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%I #36 Oct 12 2018 22:42:21
%S 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,
%T 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,
%U 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
%N Table of the prime signatures (sorted lists of exponents of distinct prime factors) of the positive integers.
%C 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)
%C MathWorld wrongly defines the prime signature of 1 as {1}, which is actually the prime signature of primes.
%C The sequences A025487, A036035, A046523 consider the prime signatures of 1 and 2 to be distinct, implying { } for 1 and {1} for 2.
%C 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).
%C 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
%H Reinhard Zumkeller, <a href="/A118914/b118914.txt">Rows n = 2..1000 of table, flattened</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeSignature.html">Prime Signature</a>
%H OEIS Wiki, <a href="/wiki/Prime_signatures">Prime signatures</a>
%H OEIS Wiki, <a href="/wiki/Ordered_prime_signatures">Ordered prime signatures</a>
%e The table starts:
%e n : prime signature of n (factorization of n)
%e 1 : {}, (empty product)
%e 2 : {1}, (2^1)
%e 3 : {1}, (3^1)
%e 4 : {2}, (2^2)
%e 5 : {1}, (5^1)
%e 6 : {1, 1}, (2^1 * 3^1)
%e 7 : {1}, (5^1)
%e 8 : {3}, (2^3)
%e 9 : {2}, (3^2)
%e 10 : {1, 1}, (2^1 * 5^1)
%e 11 : {1}, (11^1)
%e 12 : {1, 2}, (2^2 * 3^1, but exponents are sorted increasingly)
%e etc.
%t primeSignature[n_] := Sort[ FactorInteger[n] , #1[[2]] < #2[[2]]&][[All, 2]]; Flatten[ Table[ primeSignature[n], {n, 2, 65}]](* _Jean-François Alcover_, Nov 16 2011 *)
%o (Haskell)
%o import Data.List (sort)
%o a118914 n k = a118914_tabf !! (n-2) !! (k-1)
%o a118914_row n = a118914_tabf !! (n-2)
%o a118914_tabf = map sort $ tail a124010_tabf
%o -- _Reinhard Zumkeller_, Mar 23 2014
%o (PARI) A118914_row(n)=vecsort(factor(n)[,2]~) \\ _M. F. Hasler_, Oct 12 2018
%Y Cf. A025487, A036035, A046523, A095904.
%Y Cf. A124010.
%Y Cf. A001221 (row lengths), A001222 (row sums).
%K nonn,tabf
%O 2,3
%A _Eric W. Weisstein_, May 05 2006
%E Corrected and edited by _Daniel Forgues_, Dec 22 2010
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