A252463 -id:A252463 - cp's OEIS frontend

A297113 a(1) = 0, a(2) = 1, after which, a(n) = a(n/2) if n is of the form 4k+2, and otherwise a(n) = 1+a(A252463(n)).

0, 1, 2, 2, 3, 2, 4, 3, 3, 3, 5, 3, 6, 4, 3, 4, 7, 3, 8, 4, 4, 5, 9, 4, 4, 6, 4, 5, 10, 3, 11, 5, 5, 7, 4, 4, 12, 8, 6, 5, 13, 4, 14, 6, 4, 9, 15, 5, 5, 4, 7, 7, 16, 4, 5, 6, 8, 10, 17, 4, 18, 11, 5, 6, 6, 5, 19, 8, 9, 4, 20, 5, 21, 12, 4, 9, 5, 6, 22, 6, 5, 13, 23, 5, 7, 14, 10, 7, 24, 4, 6, 10, 11, 15, 8, 6, 25

Offset: 1

Antti Karttunen, Dec 26 2017

nonn

From Gus Wiseman, Apr 06 2019: (Start)
Also the number of squares in the Young diagram of the integer partition with Heinz number n that are graph-distance 1 from the lower-right boundary, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). For example, the partition (6,5,5,3) with Heinz number 7865 has diagram
  o o o o o o
  o o o o o
  o o o o o
  o o o
with inner rim
            o
          o
        o o
  o o o
of size 7, so a(7867) = 7.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..12721
Index entries for sequences computed from indices in prime factorization

One more than A297167 (after the initial term).
Cf. A001511, A008683, A033265, A064989, A156552, A252463, A252464, A297112, A297155.
Cf. also A297157, A297161, A297162.
Cf. A052126, A065770, A112798, A115994, A174090, A325166, A325167, A325169.

Table[If[n==1,0,PrimePi[FactorInteger[n][[-1,1]]]+PrimeOmega[n]-PrimeNu[n]],{n,100}] (* Gus Wiseman, Apr 06 2019 *)

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)};
A297113(n) = if(n<=2,n-1,if(n%2,1+A297113(A064989(n)), !(n%4)+A297113(n/2)));

\\ More complex way, after Moebius transform:
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A297112(n) = sumdiv(n,d,moebius(n/d)*A156552(d));
A297113(n) = if(1==n,0,1+valuation(A297112(n),2));

;; With memoization-macro definec.
(definec (A297113 n) (cond ((<= n 2) (- n 1)) ((= 2 (modulo n 4)) (A297113 (/ n 2))) (else (+ 1 (A297113 (A252463 n))))))

a(1) = 0, a(2) = 1, after which, a(n) = a(n/2) if n is of the form 4k+2, and otherwise a(n) = 1+a(A252463(n)) .
For n > 1, a(n) = A001511(A297112(n)), where A297112(n) = Sum_{d|n} moebius(n/d)*A156552(d).
a(n) = A252464(n) - A297155(n).
For n > 1, a(n) = 1+A033265(A156552(n)) = 1+A297167(n) = A046660(n) + A061395(n). - Last two sums added by Antti Karttunen, Sep 02 2018
Other identities. For all n >= 1:
a(A000040(n)) = n. [Each n occurs for the first time at the n-th prime.]

A292383 Base-2 expansion of a(n) encodes the steps where numbers of the form 4k+3 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 5, 0, 0, 4, 11, 4, 22, 10, 5, 0, 44, 0, 89, 8, 8, 22, 179, 8, 0, 44, 1, 20, 358, 10, 717, 0, 20, 88, 11, 0, 1434, 178, 45, 16, 2868, 16, 5737, 44, 8, 358, 11475, 16, 0, 0, 89, 88, 22950, 2, 17, 40, 176, 716, 45901, 20, 91802, 1434, 17, 0, 40, 40, 183605, 176, 356, 22, 367211, 0, 734422, 2868, 1, 356, 22, 90, 1468845, 32, 0, 5736, 2937691, 32

Offset: 1

Antti Karttunen, Sep 15 2017

			For n = 3, the starting value is of the form 4k+3, after which follows A252463(3) = 2, and A252463(2) = 1, the end point of iteration, and neither 2 nor 1 is of the form 4k+3, thus a(3) = 1*(2^0) + 0*(2^1) + 0*(2^2) = 1.
For n = 5, the starting value is not of the form 4k+3, after which follows A252463(5) = 3 (which is), continuing as before as 3 -> 2 -> 1, thus a(5) = 0*(2^0) + 1*(2^1) + 0*(2^2) + 0*(2^3) = 2.
For n = 10, the starting value is not of the form 4k+3, after which follows A252463(10) = 5 (also not 4k+3), and then A252463(5) = 3 (which is), continuing as before as 3 -> 2 -> 1, thus a(10) = 0*(2^0) + + 0*(2^1) + 1*(2^2) + 0*(2^3) + 0*(2^4) = 4.

Cf. A163511, A243071, A292274, A292373, A292377, A292380, A292381, A292382, A292384, A292385.

Table[FromDigits[Reverse@ NestWhileList[Function[k, Which[k == 1, 1, EvenQ@ k, k/2, True, Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ k]], n, # > 1 &] /. k_ /; IntegerQ@ k :> If[Mod[k, 4] == 3, 1, 0], 2], {n, 84}] (* Michael De Vlieger, Sep 21 2017 *)

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));
A292383(n) = if(1==n,0,(if(3==(n%4),1,0)+(2*A292383(A252463(n)))));

(define (A292383 n) (A292373 (A292384 n)))

a(1) = 0; for n > 1, a(n) = 2*a(A252463(n)) + [n ≡ 3 (mod 4)], where the last part of the formula is Iverson bracket, giving 1 only if n is of the form 4k+3, and 0 otherwise.
a(n) = A292373(A292384(n)).
a(n) = A292274(A243071(n)).
Other identities. For n >= 1:
a(2n) = 2*a(n).
a(n) + A292385(n) = A243071(n).
a(A163511(n)) = A292274(n).
A000120(a(n)) = A292377(n).

A292385 a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A252463(n)) + [n == 1 (mod 4)].

Original entry on oeis.org

0, 1, 2, 2, 5, 4, 10, 4, 5, 10, 20, 8, 41, 20, 8, 8, 83, 10, 166, 20, 21, 40, 332, 16, 11, 82, 8, 40, 665, 16, 1330, 16, 41, 166, 16, 20, 2661, 332, 80, 40, 5323, 42, 10646, 80, 17, 664, 21292, 32, 23, 22, 164, 164, 42585, 16, 42, 80, 333, 1330, 85170, 32, 170341, 2660, 40, 32, 83, 82, 340682, 332, 665, 32, 681364, 40, 1362729, 5322, 20, 664, 33, 160

Offset: 1

Antti Karttunen, Sep 16 2017

Variant of A292381. Here the most significant 1-bit is at the one step smaller position.

Antti Karttunen, Table of n, a(n) for n = 1..2048
Index entries for sequences related to binary expansion of n

Cf. A004754, A163511, A243071, A252463, A292381, A292383.

a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A252463(n)) + [n == 1 (mod 4)], where the last part of the formula is Iverson bracket, giving 1 only if n is of the form 4k+1, and 0 otherwise.
For n >= 1, a(n) + A292383(n) = A243071(n); a(A163511(n)) = A292271(n).
For n >= 2, A004754(a(n)) = A292381(n).

A292377 a(1) = 0, and for n > 1, a(n) = a(A252463(n)) + [n == 3 (mod 4)].

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 0, 0, 1, 3, 1, 3, 2, 2, 0, 3, 0, 4, 1, 1, 3, 5, 1, 0, 3, 1, 2, 5, 2, 6, 0, 2, 3, 3, 0, 6, 4, 4, 1, 6, 1, 7, 3, 1, 5, 8, 1, 0, 0, 4, 3, 8, 1, 2, 2, 3, 5, 9, 2, 9, 6, 2, 0, 2, 2, 10, 3, 4, 3, 11, 0, 11, 6, 1, 4, 3, 4, 12, 1, 0, 6, 13, 1, 4, 7, 6, 3, 13, 1, 3, 5, 5, 8, 5, 1, 13, 0, 3, 0, 13, 4, 14, 3, 2

Offset: 1

Antti Karttunen, Sep 17 2017

nonn

For numbers > 1, iterate the map x -> A252463(x) which divides even numbers by 2 and shifts every prime in the prime factorization of odd n one index step towards smaller primes. a(n) counts the numbers of the form 4k+3 encountered until 1 has been reached. The count includes also n itself if it is of the form 4k+3 (A004767).

In other words, locate the node which contains n in binary tree A005940 and traverse from that node towards the root, counting all numbers of the form 4k+3 that occur on the path.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A000120, A005940, A061395, A064989, A252463, A292375, A292376, A292378, A292383.

a[1] = 0; a[n_] := a[n] = a[Which[n == 1, 1, EvenQ@ n, n/2, True, Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n]] + Boole[Mod[n, 4] == 3]; Array[a, 105]

a(1) = 0, and for n > 1, a(n) = a(A252463(n)) + floor((n mod 4)/3).
Equivalently, a(2n) = a(n), and for odd numbers n > 1, a(n) = a(A064989(n)) + [n == 3 (mod 4)].
a(n) = A000120(A292383(n)).
Other identities. For n >= 1:
a(n) >= A292376(n).
a(A000040(n)) = A267098(n).
1 + a(n) - A292375(n) = A292378(n).
For n >= 2, a(n) + A292375(n) = A061395(n).

A292381 Base-2 expansion of a(n) encodes the steps where numbers of the form 4k+1 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

Original entry on oeis.org

1, 2, 4, 4, 9, 8, 18, 8, 9, 18, 36, 16, 73, 36, 16, 16, 147, 18, 294, 36, 37, 72, 588, 32, 19, 146, 16, 72, 1177, 32, 2354, 32, 73, 294, 32, 36, 4709, 588, 144, 72, 9419, 74, 18838, 144, 33, 1176, 37676, 64, 39, 38, 292, 292, 75353, 32, 74, 144, 589, 2354, 150706, 64, 301413, 4708, 72, 64, 147, 146, 602826, 588, 1177, 64, 1205652, 72, 2411305, 9418, 36, 1176

Offset: 1

Antti Karttunen, Sep 15 2017

			For n = 1, the starting value (which is also the ending point) is of the form 4k+1, thus a(1) = 1*(2^0) = 1.
For n = 2, the starting value is not of the form 4k+1, but its parent, A252463(2) = 1 is, thus a(2) = 0*(2^0) + 1*(2^1) = 2.
For n = 3, the starting value is not of the form 4k+1, after which follows 2 (also not 4k+1), and then 2 -> 1, and it is only the end-point of iteration which is of the form 4k+1, thus a(3) = 0*(2^0) + 0*(2^1) + 1*(2^2) = 4.
For n = 5, the starting value is of the form 4k+1, after which follows A252463(5) = 3 (which is not), and then continuing as before as 3 -> 2 -> 1, thus a(5) = 1*(2^0) + 0*(2^1) + 0*(2^2) + 1*(2^3) = 9.

Cf. A252463, A292371, A292375, A292380, A292382, A292383, A292384, A292385.

Table[FromDigits[Reverse@ NestWhileList[Function[k, Which[k == 1, 1, EvenQ@ k, k/2, True, Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ k]], n, # > 1 &] /. k_ /; IntegerQ@ k :> If[Mod[k, 4] == 1, 1, 0], 2], {n, 76}] (* Michael De Vlieger, Sep 21 2017 *)

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));
A292381(n) = if(1==n,n,(if(1==(n%4),1,0)+(2*A292381(A252463(n)))));

from sympy.core.cache import cacheit
from sympy.ntheory.factor_ import digits
from sympy import factorint, prevprime
from operator import mul
from functools import reduce
def a292371(n):
    k=digits(n, 4)[1:]
    return 0 if n==0 else int("".join(['1' if i==1 else '0' for i in k]), 2)
def a064989(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==2 else prevprime(i)**f[i] for i in f])
def a252463(n): return 1 if n==1 else n//2 if n%2==0 else a064989(n)
@cacheit
def a292384(n): return 1 if n==1 else 4*a292384(a252463(n)) + n%4
def a(n): return a292371(a292384(n))
print([a(n) for n in range(1, 111)]) # Indranil Ghosh, Sep 21 2017

(define (A292381 n) (A292371 (A292384 n)))

a(1) = 1; for n > 1, a(n) = 2*a(A252463(n)) + [n ≡ 1 (mod 4)], where the last part of the formula is Iverson bracket, giving 1 only if n is of the form 4k+1, and 0 otherwise.
a(n) = A292371(A292384(n)).
Other identities. For n >= 1:
a(2n) = 2*a(n).
A000120(a(n)) = A292375(n).
For n >= 2, a(n) = A004754(A292385(n)).

A320107 a(n) = A001227(A252463(n)).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 1, 2, 2, 4, 2, 1, 2, 2, 4, 3, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 4, 4, 2, 2, 2, 2, 4, 2, 2, 2, 1, 4, 4, 2, 2, 2, 4, 2, 3, 2, 2, 3, 2, 4, 4, 2, 2, 1, 2, 2, 4, 4, 2, 2, 2, 2, 6, 4, 2, 2, 2, 4, 2, 2, 3, 2, 3, 2, 4, 2, 2, 4

Offset: 1

Antti Karttunen, Nov 22 2018

nonn

Records 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 20, 24, 32, 36, 40, ... occur at n = 1, 5, 18, 30, 90, 210, 450, 630, 1890, 3150, 5670, 6930, 20790, 34650, 62370, ...

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A001227, A005940, A051064, A055457, A252463, A320106 (Möbius transform).

Cf. also A320111 and A319699, A319700, A319703, A319989.

A001227(n) = numdiv(n>>valuation(n, 2));
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));
A320107(n) = A001227(A252463(n));

A051064(n) = if(n<1, 0, 1+valuation(n, 3));
A320107(n) = if(n<=2, 1, my(p=A252463(n)); if(!(p%2), A320107(p), A051064(p)*A320107(p)));

A051064(n) = if(n<1, 0, 1+valuation(n, 3));
A055457(n) = if(n<1, 0, 1+valuation(n, 5));
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)};
A320107(n) = if(n<=2, 1, if(!(n%4), A320107(n/2), if(2==(n%4), A051064(n)*A320107(n/2), if(!(n%3), A320107(A064989(n)), A055457(n)*A320107(A064989(n))))));

a(n) = A001227(A252463(n)).
a(1) = a(2) = 1; for n > 2, a(n) = a(n/2) when n == 0 mod 4, a(n) = A051064(n) * a(n/2) when n == 2 mod 4, a(n) = a(A064989(n)), when n == 3 mod 6, otherwise a(n) = A055457(n) * a(A064989(n)).
For n > 2, let p = A252463(n). If p is even, then a(n) = a(p), if p is odd, then a(n) = A051064(p) * a(p).

A292375 a(1) = 1, and for n > 1, a(n) = a(A252463(n)) + [n == 1 (mod 4)].

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 2, 1, 1, 4, 2, 4, 2, 3, 2, 4, 1, 3, 3, 1, 2, 5, 1, 5, 1, 3, 4, 1, 2, 6, 4, 2, 2, 7, 3, 7, 2, 2, 4, 7, 1, 4, 3, 3, 3, 8, 1, 3, 2, 5, 5, 8, 1, 9, 5, 2, 1, 4, 3, 9, 4, 5, 1, 9, 2, 10, 6, 2, 4, 2, 2, 10, 2, 2, 7, 10, 3, 3, 7, 4, 2, 11, 2, 3, 4, 6, 7, 3, 1, 12, 4, 2, 3, 13, 3, 13, 3, 2

Offset: 1

Antti Karttunen, Sep 17 2017

nonn

For numbers > 1, iterate the map x -> A252463(x) which divides even numbers by 2, and shifts every prime in the prime factorization of odd n one index step towards smaller primes. a(n) counts the numbers of the form 4k+1 encountered until 1 has been reached, which is also included in the count. The count includes also n itself if it is of the form 4k+1 (A016813), thus a(1) = 1.

In other words, locate the node which contains n in binary tree A005940 and traverse from that node towards the root, counting all numbers of the form 4k+1 that occur on the path.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A000120, A001248, A005940, A061395, A064989, A252463, A292374, A292377, A292378, A292381, A292583.

a[1] = 1; a[n_] := a[n] = a[Which[n == 1, 1, EvenQ@n, n/2, True, Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n]] + Boole[Mod[n, 4] == 1]; Array[a, 105] (* Michael De Vlieger, Sep 17 2017 *)

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)};
A292375(n) = if(1==n,n,if(!(n%2),A292375(n/2),(if(1==(n%4),1,0)+A292375(A064989(n)))));

;; With memoization-macro definec.
(definec (A292375 n) (if (= 1 n) 1 (+ (if (= 1 (modulo n 4)) 1 0) (A292375 (A252463 n)))))

a(1) = 1, a(2n) = a(n), and for odd numbers n > 1, a(n) = a(A064989(n)) + [n == 1 (mod 4)].
a(n) = A000120(A292381(n)).
Other identities and observations. For n >= 1:
a(n) >= A292374(n).
a(A000040(n))-1 = A267097(n).
1 + A292377(n) - a(n) = A292378(n).
For n >= 2, a(n) + A292377(n) = A061395(n).
From Antti Karttunen, Apr 22 2022: (Start)
For n >= 2, a(n^2) = A061395(n). [Because A292377(n^2) = 0]
For n >= 1, a(A001248(n)) = n. [See comments in A292583]
(End)

A292943 a(n) = A292944(A243071(n)); Base-2 expansion of a(n) encodes the steps where numbers of the form 6k+3 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 4, 0, 1, 4, 8, 4, 16, 8, 5, 0, 32, 2, 64, 8, 9, 16, 128, 8, 2, 32, 1, 16, 256, 10, 512, 0, 17, 64, 10, 4, 1024, 128, 33, 16, 2048, 18, 4096, 32, 9, 256, 8192, 16, 4, 4, 65, 64, 16384, 2, 18, 32, 129, 512, 32768, 20, 65536, 1024, 17, 0, 34, 34, 131072, 128, 257, 20, 262144, 8, 524288, 2048, 5, 256, 20, 66, 1048576, 32, 1, 4096

Offset: 1

Antti Karttunen, Sep 28 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2048
Index entries for sequences related to binary expansion of n

Cf. A005940, A163511, A243071, A292941, A292944, A292945.

Cf. also A292253, A292255, A292383.

(define (A292943 n) (A292944 (A243071 n)))
(define (A292943 n) (if (<= n 1) 0 (+ (if (= 3 (modulo n 6)) 1 0) (* 2 (A292943 (A252463 n))))))

a(n) = A292944(A243071(n)).
a(1) = 0, and for n > 1, a(n) = 2*a(A252463(n)) + [n == 3 (mod 6)], where the last part of the formula is Iverson bracket, giving 1 only if n is of the form 6k+3, and 0 otherwise.
For n >= 0, a(A163511(n)) = A292944(n).
For n >= 1, A292941(n) + a(n) + A292945(n) = a(n) + A292253(n) + A292255(n) = A243071(n).

A319703 a(n) = A003415(A252463(n)).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 4, 4, 1, 1, 5, 1, 1, 5, 12, 1, 6, 1, 7, 7, 1, 1, 16, 6, 1, 12, 9, 1, 8, 1, 32, 9, 1, 8, 21, 1, 1, 13, 24, 1, 10, 1, 13, 16, 1, 1, 44, 10, 10, 15, 15, 1, 27, 10, 32, 19, 1, 1, 31, 1, 1, 24, 80, 14, 14, 1, 19, 21, 12, 1, 60, 1, 1, 21, 21, 12, 16, 1, 68, 32, 1, 1, 41, 16, 1, 25, 48, 1, 39, 16, 25, 31, 1

Offset: 1

Antti Karttunen, Nov 22 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A003415, A252463.

Cf. also A319699, A319700, A319989, A320107, A320111.

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
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));
A319703(n) = A003415(A252463(n));

a(n) = A003415(A252463(n)).

A348045 Möbius transform of A252463, where A252463 shifts the prime factorization of odd numbers one step towards smaller primes and divides even numbers by two.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 4, 2, 2, 2, 6, 2, 10, 2, 2, 4, 12, 4, 16, 4, 4, 4, 18, 4, 6, 2, 4, 6, 22, 6, 28, 8, 6, 4, 8, 6, 30, 2, 10, 8, 36, 8, 40, 10, 4, 4, 42, 8, 20, 14, 12, 12, 46, 14, 12, 12, 16, 6, 52, 8, 58, 2, 8, 16, 20, 14, 60, 16, 18, 16, 66, 12, 70, 6, 6, 18, 24, 14, 72, 16, 8, 4, 78, 12, 24, 2, 22, 20, 82, 20

Offset: 1

Antti Karttunen, Oct 12 2021

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A008683, A064989, A252463, A285702 (odd bisection), A348046 (positions of 2's).

Cf. also A023022, A326305, A347115.

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));
A348045(n) = sumdiv(n,d,moebius(n/d)*A252463(d));

a(n) = Sum_{d|n} A008683(n/d) * A252463(d).

cp's OEIS Frontend

A297113 a(1) = 0, a(2) = 1, after which, a(n) = a(n/2) if n is of the form 4k+2, and otherwise a(n) = 1+a(A252463(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Scheme

Formula

A292383 Base-2 expansion of a(n) encodes the steps where numbers of the form 4k+3 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Scheme

Formula

A292385 a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A252463(n)) + [n == 1 (mod 4)].

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A292377 a(1) = 0, and for n > 1, a(n) = a(A252463(n)) + [n == 3 (mod 4)].

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A292381 Base-2 expansion of a(n) encodes the steps where numbers of the form 4k+1 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Scheme

Formula

A320107 a(n) = A001227(A252463(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

PARI

Formula

A292375 a(1) = 1, and for n > 1, a(n) = a(A252463(n)) + [n == 1 (mod 4)].

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Scheme