A039634 -id:A039634 - cp's OEIS frontend

A346063 a(n) = primepi(A039634(prime(n)^2-1)).

2, 1, 2, 2, 4, 3, 1, 5, 1, 6, 4, 3, 6, 4, 7, 14, 6, 10, 7, 37, 23, 25, 28, 18, 21, 22, 67, 24, 9, 46, 11, 19, 62, 12, 40, 24, 2, 27, 6, 91, 11, 31, 20, 1, 36, 203, 69, 25, 2, 80, 16, 48, 155, 18, 1, 326, 7, 20, 109, 365, 8, 39, 9, 240, 438, 2, 16, 154, 10, 17

Offset: 1

Ya-Ping Lu, Jul 03 2021

nonn

This sequence looks at the effect on p^2 - 1 of A039634 with the primes represented by their indices. It seems that primes obtained by iterating the map A039634 on p^2 - 1 never fall into a cycle before reaching 2. Conjecture: Iterating the map k -> a(k) eventually reaches 1. For example, 1 -> 2 -> 1; 5 -> 4 -> 2 -> 1; and 27 -> 67 -> 16 -> 14 -> 4 -> 2 -> 1.

If the conjecture holds, then A339991(n) != -1 and A340419 is a finite sequence.

Ya-Ping Lu, Plot of a(n) vs n for n up to 10000

Cf. A000040, A000720, A039634, A339991, A340008, A340418, A340419.

Array[PrimePi@ FixedPoint[If[EvenQ[#] && # > 2, #/2, If[PrimeQ[#] || (# === 1), #, (# - 1)/2]] &, Prime[#]^2 - 1] &, 70] (* Michael De Vlieger, Jul 06 2021 *)

from sympy import prime, isprime, primepi
def a(n):
    p = prime(n); m = p*p - 1
    while not isprime(m): m = m//2
    return primepi(m)
for n in range(1, 71): print(a(n))

a(n) = A000720(A039634(A000040(n)^2-1)). - Pontus von Brömssen, Jul 03 2021

A346161 Prime numbers p such that the number of iterations of map A039634 required for p to reach 2 sets a new record.

Original entry on oeis.org

2, 3, 7, 23, 47, 191, 383, 1439, 2879, 11519, 23039, 261071, 1044287, 2949119, 31426559, 194224127, 1069493759, 8554807007, 31337349119, 68438456063, 136876912127, 547507648511, 8760122376191

Offset: 1

Ya-Ping Lu, Jul 08 2021

It seems that the record number of iterations for a(n) is n-1.

Alternatively, prime numbers p such that the number of odd primes encountered under iteration of A004526 sets a new record. - Martin Ehrenstein, Aug 16 2021

			Terms in this sequence are indicated in square brackets in the tree below for primes up to 97. Note that a(n) is the smallest prime of depth n-1.
                 1                 ___________[2]____________
                 |                /        /   |   \    \    \
         _______[3]__       ____ 5 _     17   19   37   67   73
        /        |   \     /     |  \     |    |
     _[7]_      13   97   11    41  43   71   79
    /  |  \      |       /  \    |
  29  31  61    53    [23]  89  83
   |                    |
  59                  [47]

Cf. A005384, A058000, A039634, A346063, A346163.

from sympy import nextprime, isprime
rec = -1; p1 = 1
while p1 < 1000000000:
    p = nextprime(p1); m = p; ct = 0
    while m > 2:
        if isprime(m): ct += 1
        m //= 2
    if ct > rec: print(p); rec = ct
    p1 = p

a(19)-a(23) from Martin Ehrenstein, Aug 22 2021

A039645 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, 3, 3, 4, 5, 5, 2, 3, 3, 5, 4, 6, 2, 3, 5, 2, 5, 4, 3, 4, 4, 4, 5, 5, 3, 7, 4, 6, 6, 4, 4, 2, 3, 5, 5, 4, 4, 7, 2, 5, 5, 3, 6, 4, 4, 3, 8, 3, 8, 5, 5, 5, 5, 2, 3, 3, 4, 7, 7, 2, 7, 3, 4, 6, 6, 3, 5, 5, 4, 8, 8, 6, 2, 3, 3, 4, 2, 7, 3, 7, 7, 3, 2, 5, 5, 4, 9, 4, 5, 9, 9, 9, 3, 3, 2, 3, 8

Offset: 1

Wouter Meeussen

nonn

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

Cf. A039634-A039644.

Cf. A010051, A000040, A039641, A039643, A039637.

a039645 n = snd $ until ((== 1) . a010051 . fst)
            (\(x, i) -> ((x + 1) `div` 2 , i + 1)) (a000040 n + 1, 1)
-- Reinhard Zumkeller, Nov 17 2013

f[k_] := Which[PrimeQ[k], k, EvenQ[k], k/2, True, (k+1)/2];
a[n_] := Length[FixedPointList[f, Prime[n] + 1]] - 1;
Array[a, 102] (* Jean-François Alcover, Mar 01 2019 *)

Offset corrected by Reinhard Zumkeller, Nov 17 2013

A039636 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, 3, 2, 1, 3, 1, 2, 2, 4, 1, 4, 1, 3, 3, 2, 1, 4, 4, 2, 2, 3, 1, 3, 1, 5, 5, 2, 2, 5, 1, 2, 2, 4, 1, 4, 1, 3, 3, 2, 1, 5, 5, 5, 5, 3, 1, 3, 3, 4, 4, 2, 1, 4, 1, 2, 2, 6, 6, 6, 1, 3, 3, 3, 1, 6, 1, 2, 2, 3, 3, 3, 1, 5, 5, 2, 1, 5, 5, 2, 2, 4, 1, 4, 4, 3, 3, 2, 2, 6, 1, 6, 6, 6, 1, 6

Offset: 1

Wouter Meeussen

nonn

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

Cf. A039634-A039645.

Cf. A010051, A039634, A039637, A039642, A039643.

a039636 1 = 1
a039636 n = snd $ until ((== 1) . a010051 . fst)
                        (\(x, i) -> (x `div` 2 , i + 1)) (n, 1)
-- Reinhard Zumkeller, Nov 17 2013

nerlist[ 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

A039635 Fixed point of "n -> n/2 or (n+1)/2 until result is prime".

Original entry on oeis.org

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

Offset: 1

Wouter Meeussen

nonn

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

Cf. A039634-A039645.

Cf. A010051, A039637, A039634, A039640, A039641.

a039635 1 = 1
a039635 n = until ((== 1) . a010051) ((flip div 2) . (+ 1)) n
-- Reinhard Zumkeller, Nov 17 2013

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

Offset corrected by Reinhard Zumkeller, Nov 17 2013

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

A039644 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, 3, 2, 3, 5, 4, 4, 3, 2, 3, 2, 6, 3, 5, 5, 5, 2, 4, 6, 4, 4, 2, 5, 5, 7, 4, 4, 6, 3, 4, 6, 3, 2, 3, 2, 4, 7, 7, 5, 5, 3, 6, 2, 4, 4, 8, 8, 8, 8, 2, 3, 5, 7, 7, 3, 3, 7, 7, 7, 3, 3, 6, 2, 6, 6, 2, 5, 4, 8, 2, 3, 6, 6, 6, 4, 4, 7, 7, 7, 7, 7, 5, 5, 5, 2, 2, 4, 5, 9, 2, 3, 6, 3, 6, 3, 3

Offset: 1

Wouter Meeussen

nonn

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

Cf. A039634-A039645.

Cf. A010051, A000040, A039640, A039642, A039637.

a039644 1 = 1
a039644 n = snd $ until ((== 1) . a010051 . fst)
            (\(x, i) -> ((x + 1) `div` 2 , i + 1)) (a000040 n - 1, 1)
-- Reinhard Zumkeller, Nov 17 2013

Mathematica
```
(* See A039637. *)
```

Offset corrected by Reinhard Zumkeller, Nov 17 2013

A078833 Greatest prime contained as binary substring in binary representation of n>1, a(1)=1.

Original entry on oeis.org

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, 13, 23, 47, 3, 17, 3, 19, 13, 53, 13, 23, 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, 17

Offset: 1

Reinhard Zumkeller, Dec 08 2002

a(n) = A039634(n) for n<=44, but a(45) = 13 <> 11 = A039634(45);
for n>1: a(n) = n iff n is prime.
a(n) = A225243(n, A078826(n)). - Reinhard Zumkeller, Aug 14 2013

			n=12 -> '1100' contains 2 binary substrings which are primes: '11' (11bb) and '10' (b11b); 3='11' is the greater one, therefore a(12)=3.

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

Cf. A078832, A078826, A078822, A007088, A006530.

a078833 = last . a225243_row  -- Reinhard Zumkeller, Aug 14 2013

A039638 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, 5, 3, 2, 2, 11, 7, 7, 2, 5, 5, 23, 13, 29, 7, 2, 17, 2, 19, 41, 11, 3, 3, 3, 53, 13, 7, 31, 2, 17, 17, 37, 37, 19, 5, 83, 43, 89, 11, 47, 3, 3, 3, 13, 13, 113, 7, 29, 59, 7, 31, 2, 131, 67, 67, 17, 17, 17, 73, 19, 19, 19, 79, 41, 5, 173, 43, 11, 179, 11, 23, 47, 191

Offset: 1

Wouter Meeussen

nonn

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

Cf. A039634-A039645.

Cf. A010051, A000040, A039642, A039640, A039634.

a039638 1 = 1
a039638 n = until ((== 1) . a010051) (flip div 2) (a000040 n - 1)
-- Reinhard Zumkeller, Nov 17 2013

(* See A039634. *)
Table[NestWhile[If[EvenQ[#],#/2,(#-1)/2]&,Prime[n]-1,CompositeQ],{n,80}] (* Harvey P. Dale, May 27 2023 *)

Offset corrected by Reinhard Zumkeller, Nov 17 2013

A039639 Fixed point of "k -> k/2 or (k-1)/2 until result is prime", starting with prime(n)+1.

Original entry on oeis.org

3, 2, 3, 2, 3, 7, 2, 5, 3, 7, 2, 19, 5, 11, 3, 13, 7, 31, 17, 2, 37, 5, 5, 11, 3, 3, 13, 13, 13, 7, 2, 2, 17, 17, 37, 19, 79, 41, 5, 43, 11, 11, 3, 97, 3, 3, 53, 7, 7, 7, 29, 7, 7, 31, 2, 2, 67, 17, 139, 17, 71, 73, 19, 19, 157, 79, 83, 5, 43, 43, 11, 11, 23, 23, 47, 3, 97, 199, 3

Offset: 1

Wouter Meeussen

nonn

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

Cf. A039634-A039645.

Cf. A010051, A000040, A039643, A039641, A039634.

a039639 = until ((== 1) . a010051) (flip div 2) . (+ 1) . a000040
-- Reinhard Zumkeller, Nov 17 2013

Mathematica
```
(* See A039634. *)
```

Offset corrected by Reinhard Zumkeller, Nov 17 2013

cp's OEIS Frontend

A346063 a(n) = primepi(A039634(prime(n)^2-1)).

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A346161 Prime numbers p such that the number of iterations of map A039634 required for p to reach 2 sets a new record.

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Python

Extensions

A039645 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

A039636 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

A039635 Fixed point of "n -> n/2 or (n+1)/2 until result is prime".

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Author

Keywords

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Crossrefs

Programs

Haskell

Mathematica

Extensions

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

A039644 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

A078833 Greatest prime contained as binary substring in binary representation of n>1, a(1)=1.

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