A060461 -id:A060461 - cp's OEIS frontend

A161996 A (negated) characteristic function of twin composite odd numbers.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1

Offset: 1

Pierre CAMI, Jun 24 2009

a(n) = 0 if n is a member of A060461, otherwise a(n) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..37317
Index entries for characteristic functions

Cf. A060461, A100923.

If[AllTrue[6#+{1,-1},CompositeQ],0,1]&/@Range[300] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 15 2015 *)

A161996(n) = (isprime((6*n)-1)||isprime((6*n)+1)); \\ Antti Karttunen, Dec 15 2017

Definition rephrased by R. J. Mathar, Aug 14 2009

A345124 a(n) is the smallest k such that f(k) is composite for all m-fold compositions f of the functions 6*x +- 1, 1 <= m <= n.

Original entry on oeis.org

20, 50, 284, 1868, 47951, 6245927, 15932178151

Offset: 1

Marc Morgenegg, Oct 06 2021

Proof that a(n) exists for all n: The numbers that are equal to f(k) for some m-fold composition f of the functions 6*x +- 1 can be written as 6^m*k +- c, where c is in the set C_m, defined by C_1 = {1} and C_{m+1} = {6*c +- 1 for c in C_m}. Choose a positive integer k_0 that is divisible by all numbers in C_m for 1 <= m <= n. Then 6^m*k_0 +- c is divisible by (and greater than) c, so it is composite if c > 1. (In fact, the largest number in C_m is A003464(m).) Since there are arbitrarily long prime gaps, we can choose a positive integer r such that 6*k_0*r +- 1 are both composite. With k = k_0*r, the numbers 6^m*k +- c will all be composite for c in C_m, 1 <= m <= n, as desired. - Pontus von Brömssen, Nov 01 2021

			Formula for the twin composites by iteration n:
n=1: 6*k+-1.
n=2: 6*(6*k+-1)+-1.
n=3: 6*(6*(6*k+-1)+-1)+-1.
Term a(n) example for smallest number k for iteration n:
a(1)=20, 6*20-1=119, 6*20+1=121, all {119,121} are composite numbers.
a(2)=50, 6*50-1=299, 6*50+1=301, 6*(6*50-1)-1=1793, 6*(6*50-1)+1=1795, 6*(6*50+1)-1=1805, 6*(6*50+1)+1=1807, all {299,301,1793,1795,1805,1807} are composite numbers.

Cf. A003464, A060461 (numbers k such that 6*k+-1 are twin composites).

a[n_] := Module[{k = 1}, While[!AllTrue[Flatten@ Rest@ NestList[Flatten@ Join[{6*# - 1, 6*# + 1}] &, k, n], CompositeQ], k++]; k]; Array[a, 5] (* Amiram Eldar, Oct 25 2021 *)

from sympy import isprime
def A345124(n):
    C = [[1]]
    for i in range(n-1):
        C.append(sum(([6*c-1,6*c+1] for c in C[-1]),[]))
    k = 1
    while 1:
        k6 = 6*k
        for i in range(n):
            if any(isprime(k6-c) or isprime(k6+c) for c in C[i]):
                break
            k6 *= 6
        else:
            return k
        k += 1 # Pontus von Brömssen, Nov 01 2021

More terms from Pontus von Brömssen, Oct 06 2021
Name edited by Pontus von Brömssen, Nov 01 2021
a(7) from Martin Ehrenstein, Nov 13 2021

A217707 Numbers n such that both 4n-1 and 4n+1 are composite.

Original entry on oeis.org

14, 16, 19, 23, 29, 30, 31, 36, 40, 44, 46, 47, 51, 52, 54, 55, 59, 61, 62, 65, 72, 74, 75, 76, 80, 81, 82, 85, 86, 89, 91, 94, 98, 101, 103, 104, 106, 107, 109, 113, 118, 119, 121, 124, 128, 129, 132, 133, 134, 136, 138, 140, 145, 146, 149, 151, 156, 157, 159

Offset: 1

Jayanta Basu, Mar 20 2013

Cf. A060461, A045751.

Select[Range[200], ! PrimeQ[4 # - 1] && ! PrimeQ[4 # + 1] &]
Select[Range[200],AllTrue[4#+{1,-1},CompositeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 11 2015 *)

A377540 Numbers k such that at least one of the numbers 6k-1 or 6k+1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 28, 29, 30, 32, 33, 35, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 87

Offset: 1

Michel Eduardo Beleza Yamagishi, Oct 31 2024

Union of A024898 and A024899.
Cf. A002476, A007528.
Complement of A060461 (with respect to the positive integers) or A171696 (with respect to the nonnegative integers).

Select[Range[100], PrimeQ[6 # - 1] || PrimeQ[6 # + 1] &]

isok(k) = isprime(6*k-1) || isprime(6*k+1); \\ Michel Marcus, Oct 31 2024

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A161996 A (negated) characteristic function of twin composite odd numbers.

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A345124 a(n) is the smallest k such that f(k) is composite for all m-fold compositions f of the functions 6*x +- 1, 1 <= m <= n.

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A217707 Numbers n such that both 4n-1 and 4n+1 are composite.

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A377540 Numbers k such that at least one of the numbers 6k-1 or 6k+1 is prime.

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PARI

cp's OEIS Frontend

A161996 A (negated) characteristic function of twin composite odd numbers.

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Mathematica

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A345124 a(n) is the smallest k such that f(k) is composite for all m-fold compositions f of the functions 6*x +- 1, 1 <= m <= n.

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Author

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Comments

Examples

Crossrefs

Programs

Mathematica

Python

Extensions

A217707 Numbers n such that both 4*n-1 and 4*n+1 are composite.

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Author

Keywords

Crossrefs

Programs

Mathematica

A377540 Numbers k such that at least one of the numbers 6k-1 or 6k+1 is prime.

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Author

Keywords

Crossrefs

Programs

Mathematica

PARI

A217707 Numbers n such that both 4n-1 and 4n+1 are composite.