A292377 -id:A292377 - cp's OEIS frontend

A292603 Doudna-tree reduced modulo 4: a(n) = A005940(1+n) mod 4.

1, 2, 3, 0, 1, 2, 1, 0, 3, 2, 3, 0, 1, 2, 3, 0, 3, 2, 1, 0, 3, 2, 1, 0, 1, 2, 3, 0, 1, 2, 1, 0, 1, 2, 1, 0, 3, 2, 3, 0, 1, 2, 1, 0, 3, 2, 3, 0, 1, 2, 3, 0, 1, 2, 1, 0, 3, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 3, 0, 1, 2, 3, 0, 3, 2, 1, 0, 3, 2, 1, 0, 1, 2, 3, 0, 1, 2, 1, 0, 3, 2, 3, 0, 1, 2, 3, 0, 3, 2, 1, 0, 3, 2, 1, 0, 1

Offset: 0

Antti Karttunen, Dec 01 2017

nonn

			The first six levels of the binary tree (compare also to the illustrations given at A005940 and A292602):
                               1
                               |
                               2
                ............../ \..............
               3                               0
        ....../ \......                 ....../ \......
       1               2               1               0
      / \             / \             / \             / \
     /   \           /   \           /   \           /   \
    3     2         3     0         1     2         3     0
   / \   / \       / \   / \       / \   / \       / \   / \
  3   2 1   0     3   2 1   0     1   2 3   0     1   2 1   0

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences computed from indices in prime factorization

Cf. A003961, A005940, A292602.
Cf. A004767 (gives the positions of 0's), A016813 (of 2's).
Cf. also A246261, A246263, A246271, A292271, A292274, A292375, A292377, A292381, A292383, A292384, A292583.

(define (A292603 n) (modulo (A005940 (+ 1 n)) 4))

a(n) = A010873(A005940(1+n)).
a(n) + 4*A292602(n) = A005940(1+n).
a(2n+1) = 2*a(n) mod 4.
a(A004767(n)) = 0.
a(A016813(n)) = 2.
a(2*A156552(A246261(n))) = 1.
a(2*A156552(A246263(n))) = 3.
a(n * 2^(1+A246271(A005940(1+n)))) = 1.

A331302 Number of 4k+3 composites encountered when traversing from n to the root of A005940-tree.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 2, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1

Offset: 1

Antti Karttunen, Feb 07 2020

nonn

For numbers > 1, iterate the map x -> A252463(x) which divides even numbers by 2 and for odd numbers shifts every prime in the prime factorization one index step towards smaller primes with A064989. a(n) counts the composite numbers of the form 4k+3 (A091236) encountered until 1 has been reached, including in the count also n itself if it is of the same form.

			Here -> stands for transition x -> A252463(x):
For n = 35, 35 mod 4 = 3, 35 -> 15 and 15 mod 4 = 3 also, but then 15 -> 6 (with 6 mod 4 = 2), and 6 -> 3, a prime, after which only noncomposites occur in the trajectory -> 2 -> 1, thus a(35) = 2 as there were exactly two 4k+3 composites on the whole path.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A005940, A064989, A091236, A252463, A292377.

Subsequences of the indices of zeros: A093641, A028982 (see A292583 for the explanation of the latter).

Array[Count[FixedPointList[Which[# == 1, 1, EvenQ@ #, #/2, True, Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ #] &, #], ?(And[CompositeQ@ #, Mod[#, 4] == 3] &)] &, 105] (* _Michael De Vlieger, Feb 08 2020 *)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A252463(n) = if(!(n%2),n/2,A064989(n));
A331302(n) = if((1==n)||isprime(n),0,(3==(n%4))+A331302(A252463(n)));

a(1) = 0, a(p) = 0 for all primes, otherwise a(n) = [n == 3 (mod 4)] + a(A252463(n)).
a(2^k * p) [with k>=0, p prime] = a(n^2) = a(2*(n^2)) = 0, with zeros occurring also on some other positions.
a(n) <= A292377(n).

cp's OEIS Frontend

A292603 Doudna-tree reduced modulo 4: a(n) = A005940(1+n) mod 4.

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