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%I A212171 #30 Sep 08 2022 08:46:02
%S A212171 1,1,2,1,1,1,1,3,2,1,1,1,2,1,1,1,1,1,1,4,1,2,1,1,2,1,1,1,1,1,1,3,1,2,
%T A212171 1,1,3,2,1,1,1,1,1,1,5,1,1,1,1,1,1,2,2,1,1,1,1,1,3,1,1,1,1,1,1,2,1,2,
%U A212171 1,1,1,1,4,1,2,2,1,1,1,2,1,1,3,1,1,1,3,1
%N A212171 Prime signature of n (nonincreasing version): row n of table lists positive exponents in canonical prime factorization of n, in nonincreasing order.
%C A212171 Length of row n equals A001221(n).
%C A212171 The multiset of positive exponents in n's prime factorization completely determines a(n) for a host of OEIS sequences, including several "core" sequences.  Of those not cross-referenced here or in A212172, many can be found by searching the database for A025487.
%C A212171 (Note: Differing opinions may exist about whether the prime signature of n should be defined as this multiset itself, or as a symbol or collection of symbols that identify or "signify" this multiset. The definition of this sequence is designed to be compatible with either view, as are the original comments. When n >= 2, the customary ways to signify the multiset of exponents in n's prime factorization are to list the constituent exponents in either nonincreasing or nondecreasing order; this table gives the nonincreasing version.)
%C A212171 Table lists exponents in the order in which they appear in the prime factorization of a member of A025487.  This ordering is common in database comments (e.g., A008966).
%C A212171 Each possible multiset of an integer's positive prime factorization exponents corresponds to a unique partition that contains the same elements (cf. A000041). This includes the multiset of 1's positive exponents, { } (the empty multiset), which corresponds to the partition of 0.
%C A212171 Differs from A124010 from a(23) on, corresponding to the factorization of 18 = 2^1*3^2 which is here listed as row 18 = [2, 1], but as [1, 2] (in the order of the prime factors) in A124010 and also in A118914 which lists the prime signatures in nondecreasing order (so that row 12 = 2^2*3^1 is also [1, 2]). - _M. F. Hasler_, Apr 08 2022
%H A212171 Jason Kimberley, <a href="/A212171/b212171.txt">Table of i, a(i) for i = 2..24301 (n = 2..10000)</a>
%F A212171 Row n of A118914, reversed.
%F A212171 Row n of A124010 for n > 1, with exponents sorted in nonincreasing order. Equivalently, row A046523(n) of A124010 for n > 1.
%e A212171 First rows of table read:
%e A212171   1;
%e A212171   1;
%e A212171   2;
%e A212171   1;
%e A212171   1,1;
%e A212171   1;
%e A212171   3;
%e A212171   2;
%e A212171   1,1;
%e A212171   1;
%e A212171   2,1;
%e A212171   ...
%e A212171 The multiset of positive exponents in the prime factorization of 6 = 2*3 is {1,1} (1s are often left implicit as exponents). The prime signature of 6 is therefore {1,1}.
%e A212171 12 = 2^2*3 has positive exponents 2 and 1 in its prime factorization, as does 18 = 2*3^2. Rows 12 and 18 of the table both read {2,1}.
%o A212171 (Magma) &cat[Reverse(Sort([pe[2]:pe in Factorisation(n)])):n in[1..76]]; // _Jason Kimberley_, Jun 13 2012
%o A212171 (PARI) apply( {A212171_row(n)=vecsort(factor(n)[,2]~,,4)}, [1..40])\\ _M. F. Hasler_, Apr 19 2022
%Y A212171 Cf. A025487, A001221 (row lengths), A001222 (row sums). A118914 gives the nondecreasing version. A124010 lists exponents in n's prime factorization in natural order, with A124010(1) = 0.
%Y A212171 A212172 cross-references over 20 sequences that depend solely on n's prime exponents >= 2, including the "core" sequence A000688.  Other sequences determined by the exponents in the prime factorization of n include:
%Y A212171 Multiplicative: A000005, A007425, A008683, A008836, A034444, A037445, A181819.
%Y A212171 Additive: A001221, A001222, A056169.
%Y A212171 Other: A001055, A008480, A010553, A038548, A050320, A051707, A071625, A074206, A076078, A085082, A088873.
%Y A212171 A highly incomplete selection of sequences, each definable by the set of prime signatures possessed by its members: A000040, A000290, A000578, A000583, A000961, A001248, A001358, A001597, A001694, A002808, A004709, A005117, A006881, A013929, A030059, A030229, A052486.
%K A212171 nonn,easy,tabf
%O A212171 2,3
%A A212171 _Matthew Vandermast_, Jun 03 2012

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