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A212171
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Prime signature of n (nonincreasing version): row n of table lists positive exponents in canonical prime factorization of n, in nonincreasing order.
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17
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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, 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, 1, 1, 1, 1, 4, 1, 2, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 3, 1
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OFFSET
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2,3
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COMMENTS
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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.
(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.)
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).
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.
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
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LINKS
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FORMULA
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Row n of A124010 for n > 1, with exponents sorted in nonincreasing order. Equivalently, row A046523(n) of A124010 for n > 1.
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EXAMPLE
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First rows of table read:
1;
1;
2;
1;
1,1;
1;
3;
2;
1,1;
1;
2,1;
...
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}.
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}.
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PROG
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(Magma) &cat[Reverse(Sort([pe[2]:pe in Factorisation(n)])):n in[1..76]]; // Jason Kimberley, Jun 13 2012
(PARI) apply( {A212171_row(n)=vecsort(factor(n)[, 2]~, , 4)}, [1..40])\\ M. F. Hasler, Apr 19 2022
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CROSSREFS
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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:
Other: A001055, A008480, A010553, A038548, A050320, A051707, A071625, A074206, A076078, A085082, A088873.
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.
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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STATUS
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approved
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