A023149 -id:A023149 - cp's OEIS frontend

A004648 a(n) = prime(n) mod n.

0, 1, 2, 3, 1, 1, 3, 3, 5, 9, 9, 1, 2, 1, 2, 5, 8, 7, 10, 11, 10, 13, 14, 17, 22, 23, 22, 23, 22, 23, 3, 3, 5, 3, 9, 7, 9, 11, 11, 13, 15, 13, 19, 17, 17, 15, 23, 31, 31, 29, 29, 31, 29, 35, 37, 39, 41, 39, 41, 41, 39, 45, 55, 55, 53, 53, 63, 65, 2, 69, 69, 71, 2, 3, 4, 3

Offset: 1

N. J. A. Sloane, Daniel Wild (wild(AT)edumath.u-strasbg.fr)

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
Labos Elemer, Graph of first 50000 terms

1's occur at A023143, 2's at A023144, 3's at A023145, 4's at A023146, 5's at A023147, 6's at A023148, 7's at A023149, 8's at A023150, 9's at A023151, 10's at A023152, == -1's at A045924.

For records see A127149, A127150.

a004648 n = a004648_list !! (n-1)
a004648_list = zipWith mod a000040_list [1..]
-- Reinhard Zumkeller, Jul 30 2012

[(NthPrime(n) mod n): n in [1..100]]; // Vincenzo Librandi, Apr 06 2011

A004648 := proc(n)
    modp(ithprime(n),n) ;
end proc: # R. J. Mathar, Dec 02 2014

Table[Mod[Prime[n], n], {n, 100}] (* Zak Seidov, Apr 25 2005 *)

PARI
```
for(n=1,100,print1(prime(n)%n,","))
```

from sympy import prime; print([prime(i) % i for i in range(1, 101)]) # Jwalin Bhatt, Jul 29 2025

def A004648(n): return (nth_prime(n)%n)
[A004648(n) for n in range(1,101)] # G. C. Greubel, Apr 20 2023

a(n) = prime(n) - n*floor(prime(n)/n)

More terms from Clark Kimberling

Corrected by Jaroslav Krizek, Dec 16 2009

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

Original entry on oeis.org

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

A023144 Numbers k such that prime(k) == 2 (mod k).

Original entry on oeis.org

1, 3, 13, 15, 69, 73, 637321, 27066989, 27067095, 27067133, 179992909, 1208198629, 1208198853, 8179002209, 382465573515, 2636913003191, 18262325820323, 18262325820339, 6201265271239163, 6201265271239191, 6201265271239431, 43525513764815041, 43525513764815125

Offset: 1

David W. Wilson

nonn

a(15) > 10^13. - Charles R Greathouse IV, Mar 17 2011

Giovanni Resta, Table of n, a(n) for n = 1..28 (terms < 3*10^20)

Cf. A092044, A023143, A023145, A023146, A023147, A023148, A023149, A023150, A023151, A023152.

Cf. A105290.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 1; Do[ If[ Mod[p = NextPrim[p], n] == 2, Print[n]], {n, 1, 10^9}] (* Robert G. Wilson v, Feb 18 2004 *)

More terms from Robert G. Wilson v, Feb 18 2004
a(10)-a(13) from Giovanni Resta and Robert G. Wilson v, Feb 22 2006
Missing first term inserted by M. F. Hasler, Feb 04 2009
a(15)-a(23) from Giovanni Resta, Oct 02 2019

A023152 Numbers k such that prime(k) == 10 (mod k).

Original entry on oeis.org

1, 7, 19, 21, 2703, 15929, 4124583, 27067051, 179992913, 179993011, 179993159, 1208198559, 1208198859, 55762149031, 382465573511, 382465573547, 18262325820337, 6201265271239229, 6201265271239409, 6201265271239433, 43525513764814941, 43525513764816369, 43525513764816411, 43525513764816437, 306268030480171419

Offset: 1

David W. Wilson

nonn

Cf. A092052, A023143, A023144, A023145, A023146, A023147, A023148, A023149, A023150, A023151.

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

n=0; forprime(p=2,1e9, if(Mod(p,n++)==10, print1(n", "))) \\ Charles R Greathouse IV, Apr 29 2015

def A023152(max) :
    terms = []
    p = 2
    for n in range(1,max+1) :
        if (p - 10) % n == 0 : terms.append(n)
        p = next_prime(p)
    return terms
# Eric M. Schmidt, Feb 05 2013

Extended by Robert G. Wilson v, Feb 18 2004
a(9)-a(14) from Robert G. Wilson v, Feb 22 2006
First two terms inserted by Eric M. Schmidt, Feb 05 2013
a(15)-a(25) from Giovanni Resta, Feb 23 2020

A023150 Numbers k such that prime(k) == 8 (mod k).

Original entry on oeis.org

1, 3, 17, 443, 2701, 100365, 637235, 637325, 4124455, 4124473, 4124587, 27067125, 27067137, 27067521, 8179002131, 8179002135, 8179002153, 55762149069, 382465573521, 6201265271239145, 6201265271240411, 43525513764814933, 43525513764815131, 43525513764816415

Offset: 1

David W. Wilson

nonn

Giovanni Resta, Table of n, a(n) for n = 1..30

Cf. A092050, A023143, A023144, A023145, A023146, A023147, A023148, A023149, A023151, A023152.

p = 1; Do[ If[ Mod[p = NextPrime[p], n] == 8, Print[n]], {n, 1, 10^9}] (* Robert G. Wilson v, Feb 18 2004 *)

def A023150(max):
    terms = []
    p = 2
    for n in range(1, max+1) :
        if (p - 8) % 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
a(15)-a(18) from Robert G. Wilson v, Feb 22 2006
First two terms inserted by Eric M. Schmidt, Feb 05 2013
Terms a(19) and beyond from Giovanni Resta, Feb 23 2020

A023145 Numbers k such that prime(k) == 3 (mod k).

Original entry on oeis.org

1, 2, 4, 7, 8, 31, 32, 34, 74, 76, 1052, 6455, 15928, 251707, 251765, 4124458, 27067012, 27067120, 69709718, 69709871, 69709877, 69709934, 69709943, 69709954, 69709963, 69709964, 465769810, 8179002124, 145935689390, 382465573486, 885992692751818, 885992692751822

Offset: 1

David W. Wilson

nonn

			204475053103 = prime(8179002124) and 204475053103 = 25*8179002124 + 3.

Giovanni Resta, Table of n, a(n) for n = 1..46

Cf. A171430, A092045, A023143, A023144, A023146, A023147, A023148, A023149, A023150, A023151, A023152.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 1; Do[ If[ Mod[p = NextPrim[p], n] == 3, Print[n]], {n, 1, 10^9}] (* Robert G. Wilson v, Feb 18 2004 *)
Select[Range[100000], Mod[Prime[#] - 3, #] == 0 &] (* T. D. Noe, Feb 05 2013 *)

def A023145(max) :
    terms = []
    p = 2
    for n in range(1, max+1) :
        if (p - 3) % 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, Feb 22 2006
a(29) from Robert G. Wilson v, Feb 22 2006
First two terms inserted by Eric M. Schmidt, Feb 05 2013
Terms a(30) and beyond from Giovanni Resta, Feb 23 2020

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

Original entry on oeis.org

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

A023147 Numbers k such that prime(k) == 5 (mod k).

Original entry on oeis.org

1, 2, 3, 9, 16, 33, 40073, 40078, 40082, 40083, 40122, 251721, 251722, 251734, 251736, 4124584, 10553433, 10553439, 10553444, 10553454, 10553478, 10553552, 10553553, 10553818, 27067098, 27067524, 179992912, 465769814, 465769832, 465769839

Offset: 1

David W. Wilson

nonn

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

Cf. A171432, A092047, A023143, A023144, A023145, A023146, A023148, A023149, A023150, A023151, A023152.

p = 1; Do[ If[ Mod[p = NextPrime[p], n] == 5, Print[n]], {n, 1, 10^9}] (* Robert G. Wilson v, Feb 18 2004 *)

More terms from Robert G. Wilson v, Feb 18 2004
a(24)-a(27) from Robert G. Wilson v, Feb 18 2004
Three missing initial terms at start of sequence inserted by Zak Seidov, Feb 05 2009

A023148 Numbers k such that prime(k) == 6 (mod k).

Original entry on oeis.org

1, 5, 79, 445, 16055, 27067045, 1208198549, 1208198609, 1208198615, 2636913002983, 885992692751479, 885992692751483, 885992692751833, 43525513764815035, 43525513764815057, 43525513764815173, 43525513764816367

Offset: 1

David W. Wilson

a(18) > 2.8*10^20. - Giovanni Resta, Feb 23 2020

Cf. A092048, A023143, A023144, A023145, A023146, A023147, A023149, A023150, A023151, A023152.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 1; Do[ If[ Mod[p = NextPrim[p], n] == 6, Print[n]], {n, 1, 10^9}] (* Robert G. Wilson v, Feb 18 2004 *)

def A023148(max) :
    terms = []
    p = 2
    for n in range(1, max+1) :
        if (p - 6) % n == 0 : terms.append(n)
        p = next_prime(p)
    return terms
# Eric M. Schmidt, Feb 05 2013

Extended by Robert G. Wilson v, Feb 18 2004
a(7)-a(9) from Robert G. Wilson v, Feb 22 2006
First two terms inserted by Eric M. Schmidt, Feb 05 2013
a(10)-a(17) from Giovanni Resta, Feb 23 2020

A023151 Numbers k such that prime(k) == 9 (mod k).

Original entry on oeis.org

1, 2, 10, 11, 35, 37, 80, 100364, 251711, 251717, 251731, 251735, 251741, 251770, 4124456, 4124582, 27067096, 27067520, 69709706, 69709717, 69709723, 69709868, 69709931, 69709933, 465769825, 465769826, 465769831, 1208198548, 8179002130

Offset: 1

David W. Wilson

nonn

Giovanni Resta, Table of n, a(n) for n = 1..46

Cf. A092051, A023143, A023144, A023145, A023146, A023147, A023148, A023149, A023150, A023152.

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

def A023151(max) :
    terms = []
    p = 2
    for n in range(1, max+1) :
        if (p - 9) % 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
a(25)-a(29) from Robert G. Wilson v, Feb 22 2006
First two terms inserted by Eric M. Schmidt, Feb 05 2013

cp's OEIS Frontend

A004648 a(n) = prime(n) mod n.

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

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A023144 Numbers k such that prime(k) == 2 (mod k).

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A023152 Numbers k such that prime(k) == 10 (mod k).

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A023150 Numbers k such that prime(k) == 8 (mod k).

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A023145 Numbers k such that prime(k) == 3 (mod k).

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

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