The OEIS is supported by the many generous donors to the OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A212172 Row n of table represents second signature of n: list of exponents >= 2 in canonical prime factorization of n, in nonincreasing order, or 0 if no such exponent exists. 24
 0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 3, 2, 0, 3, 2, 0, 0, 0, 5, 0, 0, 0, 2, 2, 0, 0, 0, 3, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 3, 0, 3, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Length of row n equals A056170(n) if A056170(n) is positive, or 1 if A056170(n) = 0. The multiset of exponents >=2 in the prime factorization of n completely determines a(n) for over 20 sequences in the database (see crossreferences). It also determines the fractions A034444(n)/A000005(n) and A037445(n)/A000005(n). For squarefree numbers, this multiset is { } (the empty multiset). The use of 0 in the table to represent each n with no exponents >=2 in its prime factorization accords with the usual OEIS practice of using 0 to represent nonexistent elements when possible. In comments, the second signature of squarefree numbers will be represented as { }. For each second signature {S}, there exist values of j and k such that, if the second signature of n is {S}, then A085082(n) is congruent to j modulo k.  These values are nontrivial unless {S} = { }. Analogous (but not necessarily identical) values of j and k also exist for each second signature with respect to A088873 and A181796. Each sequence of integers with a given second signature {S} has a positive density, unlike the analogous sequences for prime signatures. The highest of these densities is 6/Pi^2 = 0.607927... for A005117 ({S} = { }). LINKS Jason Kimberley, Table of i, a(i) for i = 1..10575 (n = 1..10000) M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Primefan, The First 2500 Integers Factored (1st of 5 pages) FORMULA For nonsquarefree n, row n is identical to row A057521(n) of table A212171. EXAMPLE First rows of table read: 0; 0; 0; 2; 0; 0; 0; 3; 2; 0; 0; 2;... 12 = 2^2*3 has positive exponents 2 and 1 in its canonical prime factorization (1s are often left implicit as exponents). Since only exponents that are 2 or greater appear in a number's second signature, 12's second signature is {2}. 30 = 2*3*5 has no exponents greater than 1 in its prime factorization. The multiset of its exponents >= 2 is { } (the empty multiset), represented in the table with a 0. 72 = 2^3*3^2 has positive exponents 3 and 2 in its prime factorization, as does 108 = 2^2*3^3. Rows 72 and 108 both read {3,2}. MATHEMATICA row[n_] := Select[ FactorInteger[n][[All, 2]], # >= 2 &] /. {} -> 0 /. {k__} -> Sequence[k]; Table[row[n], {n, 1, 100}] (* Jean-François Alcover, Apr 16 2013 *) PROG (Magma) &cat[IsEmpty(e)select else Reverse(Sort(e))where e is[pe:pe in Factorisation(n)|pegt 1]:n in[1..102]]; // Jason Kimberley, Jun 13 2012 CROSSREFS A181800 gives first integer of each second signature. Also see A212171, A212173-A212181, A212642-A212644. Functions determined by exponents >=2 in the prime factorization of n: Multiplicative: A000688, A005361, A008966, A038538, A046951, A049419, A050361, A050377, A056624, A061704, A063775, A162510, A162511, A212181. Additive: A046660, A056170. Other: A007424, A051903 (for n > 1), A056626, A066301, A071325, A072411, A091050, A107078, A185102 (for n > 1), A212180. Sequences that contain all integers of a specific second signature: A005117 (second signature { }), A060687 ({2}), A048109 ({3}). Sequence in context: A132349 A216226 A123391 * A275812 A280683 A171871 Adjacent sequences:  A212169 A212170 A212171 * A212173 A212174 A212175 KEYWORD nonn,easy,tabf AUTHOR Matthew Vandermast, Jun 03 2012 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 5 14:35 EDT 2022. Contains 357258 sequences. (Running on oeis4.)