A236190 -id:A236190 - cp's OEIS frontend

A236175 Prime gap pattern of compacting Eratosthenes sieve for prime(4) = 7.

11, 6, 3, 6, 3, 6, 11, 2, 11, 6, 3, 6, 3, 6, 11, 2, 11, 6, 3, 6, 3, 6, 11, 2, 11, 6, 3, 6, 3, 6, 11, 2, 11, 6, 3, 6, 3, 6, 11, 2, 11, 6, 3, 6, 3, 6, 11, 2, 11, 6, 3, 6, 3, 6, 11, 2, 11, 6, 3, 6, 3, 6, 11, 2, 11, 6, 3, 6, 3, 6, 11, 2, 11, 6, 3, 6, 3, 6, 11, 2

Offset: 1

Christopher J. Hanson, Jan 19 2014

P(x) is a function which produces a prime number at a particular ordinal x (A000040).  This pattern, p(x), describes the number of values emitted as potentially prime by a reductive sieve before a value is marked "not prime" when processing the prime at ordinal x. p(x) represents only the unique portion of the pattern and terminates when the pattern repeats.  The first digit of p(x) corresponds to A079047 for index x.
In this sequence, x = 4 and thus a(1) = A079047(4) = 11. - Michael Somos, Mar 09 2014
The Eratosthenes sieve can be expressed as follows. Start with S1 = [2, 3, 4, 5, ...] the list of numbers bigger than 1. Removing all multiples of the first element 2 yields the list S2 = [3, 5, 7, 9, ...]. Removing all multiples of the first element 3 yields S3 = [5, 7, 11, 13, 17, 19, ...], Removing all multiples of the first element 5 yields S4 = [7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, ...], and so on. The list of first differences of S4 is [4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, ...] which is A236185. The list of indices of all multiples of S4(1) = 7 is [1, 13, 20, 24, 31, 35, 42, 54, 57, 69, 76, 80, ...]. The list of first differences of this list is [12, 7, 4, 7, 4, 7, 12, 3, 12, 7, 4, ...]. Subtract one from each element yields [11, 6, 3, 6, 3, 6, 11, 2, 11, 6, 3, ...] which is this sequence. - Michael Somos, Mar 12 2014

Michael Somos, Table of n, a(n) for n = 1..80
Christopher J. Hanson, The structure of prime numbers and twin prime gaps
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1).

Equivalent sequences for prime(k): A236176 (k=5), A236177 (k=6), A236178 (k=7), A236179 (k=8), A236180 (k=9).

Cf. A079047, A236185-A236190, A359632.

PadRight[{}, 100, {11, 6, 3, 6, 3, 6, 11, 2}] (* Paolo Xausa, Jun 30 2024 *)

{a(n) = my(A); if( n<1, 0, A = vector( (n+1) * 1024 \ 37, k, k+1); for( i = 1, 3, A = select( k -> k%prime(i), A) ); polcoeff( (1 - x) * Ser( select( k -> (k%7) == 0, A, 1)), n) - 1) }; /* Michael Somos, Mar 09 2014 */
(C#)
// p(4) = GeneratePrimePattern( 4 );
static void GeneratePrimePattern( int ordinal )
{
    // Contract
    if( ordinal < 1 )
        throw new ArgumentOutOfRangeException( "ordinal" );
    // Local data
    int size = 1 << 18;
    int[] numberLine = Enumerable.Range( 2, size ).ToArray();
    int pointer = 0;
    // Apply sieve: for each integer greater than 1
    while( pointer < numberLine.Length )
    {
        // Locals
        int x = numberLine[pointer];
        int index = pointer;
        List pattern = new List();
        int skips = 0;
        // Find all products
        for( int n = x + x; n < size; n += x )
        {
            // Fast forward through number-line
            while( numberLine[++index] < n )
                skips++;
            // If the number was not already removed
            if( numberLine[index] == n )
            {
                // Mark as not prime
                numberLine[index] = 0;
                // Add skip count to pattern
                pattern.Add( skips );
                // Reset skips
                skips = 0;
            }
            // Otherwise we've skipped again
            else skips++;
        }
        // Reduce number-line
        numberLine = numberLine.Where( n => n > 0 ).ToArray();
        // If we have a pattern we want
        if( pattern.Any() && pointer == ordinal - 1 )
        {
            // Report pattern
            int previousValue = 3; // > 2
            System.Console.WriteLine( "Pattern P({0}) = {1} :: p({0}) = {2}", pointer + 1, numberLine[pointer], String.Join( ", ", pattern.TakeWhile( value => previousValue > 2 && ( previousValue = value ) > 0 ) ) );
            return;
        }
        // Move number-line pointer forward
        pointer++;
    }
}

a(n + 8) = a(n). - Michael Somos, Mar 09 2014

a(n) = A359632(n) - 1. - Peter Munn, Jan 21 2023

Edited by Michael Somos, Mar 09 2014. Made sequence periodic.

A236180 Prime gap pattern of compacting Eratosthenes sieve for prime(9) = 23.

Original entry on oeis.org

90, 22, 6, 20, 15, 7, 15, 21, 21, 7, 23, 17, 6, 22, 15, 21, 29, 17, 6, 17, 7, 13, 55, 15, 21, 8, 37, 8, 23, 23, 14, 24, 21, 8, 39, 6, 16, 6, 46, 46, 14, 7, 14, 22, 9, 39, 24, 21, 24, 8, 23, 13, 8, 39, 54, 14, 7, 14, 55, 22, 38, 7, 15, 24, 28, 22, 25, 14, 22

Offset: 1

Christopher J. Hanson, Jan 19 2014

nonn

See A236175 for details.

Christopher J. Hanson, Table of n, a(n) for n = 1..10000

Cf. A236175-A236180, A236185-A236190.

A236185 Differences between terms of compacting Eratosthenes sieve for prime(4) = 7.

Original entry on oeis.org

4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2

Offset: 1

Christopher J. Hanson, Jan 21 2014

nonn

P(x) is a function which represents a prime number at a ordinal x.
This pattern, dp(x) is a sequence of the differences between consecutive prime numbers as described by p(x).
For P(4), dp(4)is the relative offsets of the next 7 primes: 7, +4 = 11, +2 = 13, +4 = 17, +2 = 19, +4 = 23, +6 = 29, +2 = 31
The Eratosthenes sieve can be expressed as follows. Start with S1 = [2, 3, 4, 5, ...] the list of numbers bigger than 1. Removing all multiples of the first element 2 yields the list S2 = [3, 5, 7, 9, ...]. Removing all multiples of the first element 3 yields S3 = [5, 7, 11, 13, 17, 19, ...], Removing all multiples of the first element 5 yields S4 = [7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, ...], and so on. The list of first differences of S4 is [4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, ...] which is this sequence. - Michael Somos, Mar 12 2014

Christopher J. Hanson, The structure of prime numbers and twin prime gaps

Cf. A236175-A236180, A236185-A236190.

Cf. A007775. Essentially the same as A145011.

{a(n) = my(A); if( n<1, 0, A = vector( n*28 + 48, k, k+1); for( i = 1, 3, A = select( k -> k%prime(i), A) ); polcoeff( (1 - x) * Ser( select( k -> k>7 && (k%7) == 0, A) / 7), n)) }; /* Michael Somos, Mar 10 2014 */
(C#)
// dp(4) = GeneratePrimeDifferencialPattern( 4 );
static void GeneratePrimeDifferencialPattern( int ordinal )
{
    // Contract
    if( ordinal < 1 )
        throw new ArgumentOutOfRangeException( "ordinal" );
    // Local data
    int size = 1 << 18;
    int[] numberLine = Enumerable.Range( 2, size ).ToArray();
    int pointer = 0;
    // Apply sieve: for each integer greater than 1
    while( pointer < numberLine.Length )
    {
        // Locals
        int x = numberLine[pointer];
        int index = pointer;
        List pattern = new List();
        int skips = 0;
        int count = 0;
        bool counting = true;
        // Find all products
        for( int n = x + x; n < size; n += x )
        {
            // Fast forward through number-line
            while( numberLine[++index] < n )
                skips++;
            // If the number was not already removed
            if( numberLine[index] == n )
            {
                // Add skip count to pattern
                pattern.Add( numberLine[index] );
                // Mark as not prime
                numberLine[index] = 0;
                // Reset skips
                if( counting )
                {
                    count++;
                    if( skips <= 2 )
                        counting = false;
                }
                skips = 0;
            }
            // Otherwise we've skipped again
            else skips++;
        }
        // Reduce number-line
        numberLine = numberLine.Where( n => n > 0 ).ToArray();
        // If we have a pattern we want
        if( pattern.Any() && pointer == ordinal - 1 )
        {
            // Report pattern
            int prime = numberLine[ordinal-1];
            var d = pattern.Take( count ).ToArray();
            List dp = new List();
            for( int y = 1; y < count; y++ )
                dp.Add( ( d[y] - d[y - 1] ) / prime );
            System.Console.WriteLine( "Pattern P({0}) = {1} :: dp({0}) = {2}", pointer + 1, numberLine[pointer], String.Join( ", ", dp ) );
            return;
        }
        // Move number-line pointer forward
        pointer++;
    }
}

a(n + 8) = a(n). - Michael Somos, Mar 10 2014

a(n) = 4*((n+2) mod 2) + 2*((n+1) mod 2) + 4*(f(8,n+2)+f(8,n)) - 2*f(8,n+1), where f(x,n)= floor(n/x)-floor((n-1)/x). - Gary Detlefs, Nov 16 2020

Edited by Michael Somos, Mar 12 2014. (Added code and comments, refined description.)

A236177 Prime gap pattern of compacting Eratosthenes sieve for prime(6) = 13.

Original entry on oeis.org

33, 8, 6, 10, 14, 6, 14, 10, 4, 10, 15, 15, 4, 16, 10, 5, 15, 10, 15, 20, 11, 4, 10, 4, 10, 37, 10, 15, 4, 27, 5, 14, 15, 9, 3, 11, 15, 6, 26, 5, 9, 4, 31, 26, 4, 10, 5, 10, 14, 5, 16, 10, 15, 16, 13, 4, 17, 10, 5, 14, 11, 14, 21, 10, 5, 9, 15, 20, 16, 24, 5

Offset: 1

Christopher J. Hanson, Jan 19 2014

nonn

See A236175 for details.

Christopher J. Hanson, Table of n, a(n) for n = 1..480

Cf. A236175-A236180, A236185-A236190.

A236178 Prime gap pattern of compacting Eratosthenes sieve for prime(7) = 17.

Original entry on oeis.org

54, 5, 12, 18, 5, 18, 12, 6, 11, 20, 17, 7, 18, 11, 7, 18, 12, 19, 26, 12, 6, 11, 6, 13, 44, 10, 19, 6, 31, 6, 18, 20, 13, 19, 17, 5, 31, 6, 14, 5, 37, 38, 13, 5, 13, 19, 5, 32, 18, 20, 17, 7, 18, 11, 6, 19, 10, 44, 13, 5, 13, 18, 26, 19, 31, 7, 11, 18, 7

Offset: 1

Christopher J. Hanson, Jan 19 2014

nonn

See A236175 for details.

Christopher J. Hanson, Table of n, a(n) for n = 1..5760

Cf. A236175-A236180, A236185-A236190.

A236179 Prime gap pattern of compacting Eratosthenes sieve for prime(8) = 19.

Original entry on oeis.org

64, 12, 18, 6, 20, 12, 4, 15, 18, 21, 5, 18, 13, 7, 20, 12, 20, 25, 14, 6, 13, 6, 14, 46, 13, 18, 6, 34, 7, 18, 19, 14, 18, 21, 6, 34, 5, 15, 6, 39, 40, 13, 7, 12, 19, 7, 33, 19, 17, 21, 7, 19, 12, 6, 35, 46, 14, 6, 11, 49, 20, 33, 4, 15, 16, 7, 21, 21, 19, 11

Offset: 1

Christopher J. Hanson, Jan 19 2014

nonn

See A236175 for details.

Christopher J. Hanson, Table of n, a(n) for n = 1..92160

Cf. A236175-A236180, A236185-A236190.

A236176 Prime gap pattern of compacting Eratosthenes sieve for prime(5) = 11.

Original entry on oeis.org

25, 4, 9, 4, 10, 14, 3, 15, 9, 4, 8, 14, 15, 4, 14, 9, 4, 14, 10, 13, 19, 10, 4, 8, 4, 10, 19, 13, 10, 14, 4, 9, 14, 4, 15, 14, 8, 4, 9, 15, 3, 14, 10, 4, 9, 4, 25, 2

Offset: 1

Christopher J. Hanson, Jan 19 2014

nonn

See A236175 for details.

Cf. A236175-A236180, A236185-A236190.

A236187 Differences between terms of compacting Eratosthenes sieve for prime(6) = 13.

Original entry on oeis.org

4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 2, 4, 6, 2, 10, 2, 4, 2, 12, 10, 2, 4, 2, 4, 6, 2, 6, 4, 6, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 6, 8, 6, 10, 2, 4, 6, 2, 6, 6, 4, 2, 4, 6, 2, 6, 4, 2, 6

Offset: 1

Christopher J. Hanson, Jan 21 2014

nonn

P(x) is a function which represents a prime number at a particular ordinal x. This pattern, dp(x), describes the difference between consecutive prime numbers as described by p(x) (see A236175) and therefore the length of dp(x) is len(p(x)) - 1 and each value in dp(x) times P(x) is the difference between values determined not primed when running one pass of a reductive sieve, starting at P(x)^2. See A236185.

Christopher J. Hanson, Table of n, a(n) for n = 1..479

Cf. A236175-A236180, A236185-A236190.

A236188 Differences between terms of compacting Eratosthenes sieve for prime(7) = 17.

Original entry on oeis.org

2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 6, 4, 14, 4, 2, 4, 6, 8, 6, 10, 2, 4, 6, 2, 6, 6, 6, 4, 6, 2, 6, 4, 8, 10, 2, 10, 2, 4

Offset: 1

Christopher J. Hanson, Jan 21 2014

nonn

P(x) is a function which represents a prime number at a particular ordinal x. This pattern, dp(x), describes the difference between consecutive prime numbers as described by p(x) (see A236175) and therefore the length of dp(x) is len(p(x)) - 1 and each value in dp(x) times P(x) is the difference between values determined not primed when running one pass of a reductive sieve, starting at P(x)^2. See A236185.

Christopher J. Hanson, Table of n, a(n) for n = 1..5759

Cf. A236175-A236180, A236185-A236190.

A236189 Differences between terms of compacting Eratosthenes sieve for prime(8) = 19.

Original entry on oeis.org

4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 2, 6, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 4, 2, 4, 6, 8

Offset: 1

Christopher J. Hanson, Jan 21 2014

nonn

P(x) is a function which represents a prime number at a particular ordinal x. This pattern, dp(x), describes the difference between consecutive prime numbers as described by p(x) (see A236175) and therefore the length of dp(x) is len(p(x)) - 1 and each value in dp(x) times P(x) is the difference between values determined not primed when running one pass of a reductive sieve, starting at P(x)^2. See A236185.

Christopher J. Hanson, Table of n, a(n) for n = 1..92159

Cf. A236175-A236180, A236185-A236190.

cp's OEIS Frontend

A236175 Prime gap pattern of compacting Eratosthenes sieve for prime(4) = 7.

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A236180 Prime gap pattern of compacting Eratosthenes sieve for prime(9) = 23.

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A236185 Differences between terms of compacting Eratosthenes sieve for prime(4) = 7.

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A236177 Prime gap pattern of compacting Eratosthenes sieve for prime(6) = 13.

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A236178 Prime gap pattern of compacting Eratosthenes sieve for prime(7) = 17.

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A236179 Prime gap pattern of compacting Eratosthenes sieve for prime(8) = 19.

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A236176 Prime gap pattern of compacting Eratosthenes sieve for prime(5) = 11.

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A236187 Differences between terms of compacting Eratosthenes sieve for prime(6) = 13.

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A236188 Differences between terms of compacting Eratosthenes sieve for prime(7) = 17.

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A236189 Differences between terms of compacting Eratosthenes sieve for prime(8) = 19.

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