The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A297404 A binary representation of the positive exponents that appear in the prime factorization of a number, shown in decimal. 3
 0, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 3, 1, 1, 1, 8, 1, 3, 1, 3, 1, 1, 1, 5, 2, 1, 4, 3, 1, 1, 1, 16, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 9, 2, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 3, 1, 1, 3, 32, 1, 1, 1, 3, 1, 1, 1, 6, 1, 1, 3, 3, 1, 1, 1, 9, 8, 1, 1, 3, 1, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS This sequence is similar to A087207; here we encode the exponents, there the prime numbers appearing in the prime factorization of a number. The binary representation of a(n) shows which exponents appear in the prime factorization of n, but without multiplicities: - for any prime number p and k > 0, if p^k divides n but p^(k+1) does not divide n, then a(n) AND 2^(k-1) = 2^(k-1) (where AND denotes the bitwise AND operator), - conversely, if a(n) AND 2^(k-1) = 2^(k-1) for some k > 0, then there is prime number p such that p^k divides n but p^(k+1) does not divide n. LINKS Rémy Sigrist, Table of n, a(n) for n = 1..10000 FORMULA a(p^k) = 2^(k-1) for any prime number p and k > 0. a(n^2) = A000695(2 * a(n)) / 2 for any n > 0. a(n) <= 1 iff n is squarefree (A005117). a(n) <= 3 iff n is cubefree (A004709). a(n) is odd iff n belongs to A052485 (weak numbers). a(n) is even iff n belongs to A001694 (powerful numbers). a(n) AND 2 = 2 iff n belongs to A038109 (where AND denotes the bitwise AND operator). A000120(a(n)) <= 1 iff n belongs to A072774 (powers of squarefree numbers). A000120(a(n)) > 1 iff n belongs to A059404. If gcd(m, n) = 1, then a(m * n) = a(m) OR a(n) (where OR denotes the bitwise OR operator). EXAMPLE For n = 90: - 90 = 5^1 * 3^2 * 2^1, - the exponents appearing in the prime factorization of 90 are 1 and 2, - hence a(90) = 2^(1-1) + 2^(2-1) = 3. MATHEMATICA Array[Total@ Map[2^(# - 1) &, Union[FactorInteger[#][[All, -1]] ]] - Boole[# == 1] &, 86] (* Michael De Vlieger, Dec 29 2017 *) PROG (PARI) a(n) = my (x=Set(factor(n)[, 2]~)); sum(i=1, #x, 2^(x[i]))/2 CROSSREFS Cf. A000120, A000695, A001694, A004709, A005117, A038109, A052485, A059404, A087207. Sequence in context: A162512 A162510 A292589 * A235388 A294897 A252733 Adjacent sequences:  A297401 A297402 A297403 * A297405 A297406 A297407 KEYWORD nonn,easy,base AUTHOR Rémy Sigrist, Dec 29 2017 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified January 23 17:33 EST 2022. Contains 350514 sequences. (Running on oeis4.)