A103689 -id:A103689 - cp's OEIS frontend

A103688 a(n) = n if A103689(n)*n +/- 1 are twin primes; a(n) = 0 otherwise.

0, 0, 0, 4, 0, 6, 0, 0, 9, 0, 0, 12, 0, 0, 15, 0, 0, 18, 0, 0, 21, 0, 0, 0, 0, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 42, 0, 0, 0, 0, 47, 0, 0, 0, 51, 0, 0, 0, 0, 0, 0, 0, 0, 60, 0, 0, 0, 0, 0, 0, 0, 0, 69, 0, 0, 72, 0, 0, 75, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Pierre CAMI, Feb 12 2005

			For n=5 2*5-1=9 is not prime, 2*5+1=11 is prime so a(5)=0.
For n=6 1*6-1=5 is prime, 1*6+1=7 is prime, 5 and 7 are twin primes so a(6)=6.

Cf. A103689.

lktp[n_]:=Module[{k=1},While[NoneTrue[k*n+{1,-1},PrimeQ],k++];If[AllTrue[k*n+{1,-1},PrimeQ],n,0]]; Array[lktp,100] (* The program uses the functions NoneTrue and AllTrue from Mathematica version 10 *) (* Harvey P. Dale, Sep 09 2014 *)

a(n)=k=1;while(!isprime(k*n+1)&&!isprime(k*n-1),k++);k
for(n=1,100,if(isprime(a(n)*n+1)&&isprime(a(n)*n-1),print1(n,", "));print1(0,", ")) \\ Derek Orr, Sep 09 2014

Corrected by Harvey P. Dale, Sep 09 2014

Definition simplified by Derek Orr, Sep 09 2014

A034693 Smallest k such that k*n+1 is prime.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 2, 1, 4, 2, 2, 1, 6, 1, 10, 2, 2, 1, 2, 3, 4, 2, 4, 1, 2, 1, 10, 3, 2, 3, 2, 1, 4, 5, 2, 1, 2, 1, 4, 2, 4, 1, 6, 2, 4, 2, 2, 1, 2, 2, 6, 2, 4, 1, 12, 1, 6, 5, 2, 3, 2, 1, 4, 2, 2, 1, 8, 1, 4, 2, 2, 3, 6, 1, 4, 3, 2, 1, 2, 4, 12, 2, 4, 1, 2, 2, 6, 3, 4, 3, 2, 1, 4, 2

Offset: 1

Labos Elemer

Conjecture: for every n > 1 there exists a number k < n such that n*k + 1 is a prime. - Amarnath Murthy, Apr 17 2001
A stronger conjecture: for every n there exists a number k < 1 + n^(.75) such that n*k + 1 is a prime. I have verified this up to n = 10^6. Also, the expression 1 + n^(.74) does not work as an upper bound (counterexample: n = 19). - Joseph L. Pe, Jul 16 2002
Stronger version of the conjecture verified up to 10^9. - Mauro Fiorentini, Jul 23 2023
It is known that, for almost all n, a(n) <= n^2. From Heath-Brown's result (1992) obtained with help of the GRH, it follows that a(n) <= (phi(n)*log(n))^2. - Vladimir Shevelev, Apr 30 2012
Conjecture: a(n) = O(log(n)*log(log(n))). - Thomas Ordowski, Oct 17 2014
I conjecture the opposite: a(n) / (log n log log n) is unbounded. See A194945 for records in this sequence. - Charles R Greathouse IV, Mar 21 2016

			If n=7, the smallest prime in the sequence 8, 15, 22, 29, ... is 29, so a(7)=4.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, section 2.12, pp. 127-130.
P. Ribenboim, (1989), The Book of Prime Number Records. Chapter 4, Section IV.B.: The Smallest Prime In Arithmetic Progressions, pp. 217-223.

T. D. Noe, Table of n, a(n) for n = 1..10000
Gunther Cornelissen, David Hokken, and Berend Ringeling, The asymptotic Mahler measure of Gaussian periods, arXiv:2507.09303 [math.NT], 2025. See p. 1.
Steven R. Finch, Linnik's Constant
D. Graham, On Linnik's Constant, Acta Arithm., 39, 1981, pp. 163-179.
D. R. Heath-Brown, Zero-free regions for Dirichlet L-functions, and the least prime in an arithmetic progression, Proc. London Math. Soc. 64(3) (1992), pp. 265-338.
Pengcheng Niu and Junli Zhang, On Two Conjectures of A. Murthy, ResearchGate (2024).
Ivan Niven and Barry Powell, Primes in Certain Arithmetic Progressions, Amer. Math. Monthly, 83, 1976, pp. 467-489.
Wikipedia, Dirichlet's theorem on arithmetic progressions

Cf. A010051, A034694, A053989, A071558, A085420, A103689, A194944 (records), A194945 (positions of records), A200996.

a034693 n = head [k | k <- [1..], a010051 (k * n + 1) == 1]
-- Reinhard Zumkeller, Feb 14 2013

A034693 := proc(n)
    for k from 1 do
        if isprime(k*n+1) then
            return k;
        end if;
    end do:
end proc: # R. J. Mathar, Jul 26 2015

a[n_]:=(k=0; While[!PrimeQ[++k*n + 1]]; k); Table[a[n], {n,100}] (* Jean-François Alcover, Jul 19 2011 *)

a(n)=if(n<0,0,s=1; while(isprime(s*n+1)==0,s++); s)

from sympy import isprime
def a(n):
    k = 1
    while not isprime(k*n+1): k += 1
    return k
print([a(n) for n in range(1, 99)]) # Michael S. Branicky, May 05 2022

It seems that Sum_{k=1..n} a(k) is asymptotic to (zeta(2)-1)*n*log(n) where zeta(2)-1 = Pi^2/6-1 = 0.6449... . - Benoit Cloitre, Aug 11 2002

a(n) = (A034694(n)-1) / n. - Joerg Arndt, Oct 18 2020

A053989 Smallest k such that nk-1 is prime.

Original entry on oeis.org

3, 2, 1, 1, 4, 1, 2, 1, 2, 2, 4, 1, 8, 1, 2, 2, 4, 1, 2, 1, 2, 2, 6, 1, 6, 4, 2, 3, 6, 1, 2, 1, 4, 2, 4, 2, 2, 1, 6, 2, 4, 1, 6, 1, 2, 3, 6, 1, 2, 3, 2, 2, 4, 1, 2, 3, 2, 3, 6, 1, 8, 1, 4, 2, 6, 2, 6, 1, 2, 2, 4, 1, 14, 1, 2, 2, 4, 3, 2, 1, 8, 2, 4, 1, 6, 3, 2, 3, 16, 1, 2, 4, 6, 3, 4, 2, 2, 1, 2, 2

Offset: 1

Henry Bottomley, Apr 04 2000

			a(5)=4 because the smallest prime in the sequence 5k-1 (4,9,14,19,24...) is 19 when k=4

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

Cf. A034693, A038700, A071558, A010051, A103689, A200996.

a053989 n = head [k | k <- [1..], a010051' (k * n - 1) == 1]
-- Reinhard Zumkeller, Feb 14 2013

a(n)=my(j);while(!isprime(j++*n-1),);j \\ Charles R Greathouse IV, Apr 18 2013

a(n) = (A038700(n)+1)/n.

A200996 Least upper limit for numbers u and v such that un-1 and vn+1 are both prime, u and v not necessarily distinct.

Original entry on oeis.org

3, 2, 2, 1, 4, 1, 4, 2, 2, 2, 4, 1, 8, 2, 2, 2, 6, 1, 10, 2, 2, 2, 6, 3, 6, 4, 4, 3, 6, 1, 10, 3, 4, 3, 4, 2, 4, 5, 6, 2, 4, 1, 6, 2, 4, 3, 6, 2, 4, 3, 2, 2, 4, 2, 6, 3, 4, 3, 12, 1, 8, 5, 4, 3, 6, 2, 6, 2, 2, 2, 8, 1, 14, 2, 2, 3, 6, 3, 4, 3, 8, 2, 4, 4, 12

Offset: 1

Reinhard Zumkeller, Feb 14 2013

nonn

A103689(n) <= a(n) <= A071558(n).

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

Haskell
```
a200996 n = max (a053989 n) (a034693 n)
```

a(n) = max {A053989(n), A034693(n)}.

A348347 Smallest k such that in the pairs of numbers j*k +- 1, none is prime for 1 <= j < n but at least one is prime for j = n; or 0 if no such k exists.

Original entry on oeis.org

1, 5, 92, 13, 208, 47, 512, 149, 1688, 145, 6686, 539, 4106, 757, 9970, 1217, 16012, 881, 56194, 2441, 53576, 3343, 111992, 2917, 152734, 2053, 49376, 6791, 839522, 4985, 114118, 30097, 567302, 17209, 493618, 33613, 991976, 28097, 758932, 91099, 1898368, 36271

Offset: 1

Pontus von Brömssen, Oct 13 2021

nonn

Smallest k such that A103689(k) = n.

a(85) > 10^9 (unless a(85) = 0).

			a(3) = 92 because none of 92 +- 1 and 2*92 +- 1 are prime but 3*92 + 1 is prime; and for k < 92, either 3*k +- 1 are also both not prime, or some j*k +- 1 is prime for j < 3.

Pontus von Brömssen, Table of n, a(n) for n = 1..84

Cf. A103689.

f(n) = my(k=1); while (!isprime(k*n+1) && !isprime(k*n-1), k++); k; \\ A103689
a(n) = my(k=1); while (f(k) != n, k++); k; \\ Michel Marcus, Oct 18 2021

from sympy import isprime
def A348347(n):
    k = 1
    while 1:
        m = 1
        while m <= n and not (isprime(m*k-1) or isprime(m*k+1)): m += 1
        if m == n: return k
        k += 1

cp's OEIS Frontend

A103688 a(n) = n if A103689(n)*n +/- 1 are twin primes; a(n) = 0 otherwise.

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A034693 Smallest k such that k*n+1 is prime.

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A053989 Smallest k such that nk-1 is prime.

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A200996 Least upper limit for numbers u and v such that u*n-1 and v*n+1 are both prime, u and v not necessarily distinct.

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Haskell

Formula

A348347 Smallest k such that in the pairs of numbers j*k +- 1, none is prime for 1 <= j < n but at least one is prime for j = n; or 0 if no such k exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

A200996 Least upper limit for numbers u and v such that un-1 and vn+1 are both prime, u and v not necessarily distinct.