A212172 - OEIS

%I #21 Sep 08 2022 08:46:02

%S 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,

%T 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,

%U 2,0,0,0,3,2,0,0,2,2,0,0,0,4,4,0,0,2,0,0

%N 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.

%C Length of row n equals A056170(n) if A056170(n) is positive, or 1 if A056170(n) = 0.

%C 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).

%C 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 { }.

%C 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.

%C 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} = { }).

%H Jason Kimberley, <a href="/A212172/b212172.txt">Table of i, a(i) for i = 1..10575 (n = 1..10000)</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H Primefan, <a href="http://primefan.tripod.com/500factored.html">The First 2500 Integers Factored</a> (1st of 5 pages)

%F For nonsquarefree n, row n is identical to row A057521(n) of table A212171.

%e First rows of table read: 0; 0; 0; 2; 0; 0; 0; 3; 2; 0; 0; 2;...

%e 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}.

%e 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.

%e 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}.

%t row[n_] := Select[ FactorInteger[n][[All, 2]], # >= 2 &] /. {} -> 0 /. {k__} -> Sequence[k]; Table[row[n], {n, 1, 100}] (* _Jean-François Alcover_, Apr 16 2013 *)

%o (Magma) &cat[IsEmpty(e)select [0]else Reverse(Sort(e))where e is[pe[2]:pe in Factorisation(n)|pe[2]gt 1]:n in[1..102]]; // _Jason Kimberley_, Jun 13 2012

%Y A181800 gives first integer of each second signature. Also see A212171, A212173-A212181, A212642-A212644.

%Y Functions determined by exponents >=2 in the prime factorization of n:

%Y Multiplicative: A000688, A005361, A008966, A038538, A046951, A049419, A050361, A050377, A056624, A061704, A063775, A162510, A162511, A212181.

%Y Additive: A046660, A056170.

%Y Other: A007424, A051903 (for n > 1), A056626, A066301, A071325, A072411, A091050, A107078, A185102 (for n > 1), A212180.

%Y Sequences that contain all integers of a specific second signature: A005117 (second signature { }), A060687 ({2}), A048109 ({3}).

%K nonn,easy,tabf

%O 1,4

%A _Matthew Vandermast_, Jun 03 2012