A039636 -id:A039636 - cp's OEIS frontend

A039634 Fixed point of "n -> n/2 or (n-1)/2 until result is prime".

1, 2, 3, 2, 5, 3, 7, 2, 2, 5, 11, 3, 13, 7, 7, 2, 17, 2, 19, 5, 5, 11, 23, 3, 3, 13, 13, 7, 29, 7, 31, 2, 2, 17, 17, 2, 37, 19, 19, 5, 41, 5, 43, 11, 11, 23, 47, 3, 3, 3, 3, 13, 53, 13, 13, 7, 7, 29, 59, 7, 61, 31, 31, 2, 2, 2, 67, 17, 17, 17, 71, 2, 73, 37, 37, 19, 19, 19, 79, 5, 5

Offset: 1

Wouter Meeussen

a(n) is the largest prime whose binary expansion is an initial substring of n's binary expansion. - Charlie Neder, Oct 27 2018

a(1) = 1 by convention. - David A. Corneth, Oct 27 2018

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A039635-A039645, A010051, A039636, A039638, A039639.

a039634 1 = 1
a039634 n = until ((== 1) . a010051) (flip div 2) n
-- Reinhard Zumkeller, Nov 17 2013

ner[ n_Integer ] := FixedPoint[ If[ EvenQ[ # ]&&#>2, #/2, If[ PrimeQ[ # ]||(#=== 1), #, (#-1)/2 ] ]&, n, 20 ]

a(n)=while(n>3 && !isprime(n), n\=2); n \\ Charles R Greathouse IV, Jun 23 2017

from sympy import isprime
def a(n):
    while n>1 and not isprime(n): n>>=1
    return n
print([a(n) for n in range(1, 82)]) # Michael S. Branicky, Jul 24 2023

Offset corrected by Reinhard Zumkeller, Nov 17 2013

A322809 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -1 if n is an odd prime, and f(n) = floor(n/2) for all other numbers.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 6, 7, 3, 8, 3, 9, 9, 10, 3, 11, 3, 12, 12, 13, 3, 14, 14, 15, 15, 16, 3, 17, 3, 18, 18, 19, 19, 20, 3, 21, 21, 22, 3, 23, 3, 24, 24, 25, 3, 26, 26, 27, 27, 28, 3, 29, 29, 30, 30, 31, 3, 32, 3, 33, 33, 34, 34, 35, 3, 36, 36, 37, 3, 38, 3, 39, 39, 40, 40, 41, 3, 42, 42, 43, 3, 44, 44, 45, 45, 46, 3, 47, 47, 48, 48, 49, 49, 50, 3, 51, 51, 52, 3, 53, 3, 54, 54

Offset: 1

Antti Karttunen, Dec 26 2018

nonn

This sequence is a restricted growth sequence transform of a function f which is defined as f(n) = A004526(n), unless n is an odd prime, in which case f(n) = -1, which is a constant not in range of A004526. See the Crossrefs section for a list of similar sequences.
For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A039636(i) = A039636(j).
For all i, j: a(i) = a(j) <=> A323161(i+1) = A323161(j+1).
The shifted version of this filter, A323161, has a remarkable ability to find many sequences related to primes and prime chains. - Antti Karttunen, Jan 06 2019

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A004526, A039636, A305801, A323161.
A list of few similarly constructed sequences follows, where each sequence is an rgs-transform of such function f, for which the value of f(n) is the n-th term of the sequence whose A-number follows after a parenthesis, unless n is of the form ..., in which case f(n) is given a constant value outside of the range of that sequence:
  A322809 (A004526, unless an odd prime) [This sequence],
  A322589 (A007913, unless an odd prime),
  A322591 (A007947, unless an odd prime),
  A322805 (A252463, unless an odd prime),
  A323082 (A300840, unless an odd prime),
  A322822 (A300840, unless n > 2 and a Fermi-Dirac prime, A050376),
  A322988 (A322990, unless a prime power > 2),
  A323078 (A097246, unless an odd prime),
  A322808 (A097246, unless a squarefree number > 2),
  A322816 (A048675, unless an odd prime),
  A322807 (A285330, unless an odd prime),
  A322814 (A286621, unless an odd prime),
  A322824 (A242424, unless an odd prime),
  A322973 (A006370, unless an odd prime),
  A322974 (A049820, unless n > 1 and n is in A046642),
  A323079 (A060681, unless an odd prime),
  A322587 (A295887, unless an odd prime),
  A322588 (A291751, unless an odd prime),
  A322592 (A289625, unless an odd prime),
  A323369 (A323368, unless an odd prime),
  A323371 (A295886, unless an odd prime),
  A323374 (A323373, unless an odd prime),
  A323401 (A323372, unless an odd prime),
  A323405 (A323404, unless an odd prime).

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A322809aux(n) = if((n>2)&&isprime(n),-1,(n>>1));
v322809 = rgs_transform(vector(up_to,n,A322809aux(n)));
A322809(n) = v322809[n];

a(n) = A323161(n+1) - 1.

A039637 Number of steps to fixed point of "n -> n/2 or (n+1)/2 until result is prime".

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 4, 4, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 5, 1, 5, 2, 2, 4, 4, 1, 2, 4, 4, 1, 3, 1, 3, 2, 2, 1, 5, 3, 3, 3, 3, 1, 4, 4, 4, 2, 2, 1, 6, 1, 2, 6, 6, 3, 3, 1, 3, 5, 5, 1, 5, 1, 2, 3, 3, 5, 5, 1, 5, 2, 2, 1, 4, 2, 2, 4, 4, 1, 3, 3, 3, 2, 2, 6, 6, 1, 4, 4, 4, 1, 4

Offset: 1

Wouter Meeussen

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A039634-A039645.

Cf. A010051, A039635, A039636, A039644, A039645.

a039637 1 = 1
a039637 n = snd $ until ((== 1) . a010051 . fst)
                        (\(x, i) -> ((x + 1) `div` 2 , i + 1)) (n, 1)
-- Reinhard Zumkeller, Nov 17 2013

upplist[ n_Integer ] := Length/@Drop[ FixedPointList[ If[ EvenQ[ # ]&&#>2, #/ 2, If[ PrimeQ[ # ]||(#===1), #, (#+1)/2 ] ]&, n, 20 ], -1 ]

Offset corrected by Reinhard Zumkeller, Nov 17 2013

A039642 Number of steps to fixed point of "k -> k/2 or (k-1)/2 until result is prime", starting with prime(n)-1.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 4, 2, 3, 3, 5, 4, 4, 2, 3, 2, 4, 6, 3, 6, 3, 2, 4, 6, 6, 6, 2, 4, 5, 3, 7, 4, 4, 3, 3, 4, 6, 2, 3, 2, 5, 3, 7, 7, 7, 5, 5, 2, 6, 4, 3, 6, 4, 8, 2, 3, 3, 5, 5, 5, 3, 5, 5, 5, 3, 4, 7, 2, 4, 6, 2, 6, 5, 4, 2, 3, 8, 8, 8, 6, 6, 3, 6, 3, 6, 7, 7, 7, 7, 2, 2, 7, 4, 5, 2, 3, 9, 9, 4, 6, 3

Offset: 1

Wouter Meeussen

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A039634-A039645.

Cf. A010051, A000040, A039638, A039644, A039636.

a039642 1 = 1
a039642 n = snd $ until ((== 1) . a010051 . fst)
                  (\(x, i) -> (x `div` 2 , i + 1)) (a000040 n - 1, 1)
-- Reinhard Zumkeller, Nov 17 2013

Mathematica
```
(* See A039636. *)
```

Offset corrected by Reinhard Zumkeller, Nov 17 2013

A039643 Number of steps to fixed point of "k -> k/2 or (k-1)/2 until result is prime", starting with prime(n)+1.

Original entry on oeis.org

1, 2, 2, 3, 3, 2, 4, 3, 4, 3, 5, 2, 4, 3, 5, 3, 4, 2, 3, 6, 2, 5, 5, 4, 6, 6, 4, 4, 4, 5, 7, 7, 4, 4, 3, 4, 2, 3, 6, 3, 5, 5, 7, 2, 7, 7, 3, 6, 6, 6, 4, 6, 6, 4, 8, 8, 3, 5, 2, 5, 3, 3, 5, 5, 2, 3, 3, 7, 4, 4, 6, 6, 5, 5, 4, 8, 3, 2, 8, 8, 6, 2, 6, 6, 6, 6, 7, 2, 7, 5, 5, 7, 4, 4, 5, 5, 3, 9, 3, 2, 3, 3

Offset: 1

Wouter Meeussen

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A039634-A039645.

Cf. A010051, A000040, A039639, A039645, A039636.

a039643 n = snd $ until ((== 1) . a010051 . fst)
                  (\(x, i) -> (x `div` 2 , i + 1)) (a000040 n + 1, 1)
-- Reinhard Zumkeller, Nov 17 2013

Mathematica
```
(* See A039636. *)
```

Offset corrected by Reinhard Zumkeller, Nov 17 2013

cp's OEIS Frontend

A039634 Fixed point of "n -> n/2 or (n-1)/2 until result is prime".

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A322809 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -1 if n is an odd prime, and f(n) = floor(n/2) for all other numbers.

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A039637 Number of steps to fixed point of "n -> n/2 or (n+1)/2 until result is prime".

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

Extensions

A039642 Number of steps to fixed point of "k -> k/2 or (k-1)/2 until result is prime", starting with prime(n)-1.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

Extensions

A039643 Number of steps to fixed point of "k -> k/2 or (k-1)/2 until result is prime", starting with prime(n)+1.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

Extensions