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A306345 Absolute difference between the number of prime divisors and the number of composite divisors of n. 1
0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 0, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 5, 2, 1, 1, 5, 1, 1, 1, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,16

COMMENTS

Conjecture: a(n) = 0 iff n is a term of A280076 = union of A001248 and {1}.

Conjecture is true, since having an n with k distinct prime factors such that a(n) = 0 requires that 2k+1 can be factored into k parts > 1, and 1 is the only positive k for which this is possible. - Charlie Neder, Feb 12 2019

LINKS

David A. Corneth, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = abs(A001221(n) - A055212(n)).

a(n) = abs(2*A001221(n) - A000005(n) + 1). - Michel Marcus, Feb 12 2019

EXAMPLE

For n = 24: The set of divisors of 24 is {1, 2, 3, 4, 6, 8, 12, 24}. The prime divisors are {2, 3} and the composite divisors are {4, 6, 8, 12, 24}. The cardinalities of the sets are 2 and 5, respectively, and abs(2-5) = 3, so a(24) = 3.

MATHEMATICA

Array[Abs[2 PrimeNu@ # - DivisorSigma[0, #] + 1] &, 105] (* Michael De Vlieger, Feb 17 2019 *)

PROG

(PARI) a(n) = my(d=divisors(n), p=0, c=0); for(k=2, #d, if(ispseudoprime(d[k]), p++, c++)); abs(p-c)

(PARI) a(n) = abs(2*omega(n) - numdiv(n) + 1); \\ Michel Marcus, Feb 12 2019

CROSSREFS

Cf. A000005, A001221, A001248, A055212, A280076.

Sequence in context: A101875 A081387 A261795 * A204125 A204127 A225174

Adjacent sequences:  A306342 A306343 A306344 * A306346 A306347 A306348

KEYWORD

nonn

AUTHOR

Felix Fröhlich, Feb 08 2019

EXTENSIONS

a(1)=0 prepended by David A. Corneth, Feb 12 2019

STATUS

approved

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Last modified September 25 20:01 EDT 2020. Contains 337344 sequences. (Running on oeis4.)