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A212180 Number of distinct second signatures (cf. A212172) represented among divisors of n.

%I A212180 #29 May 04 2020 15:05:20
%S A212180 1,1,1,2,1,1,1,3,2,1,1,2,1,1,1,4,1,2,1,2,1,1,1,3,2,1,3,2,1,1,1,5,1,1,
%T A212180 1,3,1,1,1,3,1,1,1,2,2,1,1,4,2,2,1,2,1,3,1,3,1,1,1,2,1,1,2,6,1,1,1,2,
%U A212180 1,1,1,5,1,1,2,2,1,1,1,4,4,1,1,2,1,1,1,3
%N A212180 Number of distinct second signatures (cf. A212172) represented among divisors of n.
%C A212180 Completely determined by the exponents >=2 in the prime factorization of n (cf. A212172, A212173).
%C A212180 The fraction of the divisors of n which have a given second signature {S} is also a function of n's second signature. For example, if n has second signature {3,2}, it follows that 1/3 of n's divisors are squarefree. Squarefree numbers are represented with 0's in A212172, in accord with the usual OEIS custom of using 0 for nonexistent elements; in comments, their second signature is represented as { }.
%H A212180 Antti Karttunen, <a href="/A212180/b212180.txt">Table of n, a(n) for n = 1..10000</a>
%H A212180 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%e A212180 The divisors of 72 represent a total of 5 distinct second signatures (cf. A212172), as can be seen from the exponents >= 2, if any, in the canonical prime factorization of each divisor:
%e A212180 { }: 1, 2 (prime), 3 (prime), 6 (2*3)
%e A212180 {2}: 4 (2^2), 9 (3^2), 12 (2^2*3), 18 (2*3^2)
%e A212180 {3}: 8 (2^3), 24 (2^3*3)
%e A212180 {2,2}: 36 (2^2*3^2)
%e A212180 {3,2}: 72 (2^3*3^2)
%e A212180 Hence, a(72) = 5.
%t A212180 Array[Length@ Union@ Map[Sort@ Select[FactorInteger[#][[All, -1]], # >= 2 &] &, Divisors@ #] &, 88] (* _Michael De Vlieger_, Jul 19 2017 *)
%o A212180 (PARI)
%o A212180 A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ This function from _Charles R Greathouse IV_, Aug 17 2011
%o A212180 A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); } \\ This function from _Charles R Greathouse IV_, Aug 13 2013
%o A212180 A212173(n) = A046523(A057521(n));
%o A212180 A212180(n) = { my(vals = Set()); fordiv(n, d, vals = Set(concat(vals, A212173(d)))); length(vals); }; \\ _Antti Karttunen_, Jul 19 2017
%o A212180 (Python)
%o A212180 from sympy import factorint, divisors, prod
%o A212180 def P(n): return sorted(factorint(n).values())
%o A212180 def a046523(n):
%o A212180     x=1
%o A212180     while True:
%o A212180         if P(n)==P(x): return x
%o A212180         else: x+=1
%o A212180 def a057521(n): return 1 if n==1 else prod(p**e for p, e in factorint(n).items() if e != 1)
%o A212180 def a212173(n): return a046523(a057521(n))
%o A212180 def a(n):
%o A212180     l=[]
%o A212180     for d in divisors(n):
%o A212180         x=a212173(d)
%o A212180         if not x in l:l+=[x, ]
%o A212180     return len(l)
%o A212180 print([a(n) for n in range(1, 51)]) # _Indranil Ghosh_, Jul 19 2017
%Y A212180 Cf. A212172, A085082, A088873, A181796, A182860, A212173, A212642, A212643, A212644.
%K A212180 nonn
%O A212180 1,4
%A A212180 _Matthew Vandermast_, Jun 04 2012