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A212180 Number of distinct second signatures (cf. A212172) represented among divisors of n. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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 0s in A212172, in accord with the usual OEIS custom of using 0 for nonexistent elements; in comments, their second signature is represented as { }.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from exponents in factorization of n

EXAMPLE

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.

MATHEMATICA

Array[Length@ Union@ Map[Sort@ Select[FactorInteger[#][[All, -1]], # >= 2 &] &, Divisors@ #] &, 88] (* Michael De Vlieger, Jul 19 2017 *)

PROG

(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

from operator import mul

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 reduce(mul, [1 if e==1 else p**e for p, e in factorint(n).items()])

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 map(a, xrange(1, 151)) # Indranil Ghosh, Jul 19 2017

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Matthew Vandermast, Jun 04 2012

STATUS

approved

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Last modified November 22 18:55 EST 2019. Contains 329410 sequences. (Running on oeis4.)