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A212180
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Number of distinct second signatures (cf. A212172) represented among divisors of n.
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6
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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, 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, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 3
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OFFSET
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1,4
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COMMENTS
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Completely determined by the exponents >=2 in the prime factorization of n (cf. A212172, A212173).
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 { }.
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LINKS
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Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n
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EXAMPLE
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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:
{ }: 1, 2 (prime), 3 (prime), 6 (2*3)
{2}: 4 (2^2), 9 (3^2), 12 (2^2*3), 18 (2*3^2)
{3}: 8 (2^3), 24 (2^3*3)
{2,2}: 36 (2^2*3^2)
{3,2}: 72 (2^3*3^2)
Hence, a(72) = 5.
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MATHEMATICA
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Array[Length@ Union@ Map[Sort@ Select[FactorInteger[#][[All, -1]], # >= 2 &] &, Divisors@ #] &, 88] (* Michael De Vlieger, Jul 19 2017 *)
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PROG
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(PARI)
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
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
A212173(n) = A046523(A057521(n));
A212180(n) = { my(vals = Set()); fordiv(n, d, vals = Set(concat(vals, A212173(d)))); length(vals); }; \\ Antti Karttunen, Jul 19 2017
(Python)
from sympy import factorint, divisors, prod
def P(n): return sorted(factorint(n).values())
def a046523(n):
x=1
while True:
if P(n)==P(x): return x
else: x+=1
def a057521(n): return 1 if n==1 else prod(p**e for p, e in factorint(n).items() if e != 1)
def a212173(n): return a046523(a057521(n))
def a(n):
l=[]
for d in divisors(n):
x=a212173(d)
if not x in l:l+=[x, ]
return len(l)
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 19 2017
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CROSSREFS
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Cf. A212172, A085082, A088873, A181796, A182860, A212173, A212642, A212643, A212644.
Sequence in context: A157754 A072411 A290107 * A091050 A005361 A303915
Adjacent sequences: A212177 A212178 A212179 * A212181 A212182 A212183
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KEYWORD
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nonn
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AUTHOR
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Matthew Vandermast, Jun 04 2012
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STATUS
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approved
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