A171431 -id:A171431 - cp's OEIS frontend

A023146 Numbers k such that prime(k) == 4 (mod k).

1, 75, 77, 637331, 637333, 637341, 637343, 27067053, 179992917, 8179002205, 2636913002917, 6201265271239157, 6201265271239347, 6201265271239413, 6201265271239981, 6201265271240331, 6201265271240341, 2159986889494445405, 2159986889494445525, 2159986889494445615

Offset: 1

David W. Wilson

nonn

			The 75th prime is 379 and 379 == 4 (mod 75). Hence 75 is in the sequence.
The 76th prime is 383, but 383 == 3, not 4, (mod 76). So 76 is not in the sequence.

Cf. A171431, A092046.

Cf. A023143, A023144, A023145, A023147, A023148, A023149, A023150, A023151, A023152.

nextPrime[n_] := Block[{k = n + 1}, While[!PrimeQ[k], k++]; k]; p = 1; Do[If[Mod[p = nextPrime[p], n] == 4, Print[n]], {n, 1, 10^9}] (* Robert G. Wilson v, Feb 18 2004 *)
Select[Range[1000], Mod[Prime[#], #] == 4 &] (* Alonso del Arte, Nov 16 2018 *)

def A023146(max) :
    terms = []
    p = 2
    for n in range(1, max+1) :
        if (p - 4) % n == 0 : terms.append(n)
        p = next_prime(p)
    return terms
# Eric M. Schmidt, Feb 05 2013

More terms from Robert G. Wilson v, Feb 18 2004
2 more terms from Giovanni Resta and Robert G. Wilson v, Feb 22 2006
First term inserted by Eric M. Schmidt, Feb 05 2013
a(11)-a(20) from Giovanni Resta, Feb 23 2020

A171432 Primes that are congruent to 5 mod n, where n is the index of the prime.

Original entry on oeis.org

2, 3, 5, 23, 53, 137, 480881, 480941, 480989, 481001, 481469, 3524099, 3524113, 3524281, 3524309, 70117933, 189961799, 189961907, 189961997, 189962177, 189962609, 189963941, 189963959, 189968729, 514274867, 514282961, 3779851157, 10246935913, 10246936309, 10246936463

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 08 2009

nonn

Giovanni Resta, Table of n, a(n) for n = 1..68

Cf. A023147, A048891, A171430, A171431.

Prime@ Select[Range[10^5], Mod[Prime[#] - 5, #] == 0 &] (* Giovanni Resta, Feb 25 2020 *)

a(13) from Harvey P. Dale, Sep 21 2014

Missing a(1)-a(3), a(17) and beyond from Giovanni Resta, Feb 23 2020

A171434 Primes that are congruent to 7 mod n, where n is the index of the prime.

Original entry on oeis.org

2, 3, 7, 13, 61, 151, 397, 1153, 64567, 64577, 480967, 480979, 9559867, 9559897, 9560167, 189961657, 189961819, 189962593, 189963043, 189964321, 189969379, 514272779, 514275401, 1394193607, 1394194727, 1394198807, 10246936003, 75370121191, 75370121767, 75370121839

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 09 2009

nonn

Giovanni Resta, Table of n, a(n) for n = 1..71

Cf. A023149, A048891, A171430, A171431, A171432.

Prime@ Select[Range[10^5], Mod[Prime[#] - 7, #] == 0 &] (* Giovanni Resta, Feb 25 2020 *)

Missing a(1)-a(4), a(16) and beyond from Giovanni Resta, Feb 23 2020

A360789 Least prime p such that p mod primepi(p) = n.

Original entry on oeis.org

2, 3, 5, 7, 379, 23, 401, 61, 59, 29, 67, 71, 467, 79, 83, 179, 431, 89, 176557, 191, 24419, 491, 97, 101, 499, 1213, 3169, 3191, 523, 229, 3187, 223, 3203, 8609, 3163, 251, 176509, 257, 24509, 263, 3253, 269, 547, 3347, 1304867, 293

Offset: 0

Robert G. Wilson v, Feb 20 2023

nonn

Inspired by A048891.

			For n=0, prime p=2 has p mod primepi(p) = 2 mod 1 = 0 so that a(0) = 2.
For n=4, no prime has p mod primepi(p) = 4 until reaching p=379 which is 379 mod 75 = 4, so that a(4) = 379.

Robert G. Wilson v, Table of n, a(n) for n = 0..19217

Cf. A038625, A004648, A048891, A052013, A073325, A073436, A162567, A171430, A171431, A171432, A171434.

V:= Array(0..100): count:= 0:
p:= 1:
for k from 1 while count < 101 do
  p:= nextprime(p);
  v:= p mod k;
  if v <= 100 and V[v] = 0 then V[v]:= p; count:= count+1 fi;
od:
convert(V,list); # Robert Israel, Feb 28 2023

t[_] := 0; p = 2; pi = 1; While[p < 1400000, m = Mod[p, pi]; If[m < 100 && t[m] == 0, t[m] = p]; p = NextPrime@p; pi++]; t /@ Range[0, 99]

a(n)={my(k=n); forprime(p=prime(n+1), oo, k++; if(p%k ==n, return(p)))} \\ Andrew Howroyd, Feb 21 2023

from sympy import prime, nextprime
def A360789(n):
    p, m = prime(n+1), n+1
    while p%m != n:
        p = nextprime(p)
        m += 1
    return p # Chai Wah Wu, Mar 18 2023

a(n) = prime(A073325(n+1)). - Kevin Ryde, Feb 21 2023

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A023146 Numbers k such that prime(k) == 4 (mod k).

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A171432 Primes that are congruent to 5 mod n, where n is the index of the prime.

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A171434 Primes that are congruent to 7 mod n, where n is the index of the prime.

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A360789 Least prime p such that p mod primepi(p) = n.

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