A088732 -id:A088732 - cp's OEIS frontend

A363997 Position in A088732 of the n-th prime.

1, 2, 3, 4, 6, 7, 9, 5, 10, 8, 12, 15, 16, 19, 14, 21, 11, 22, 24, 18, 27, 20, 30, 31, 17, 34, 36, 37, 40, 28, 42, 45, 49, 51, 13, 26, 52, 54, 55, 38, 57, 32, 64, 33, 44, 66, 23, 46, 69, 35, 70, 25, 50, 75, 76, 79, 41, 82, 56, 84, 29, 58, 87, 60, 90, 91, 48

Offset: 1

Clark Kimberling, Jul 11 2023

nonn

Every positive integer occurs exactly once.

			a(7) = 5 because the 7th prime, 19, is the 5th term in A088732.

Cf. A000040, A088732, A038780, A363998.

z= 200; t = Table[k = 1; While[p = n + k*(n + 1); ! PrimeQ[p], k++];
  p, {n, 0, z}];   (* A088732, after Frank M Jackson *)
Flatten[Table[Position[t, Prime[n]], {n, 1, z}]]  (* this sequence *)

A032448 Smallest prime == -1 modulo prime(n).

Original entry on oeis.org

3, 2, 19, 13, 43, 103, 67, 37, 137, 173, 61, 73, 163, 257, 281, 211, 353, 487, 401, 283, 1021, 157, 331, 1423, 193, 1009, 617, 641, 653, 677, 761, 523, 547, 277, 1489, 1811, 313, 977, 1669, 691, 1789, 1447, 4201, 1543, 787, 397, 421, 1783, 907, 457

Offset: 1

Reinhard Zumkeller, Jun 25 2003

nonn

It appears that a(n) <= prime(n)^2-1, where prime(n) = A000040(n) is the n-th prime; see A035095 for a related conjecture. If correct, this implies A006530(a(n)+1)=prime(n), which in turn implies that there are no duplicated values in the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A035095, A088732, A006530, A000040.

a032448 n = head [q | q <- a000040_list, let p = a000040 n,
                      q `mod` p == p - 1]
-- Reinhard Zumkeller, Aug 08 2013

f[n_] := Block[{p = Prime@ n}, r = p - 1; While[ !PrimeQ@ r, r += p]; r]; Array[f, 50] (* Robert G. Wilson v, Jun 20 2014 *)

a(n) = {prn = prime(n); p = 2; while(p % prn != prn - 1, p = nextprime(p+1)); p;} \\ Michel Marcus, Aug 04 2013

Edited by Franklin T. Adams-Watters, Jun 21 2010

A060940 Triangle in which n-th row gives the phi(n) terms appearing as initial primes in arithmetic progressions with difference n, with initial term equal to the smallest positive residue coprimes to n.

Original entry on oeis.org

2, 3, 7, 5, 5, 7, 11, 7, 13, 19, 7, 11, 29, 23, 17, 11, 19, 13, 17, 11, 13, 23, 19, 11, 13, 23, 43, 17, 11, 13, 17, 19, 23, 13, 47, 37, 71, 17, 29, 19, 31, 43, 13, 17, 19, 23, 53, 41, 29, 17, 31, 19, 59, 47, 61, 23, 37, 103, 29, 17, 19, 23, 53, 41, 31, 17, 19, 37, 23, 41, 43, 29

Offset: 1

Labos Elemer, May 07 2001

			For differences 1, 2, 3, 4, 5, 6, 7, .. the initial primes are 2; 3; 7, 5; 5, 7; 11, 7, 13, 19; 7, 11; 29, 23, 17, 11, 19, 13; ... etc. Suitable initial values (coprimes to difference) are in A038566. Position of end(start) of rows is given by values of A002088.
From _Seiichi Manyama_, Apr 02 2018: (Start)
   n | phi(n)|
  ---+-------+------------------------
   1 |   1   |  2;
   2 |   1   |  3;
   3 |   2   |  7,  5;
   4 |   2   |  5,  7;
   5 |   4   | 11,  7, 13, 19;
   6 |   2   |  7, 11;
   7 |   6   | 29, 23, 17, 11, 19, 13;
   8 |   4   | 17, 11, 13, 23;
   9 |   6   | 19, 11, 13, 23, 43, 17;
  10 |   4   | 11, 13, 17, 19;         (End)

Seiichi Manyama, Rows n = 1..200, flattened
Eric Weisstein's MathWorld, Dirichlet's theorem
Wikipedia, Dirichlet's theorem on arithmetic progressions

Cf. A000010 (phi), A002088, A038566, A034693, A034694, A002144, A007519, A088732, etc..

A116934 First prime in the arithmetic progression (n+k*p: k>0), where p is the smallest coprime greater than n+1.

Original entry on oeis.org

7, 7, 13, 11, 19, 17, 43, 19, 31, 23, 37, 29, 43, 31, 83, 73, 131, 41, 61, 43, 67, 47, 73, 53, 79, 113, 317, 59, 277, 67, 97, 67, 103, 71, 109, 241, 193, 79, 367, 83, 127, 89, 223, 373, 139, 193, 439, 101, 151, 103, 157, 107, 163, 113, 283, 233, 293, 241, 181, 127, 313

Offset: 1

Reinhard Zumkeller, Feb 27 2006

nonn

a(n) = n + A116933(n)*A079578(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primes in arithmetic progressions
Eric Weisstein's World of Mathematics, Dirichlet's Theorem.

Cf. A088732.

Cf. A010051, A079578, A116933.

a116934 n = head [q | k <- [1..], let q = n + k * a079578 n,
                      a010051' q == 1]
-- Reinhard Zumkeller, Oct 01 2014

A088733 n-th prime in the arithmetic progression n+k*(n+1) with k>0.

Original entry on oeis.org

3, 11, 19, 79, 41, 167, 127, 233, 179, 461, 227, 883, 433, 569, 719, 1801, 593, 1861, 859, 1553, 1319, 3863, 1103, 3499, 2027, 3671, 2239, 6089, 1499, 6323, 3583, 5147, 3739, 5879, 2843, 11173, 4597, 7253, 4799, 14923, 3779, 14533, 6599, 7919, 7589

Offset: 1

Reinhard Zumkeller, Oct 12 2003

nonn

			n=4, the progression starts: 4, 9, 14, 19, 24, 29, 34, 39, 44, 49,
54, 59, 64, 69, 74, 79, 84, 89, etc., with primes 19, 29, 59, 79, 89, etc.,
79 is the fourth prime: a(4)=79.

T. D. Noe, Table of n, a(n) for n = 1..10000
T. D. Noe, Linear Plot
Eric Weisstein's World of Mathematics, Dirichlet's Theorem
Index entries for sequences related to primes in arithmetic progressions

Cf. A088732.

Cf. A010051.

a088733 n = last $ take n $
            [q | q <- [2 * n + 1, 3 * n + 2 ..], a010051' q == 1]
-- Reinhard Zumkeller, Oct 01 2014

Table[k = 1; Do[While[p = n + k*(n + 1); ! PrimeQ[p], k++]; k++, {n}]; p, {n, 100}] (* T. D. Noe, Oct 20 2011 *)

cp's OEIS Frontend

A363997 Position in A088732 of the n-th prime.

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A032448 Smallest prime == -1 modulo prime(n).

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A060940 Triangle in which n-th row gives the phi(n) terms appearing as initial primes in arithmetic progressions with difference n, with initial term equal to the smallest positive residue coprimes to n.

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A116934 First prime in the arithmetic progression (n+k*p: k>0), where p is the smallest coprime greater than n+1.

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Haskell

A088733 n-th prime in the arithmetic progression n+k*(n+1) with k>0.

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Haskell

Mathematica