A034693 -id:A034693 - cp's OEIS frontend

A047982 a(n) = A047980(2n+1).

1, 24, 38, 184, 368, 668, 634, 512, 1028, 1468, 3382, 4106, 10012, 7628, 11282, 38032, 53630, 37274, 63334, 34108, 102296, 119074, 109474, 117206, 60664, 410942, 204614, 127942, 125618, 595358, 517882, 304702, 352022, 1549498, 651034, 506732, 5573116, 1379216, 1763144

Offset: 0

Labos Elemer

nonn

			a(2)=38 because A034693(38) = 2*2+1 = 5 is the first 5; 5*38+1 = 191 is the first prime. The successive progressions in which the first prime appears at position 5 are as follows: 38k+1, 62k+1, 164k+1. 2nd example: a(20)=102296 because. The first 41 appears in A034693 at this index. Also 102296*(2*20+1)+1 = 102296*41+1 = 4194137 is the first prime in {102296k+1}. The next progression with this position of prime emergence is 109946k+1 (the corresponding prime is 4507787).

Cf. A047980, A047981, A034782, A034783, A034784.

a(n) = min {d}: A034693(a(n)) is an odd number k such that in a(n)*k+1 progression the first prime occurs at k=2n+1 position.

More terms from Michel Marcus, Sep 01 2019

A072064 Least k>0 such that prime(n)+k*n is prime.

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 2, 3, 2, 3, 2, 2, 2, 2, 4, 3, 4, 1, 6, 3, 4, 1, 10, 1, 4, 1, 2, 2, 2, 2, 4, 1, 4, 1, 6, 2, 6, 2, 6, 3, 24, 1, 2, 2, 6, 3, 8, 1, 6, 3, 8, 5, 2, 2, 2, 3, 2, 4, 6, 2, 16, 3, 2, 2, 2, 1, 4, 3, 6, 1, 10, 1, 4, 2, 6, 6, 16, 3, 8, 2, 4, 1, 6, 2, 10, 3, 4, 4, 18, 2, 6, 1, 2

Offset: 1

Reinhard Zumkeller, Jun 12 2002

nonn

			n=3, prime(3)=5: 5+1*3=8 is not prime, but 5+2*3=11, therefore a(3)=2 and A072063(3)=11.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A034693, A000040, A072063.

A072064[n_]:=Module[{p=Prime[n],k=1},While[!PrimeQ[p+k*n],k++];k];Array[A072064,100] (* Paolo Xausa, Nov 27 2023 *)

a(n) = my(p=prime(n), k=1); while (!isprime(p+k*n), k++); k; \\ Michel Marcus, Nov 27 2023

A087376 Leading diagonal of A087377.

Original entry on oeis.org

1, 2, 6, 7, 14, 7, 30, 24, 30, 19, 60, 23, 72, 33, 46, 48, 90, 34, 124, 51, 82, 64, 144, 67, 148, 78, 118, 87, 240, 58, 232, 113, 150, 114, 200, 88, 288, 165, 202, 138, 332, 91, 352, 167, 202, 163, 384, 136, 372, 143, 258, 195, 500, 142, 360, 207, 318, 226, 552, 137

Offset: 1

Amarnath Murthy, Sep 09 2003

nonn

Ivan Neretin, Table of n, a(n) for n = 1..1000

Cf. A034693, A087377.

f = Function[n, k = cnt = 0; While[cnt < n, k++; If[PrimeQ[k n + 1], cnt++]]; k]; Table[f[n], {n, 60}] (* Ivan Neretin, May 22 2015 *)

Conjecture: a(n) < n^2.

More terms from David Wasserman, May 24 2005

A087377 Triangle read by rows: n-th row contains n smallest numbers k (say) such that nk+1 is a prime.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 1, 3, 4, 7, 2, 6, 8, 12, 14, 1, 2, 3, 5, 6, 7, 4, 6, 10, 16, 18, 28, 30, 2, 5, 9, 11, 12, 14, 17, 24, 2, 4, 8, 12, 14, 18, 20, 22, 30, 1, 3, 4, 6, 7, 10, 13, 15, 18, 19, 2, 6, 8, 18, 30, 32, 36, 38, 42, 56, 60, 1, 3, 5, 6, 8, 9, 13, 15, 16, 19, 20, 23, 4, 6, 10, 12, 24, 34

Offset: 1

Amarnath Murthy, Sep 09 2003

From Robert Israel, May 22 2015: (Start)
If n is odd, all elements in row n are even.
By Dirichlet's theorem, every positive integer occurs infinitely often in the sequence. (End)

			1
1 2
2 4 6
1 3 4 7
2 6 8 12 14
1 2 3 5 6 7
4 6 10 16 18 28 30

Ivan Neretin, Table of n, a(n) for n = 1..5050

The first column is given by A034693.

Cf. A034693, A087376.

F:= proc(n) local res,count,k;
   res:= NULL:
   count:= 0:
   for k from 1 while count < n do
      if isprime(n*k+1) then
        res:= res,k;
        count:= count+1
      fi
   od;
   res
end proc:
seq(F(n),n=1..20); # Robert Israel, May 22 2015

a = {}; Do[k = 0; ar = {}; While[Length[ar] < n, k++; If[PrimeQ[k n + 1], AppendTo[ar, k]]]; a = Join[a, ar], {n, 13}]; a (* Ivan Neretin, May 22 2015 *)

tabl(nn) = {for (n = 1, nn, j = 0; for (k = 1, n, j++; while (!isprime(n*j+1), j++); print1(j, ", ");); print(););} \\ Michel Marcus, May 23 2015

More terms from Sam Alexander, Feb 27 2004

A087952 Smallest prime == 1 (mod n) and > n^2.

Original entry on oeis.org

2, 5, 13, 17, 31, 37, 71, 73, 109, 101, 199, 157, 313, 197, 241, 257, 307, 379, 419, 401, 463, 617, 599, 577, 701, 677, 757, 953, 929, 991, 1117, 1153, 1123, 1259, 1471, 1297, 1481, 1483, 1873, 1601, 1723, 1933, 1979, 2069, 2161, 2347, 2351, 2593, 2549, 2551

Offset: 1

Ray Chandler, Sep 16 2003

nonn

Primes arising in A087554.

Since A014085(n) ~ n/log(n) one may conjecture that a(n) < 2*n^2 for all n > 1. Numerically we find a(n) = n^2*(1 + O(1/sqrt(n))). - M. F. Hasler, Feb 27 2020

			For n=1, a(1) = 2, because 2 == 1 mod 1 and 2 > 1^2.
For n=2, a(2) = 5, because 5 == 1 mod 2 and 5 > 2^2.

M. F. Hasler, Table of n, a(n) for n = 1..10000, Feb 27 2020

Cf. A034693, A034694, A087554.

Cf. A014085 (number of primes between n^2 and (n+1)^2).

spr[n_]:=Module[{p=NextPrime[n^2]},While[Mod[p,n]!=1,p=NextPrime[p]];p]; Join[ {2},Array[spr,50,2]] (* Harvey P. Dale, Jun 21 2021 *)

apply( {A087952(n)=forprime(p=n^2+1,,(p-1)%n||return(p))}, [1..66]) \\ M. F. Hasler, Feb 27 2020

Examples added by N. J. A. Sloane, Jun 21 2021

A231818 Least positive k such that k*n^n - 1 is a prime, or 0 if no such k exists.

Original entry on oeis.org

3, 1, 2, 5, 6, 3, 6, 39, 18, 6, 12, 19, 8, 23, 10, 3, 76, 13, 90, 26, 52, 45, 124, 12, 60, 27, 10, 99, 126, 11, 50, 27, 28, 59, 6, 80, 122, 71, 110, 21, 72, 111, 590, 147, 178, 84, 238, 12, 138, 236, 10, 53, 6, 60, 98, 72, 620, 30, 166, 5, 98, 18, 22, 384, 126

Offset: 1

Alex Ratushnyak, Nov 13 2013

nonn

Cf. A035092 (least k such that k*(n^2)+1 is a prime).
Cf. A175763 (least k such that k*(n^n)+1 is a prime).
Cf. A035093 (least k such that k*n!+1    is a prime).
Cf. A193807 (least k such that n*(k^2)+1 is a prime).
Cf. A231119 (least k such that n*(k^k)+1 is a prime).
Cf. A057217 (least k such that n*k!+1    is a prime).
Cf. A034693 (least k such that n*k +1    is a prime).
Cf. A231819 (least k such that k*(n^2)-1 is a prime).
Cf. A083663 (least k such that k*n!-1    is a prime).
Cf. A231734 (least k such that n*(k^2)-1 is a prime).
Cf. A231735 (least k such that n*(k^k)-1 is a prime).
Cf. A231820 (least k such that n*k!-1    is a prime).
Cf. A053989 (least k such that n*k -1    is a prime).

Table[k = 1; While[! PrimeQ[k*n^n - 1], k++]; k, {n, 65}] (* T. D. Noe, Nov 15 2013 *)

A238847 Smallest k such that k*n^3 + 1 is prime.

Original entry on oeis.org

1, 2, 4, 3, 2, 2, 4, 15, 2, 3, 2, 2, 6, 3, 10, 3, 26, 3, 4, 2, 2, 15, 26, 7, 4, 2, 2, 6, 2, 2, 10, 2, 20, 4, 2, 3, 4, 3, 4, 6, 6, 4, 10, 2, 14, 16, 12, 3, 4, 9, 10, 6, 24, 3, 4, 6, 2, 3, 2, 2, 18, 6, 6, 3, 14, 5, 16, 9, 18, 3, 2, 2, 4, 3, 10, 6

Offset: 1

Derek Orr, Mar 06 2014

nonn

			a(1) = 1 because in order for k*(1^3)+1 to be the smallest prime, k must be 1 (1*(1^3)+1 = 2).
a(2) = 2 because in order for k*(2^3)+1 to be the smallest prime, k must be 2 (2*(2^3)+1 = 17).
a(3) = 4 because in order for k*(3^3)+1 to be the smallest prime, k must be 4 (4*(3^3)+1 = 109).

Cf. A034693, A035092.

sk[n_]:=Module[{k=1,n3=n^3},While[!PrimeQ[k*n3+1],k++];k]; Array[sk, 80] (* Harvey P. Dale, Aug 27 2014 *)
Table[SelectFirst[Range[10^2], PrimeQ[# n^3 + 1] &], {n, 76}] (* Michael De Vlieger, Mar 27 2016, Version 10 *)

a(n) = {k=1; while(!isprime(k*n^3+1), k++); k;} \\ Altug Alkan, Mar 26 2016

import sympy
from sympy import isprime
def f(n):
  for k in range(1,10**3):
    if isprime(k*(n**3)+1):
      return k
n = 1
while n < 10**3:
  print(f(n))
  n += 1

A362376 a(n) is the least k such that Fibonacci(n)*Fibonacci(k) + 1 is a prime, and -1 if no such k exists.

Original entry on oeis.org

1, 1, 1, 3, 3, 3, 9, 3, 4, 9, 3, 4, 3, 27, 4, 24, 24, 4, 3, 6, 3, 3, 444, 3, 12, 9, 3, 63, 6, 8, 36, 6, 36, 12, 12, 4, 21, 60, 4, 3, 24, 73, 51, 3, 11, 51, 12, 4, 504, 12, 3, 33, 21, 6, 9, 6, 4, 384, 21, 7, 54, 3, 4, 51, 24, 63, 30, 24, 11, 45, 72, 6, 39, 9, 22, 42, 12, 16, 60, 30

Offset: 1

Jack Braxton, Apr 17 2023

nonn

The frequencies seem interesting. In the early terms, 5 appears notably rarely, i.e., not until at a(240), whereas several other numbers appear notably frequently, e.g., 24 appears 13 times before a(240). - Peter Munn, May 03 2023

			For n=4, Fibonacci(4)=3 and 3*Fibonacci(k)+1 is not prime until k reaches 3, so a(4)=3.

Chai Wah Wu, Table of n, a(n) for n = 1..448

Cf. A000040, A000045, A034693, A363533.

Table[m = Fibonacci[n]; k = 1; While[! PrimeQ[m*Fibonacci[k] + 1], k++]; k, {n, 120}] (* Michael De Vlieger, May 03 2023 *)

a(n) = my(F=fibonacci(n), k=1); while (!ispseudoprime(F*fibonacci(k) + 1), k++); k; \\ Michel Marcus, Apr 18 2023

from itertools import count
from sympy import fibonacci, isprime
def A362376(n):
    a = b = fibonacci(n)
    for k in count(1):
        if isprime(a+1):
            return k
        a, b = b, a+b # Chai Wah Wu, May 03 2023

a(n) = A363533(A000045(n)). - Pontus von Brömssen, Jun 20 2023

More terms from Michel Marcus, Apr 18 2023

A034846 a(n) = P(n,6) = 1+6K(n,6)=1+6A034783(n). P(n,6) are special primes of 6k+1. The relevant values of k are given by A034783.

Original entry on oeis.org

103, 283, 331, 367, 463, 547, 607, 619, 643, 709, 727, 739, 823, 859, 883, 907, 967, 1021, 1087, 1123, 1171, 1249, 1303, 1423, 1447, 1483, 1489, 1543, 1579, 1597, 1627, 1699, 1723, 1747, 1783

Offset: 1

Labos Elemer

nonn

Cf. A034693, A034694, A034783.

A034847 a(n) = 1 + 4*A034780(n).

Original entry on oeis.org

29, 53, 101, 109, 149, 173, 181, 197, 229, 269, 293, 317, 337, 349, 373, 389, 461, 509, 557, 569, 641, 653, 677, 701, 709, 773, 797, 821, 829, 853, 937, 941, 1013, 1021, 1033, 1061, 1069, 1109, 1117, 1181, 1193, 1217, 1229, 1277, 1297, 1301, 1373, 1429, 1481, 1493, 1549, 1597

Offset: 1

Labos Elemer

nonn

a(n) = P(n,4) = 1 + 4*K(n,4) = 1 + 4*A034780(n). P(n,4) are special primes of the form 4k+1. The relevant values of k are given by A034780.

Note that, e.g., 5 and 13 are not in this sequence.

Cf. A034693, A034694, A034780.

a034693(n) = my(s=1); while(!isprime(s*n+1), s++); s;
isok(n) = a034693(n) == 4;
lista(nn) = {for (n=1, nn, if (isok(n), print1(4*n+1, ", ")););} \\ Michel Marcus, May 13 2018

More terms from Michel Marcus, May 13 2018

cp's OEIS Frontend

A047982 a(n) = A047980(2n+1).

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A072064 Least k>0 such that prime(n)+k*n is prime.

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Mathematica

PARI

A087376 Leading diagonal of A087377.

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A087377 Triangle read by rows: n-th row contains n smallest numbers k (say) such that nk+1 is a prime.

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Maple

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A087952 Smallest prime == 1 (mod n) and > n^2.

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Mathematica

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A231818 Least positive k such that k*n^n - 1 is a prime, or 0 if no such k exists.

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Mathematica

A238847 Smallest k such that k*n^3 + 1 is prime.

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Mathematica

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Python

A362376 a(n) is the least k such that Fibonacci(n)*Fibonacci(k) + 1 is a prime, and -1 if no such k exists.

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A034846 a(n) = P(n,6) = 1+6K(n,6)=1+6A034783(n). P(n,6) are special primes of 6k+1. The relevant values of k are given by A034783.