki, Prime signatures
OEIS Wiki, Ordered prime signatures
EXAMPLE
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.
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 *)
PROG
(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
(PARI) A118914_row(n)=vecsort(factor(n)[, 2]~) \\ M. F. Hasler, Oct 12 2018
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Eric W. Weisstein, May 05 2006
EXTENSIONS
Corrected and edited by Daniel Forgues, Dec 22 2010
STATUS
approved