A052013 -id:A052013 - cp's OEIS frontend

A023143 Numbers k such that prime(k) == 1 (mod k).

1, 2, 5, 6, 12, 14, 181, 6459, 6460, 6466, 100362, 251712, 251732, 637236, 10553504, 10553505, 10553547, 10553827, 10553851, 10553852, 69709709, 69709724, 69709728, 69709869, 69709961, 69709962, 179992920, 179992922, 179993170, 465769815, 465769819, 465769840, 3140421737, 3140421744, 3140421767, 3140421892, 3140421935

Offset: 1

David W. Wilson and G. L. Honaker, Jr., Jun 14 1998

A004648(a(n)) <= 1. - Reinhard Zumkeller, Jul 30 2012

			6 is in the sequence because the 6th prime, 13, is congruent to 1 (mod 6).

Giovanni Resta, Table of n, a(n) for n = 1..94

Cf. A048891, A045924, A052013, A023144, A023145, A023146, A023147, A023148, A023149, A023150, A023151, A023152.

import Data.List (elemIndices)
a023143 n = a023143_list !! (n-1)
a023143_list = 1 : map (+ 1) (elemIndices 1 a004648_list)
-- Reinhard Zumkeller, Jul 30 2012, Jun 08 2011

[n: n in [1..10000] | IsIntegral((NthPrime(n)-1)/n)]; // Marius A. Burtea, Dec 30 2018

Do[ If[ IntegerQ[ (Prime[ n ] - 1) / n ], Print[ n ] ], {n, 1, 10^8} ]

n=0; print1(1); forprime(p=2,1e9, if(p%n++==1, print1(", "n))) \\ Charles R Greathouse IV, Apr 28 2015

def A023143(end):
    primes=[2,3]
    a023143_list=[1]
    num=3
    while len(primes)<=end:
        num+=1
        prime=False
        length=len(primes)
        for y in range(0,length):
            if num % primes[y]!=0:
                prime=True
            else:
                prime=False
                break
        if (prime):
            primes.append(num)
    for x in range(2, len(primes)):
        if (primes[x-1]%(x))==1:
            a023143_list.append(x)
    return a023143_list
# Conner L. Delahanty, Apr 19 2014

from sympy import primerange
def A023143(end): return [n+1 for n, p in enumerate(primerange(2, end)) if (p-1) % (n-1) == 0] # David Radcliffe, Jun 27 2016

More terms from Jud McCranie, Dec 11 1999
a(30)-a(37) from Zak Seidov, Apr 19 2014
Terms a(33)-a(37) sorted in correct order by Giovanni Resta, Feb 23 2020

A045924 Numbers n such that prime(n) == -1 (mod n).

Original entry on oeis.org

1, 2, 3, 4, 10, 70, 72, 182, 440, 1053, 6458, 6461, 6471, 40087, 40089, 251737, 251742, 637320, 637334, 637336, 1617173, 4124466, 10553445, 10553455, 10553569, 10553570, 10553574, 10553576, 10553819, 10553829, 27067100, 27067262, 69709705, 69709719, 69709734, 69709873

Offset: 1

Len Smiley

nonn

Same as n such that n divides A008864(n). - David James Sycamore, Jul 23 2018

Also numbers n such that prime(n) == n-1 (mod n). - Muniru A Asiru, Jul 24 2018

			10 is a member because the 10th prime, 29, is congruent to -1 mod 10.

Giovanni Resta, Table of n, a(n) for n = 1..108
E. Labos, Graph of prime(n) mod n

Cf. A004648 and esp. A023143.

Cf. A052013, A048891, A092044, A092045, A092046, A092047, A092048, A092049, A092050, A092051, A092052, A008864.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 1; Do[ If[Mod[p = NextPrim[p], n] == n - 1, Print[n]], {n, 1, 10^9}] (* Robert G. Wilson v, Feb 18 2004 *)

isok(n) = Mod(prime(n), n) == -1; \\ Michel Marcus, Jul 24 2018

More terms from Patrick De Geest, Nov 15 1999

Terms a(33) and beyond from Giovanni Resta, Feb 23 2020

A048891 Primes that are congruent to 1 mod n, where n is the index of the prime.

Original entry on oeis.org

2, 3, 11, 13, 37, 43, 1087, 64591, 64601, 64661, 1304707, 3523969, 3524249, 9558541, 189963073, 189963091, 189963847, 189968887, 189969319, 189969337, 1394194181, 1394194481, 1394194561, 1394197381, 1394199221, 1394199241, 3779851321, 3779851363, 3779856571, 10246935931, 10246936019, 10246936481, 75370121689, 75370121857, 75370122409, 75370125409, 75370126441

Offset: 1

Jud McCranie

nonn

Based on problem by G. L. Honaker, Jr.

A subsequence of A073465. - Ivan N. Ianakiev, Aug 06 2019

			13 is the 6th prime and 13 == 1 mod 6.

Giovanni Resta, Table of n, a(n) for n = 1..94

Cf. A000040, A023143, A045924, A052013.

f[p_,n_]:=Mod[p,n]==0; lst={};Do[p=Prime[n];If[f[p-1,n],AppendTo[lst,p]],{n,10!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 08 2009 *)

lista(nn) = forprime(p=1, nn, if (Mod(p, primepi(p)) == 1, print1(p, ", "))); \\ Michel Marcus, Jan 08 2015; Aug 06 2019

A087611(a(n)) = 0. - Reinhard Zumkeller, Sep 11 2003

a(n) = A000040(A023143(n)). - Zak Seidov, Feb 19 2015

More terms from Zak Seidov, Feb 19 2015

Terms a(33)-a(37) sorted into correct order by Giovanni Resta, Feb 23 2020

A162567 Primes p such that pi(p) divides p-1 and/or p+1, where pi(p) is the number of primes <= p.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 29, 37, 43, 349, 359, 1087, 1091, 3079, 8423, 64579, 64591, 64601, 64609, 64661, 64709, 481043, 481067, 1304707, 3523969, 3524249, 3524317, 3524387, 9558541, 9559799, 9560009, 9560039, 25874767, 70115921, 189962009

Offset: 1

Leroy Quet, Jul 06 2009

nonn

			The 10th prime is 29. Since 10 divides 29+1 = 30, 29 is in the sequence.
The 12th prime is 37. Since 12 divides 37-1 = 36, 37 is in the sequence.

Robert G. Wilson v, Table of n, a(n) for n = 1..199 (data obtained from A048891 and A052013; sorted by Amiram Eldar Sep 05 2024)

Union of A048891 and A052013. - Michel Marcus, Mar 04 2019

Cf. A000040, A078931.

isA162567 := proc(p) RETURN ( (p-1) mod numtheory[pi](p) = 0 or (p+1) mod numtheory[pi](p) = 0 ) ; end: for n from 1 to 50000 do p := ithprime(n) ; if isA162567(p) then printf("%d,",p) ; fi; od: # R. J. Mathar, Jul 30 2009
with(numtheory): a := proc (n) if `mod`(ithprime(n)-1, pi(ithprime(n))) = 0 or `mod`(ithprime(n)+1, pi(ithprime(n))) = 0 then ithprime(n) else end if end proc: seq(a(n), n = 1 .. 250000); # Emeric Deutsch, Jul 31 2009

Select[Prime[Range[11000000]],Or@@Divisible[{#-1,#+1},PrimePi[#]]&] (* Harvey P. Dale, Sep 08 2012 *)

a(n) = A000040(A078931(n)). - Alois P. Heinz, Feb 20 2023

a(10)-a(35) from Donovan Johnson, Jul 29 2009

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

A269022 Primes p such that sigma(p)/pi(p) is prime.

Original entry on oeis.org

2, 3, 5, 7, 29, 349, 359, 3079, 70115921, 514274899, 514277977, 11091501632311

Offset: 1

Soumadeep Ghosh, Feb 17 2016

Corresponding quotient primes are 3, 2, 2, 2, 3, 5, 5, 7, 17, 19, 19, 29.

a(13) > 8.1*10^13 if it exists. Assuming the Riemann Hypothesis, a(13) > 3.27*10^16 (if it exists). - Chai Wah Wu, May 25 2018

			7 is in the sequence because sigma(7) = 8, pi(7) = 4 and 8/4 = 2 is a prime.

Subsequence of A052013.

Cf. A000203, A000720.

Select[Prime[Range[10^6]], ProvablePrimeQ[DivisorSigma[1, #]/PrimePi[#]] &]
Select[ (* the terms of A052013 *), PrimeQ[(# + 1)/PrimePi@ #] &] (* Robert G. Wilson v, Mar 16 2016 *)

is(n)=my(t=(n+1)/primepi(n)); denominator(t)==1 && isprime(t) && isprime(n) \\ Charles R Greathouse IV, Feb 18 2016

list(lim)=my(v=List(),n,t); forprime(p=2,lim, t=(p+1)/n++; if(denominator(t)==1 && isprime(t), listput(v, p))); Vec(v) \\ Charles R Greathouse IV, Feb 18 2016

a(9)-a(11) from Charles R Greathouse IV, Feb 18 2016

a(12) from Chai Wah Wu, May 25 2018

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A023143 Numbers k such that prime(k) == 1 (mod k).

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A045924 Numbers n such that prime(n) == -1 (mod n).

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A048891 Primes that are congruent to 1 mod n, where n is the index of the prime.

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A162567 Primes p such that pi(p) divides p-1 and/or p+1, where pi(p) is the number of primes <= p.

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

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A269022 Primes p such that sigma(p)/pi(p) is prime.

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