A045924 -id:A045924 - cp's OEIS frontend

A049204 Duplicate of A045924.

1, 2, 3, 4, 10, 70, 72, 182, 440, 1053, 6458, 6461, 6471, 40087, 40089, 251737

Offset: 1

dead

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

A092044 Numbers n such that prime(n) == -2 (mod n).

Original entry on oeis.org

1, 11, 71, 637319, 637323, 637327, 179993015, 1208198861, 8179002163, 8179002189, 55762149067, 55762149103, 6201265271239787, 43525513764814971, 43525513764815121, 2159986889494444325, 2159986889494445481, 2159986889494445499, 2159986889494447181

Offset: 1

Robert G. Wilson v, Feb 18 2004

nonn

Cf. A023144, A045924, A092045, A092046, A092047, A092048, A092049, A092050, A092051, A092052.

Cf. A105290.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 1; Do[ If[ Mod[p = NextPrim[p], n] == n - 2, Print[n]], {n, 1, 10^9}]

for(i=1,10^9,if(Mod(prime(i),i)==-2,print1(i,",")));

Corrected by Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Feb 20 2004
a(7)-a(12) from Robert G. Wilson v, Feb 22 2006
a(13)-a(19) 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

A092045 Numbers n such that prime(n) == -3 (mod n).

Original entry on oeis.org

1, 2, 25, 26, 68, 1054, 1058, 6472, 251723, 4124468, 69709727, 69709942, 465769817, 465769835, 1208198860, 8179002154, 8179002176, 8179002178, 145935689360, 145935689369, 145935689392, 145935689393

Offset: 1

Robert G. Wilson v, Feb 18 2004

nonn

Cf. A023145, A045924, A092044, A092046, A092047, A092048, A092049, A092050, A092051, A092052.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 1; Do[ If[ Mod[p = NextPrim[p], n] == n - 3, Print[n]], {n, 1, 10^9}]
Join[{1,2},Select[Range[5 10^6],Mod[Prime[#],#]==#-3&]] (* Harvey P. Dale, Mar 29 2023 *)

Corrected by Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Feb 20 2004

a(13)-a(22) from Robert G. Wilson v, Feb 22 2006

A092051 Numbers n such that prime(n) == -9 (mod n).

Original entry on oeis.org

1, 2, 4, 5, 17, 19, 20, 22, 23, 64, 1055, 1057, 6463, 251708, 251743, 251744, 251755, 251758, 27067054, 27067118, 27067138, 69709681, 69709703, 69709712, 145935689366, 382465573490, 382465573498, 6935812012621, 126979448983610, 885992692751476

Offset: 1

Robert G. Wilson v, Feb 18 2004

nonn

Giovanni Resta, Table of n, a(n) for n = 1..39

Cf. A023151, A045924, A092044, A092045, A092046, A092047, A092048, A092049, A092050, A092052.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 1; Do[ If[ Mod[p = NextPrim[p], n] == n - 9, Print[n]], {n, 1, 10^9}]

Corrected by Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Feb 20 2004
a(25) from Robert G. Wilson v, Feb 22 2006
Terms a(26) and beyond from Giovanni Resta, Feb 23 2020

A092046 Numbers n such that prime(n) == -4 (mod n).

Original entry on oeis.org

1, 3, 5, 7, 9, 67, 441, 2615, 637237, 637329, 4124703, 27067119, 179993017, 1208198617, 8179002101, 55762149071, 55762149091

Offset: 1

Robert G. Wilson v, Feb 18 2004

nonn

Cf. A023146, A045924, A092044, A092045, A092047, A092048, A092049, A092050, A092051, A092052.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 1; Do[ If[ Mod[p = NextPrim[p], n] == n - 4, Print[n]], {n, 1, 10^9}]

Corrected by Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Feb 20 2004

a(13)-a(17) from Robert G. Wilson v, Feb 22 2006

A092047 Numbers n such that prime(n) == -5 (mod n).

Original entry on oeis.org

1, 2, 4, 6, 8, 27, 28, 169, 183, 187, 188, 189, 438, 442, 1056, 1059, 40084, 40088, 40091, 40114, 40121, 100348, 251709, 4124588, 10553499, 10553853, 27066972, 179992914, 179992932, 179993012, 465769812, 1208198618, 1208198852

Offset: 1

Robert G. Wilson v, Feb 18 2004

nonn

Cf. A023147, A045924, A092044, A092045, A092046, A092048, A092049, A092050, A092051, A092052.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 1; Do[ If[ Mod[p = NextPrim[p], n] == n - 5, Print[n]], {n, 1, 10^9}]

Corrected by Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Feb 20 2004

a(28)-a(34) from Robert G. Wilson v, Feb 22 2006

A092048 Numbers n such that prime(n) == -6 (mod n).

Original entry on oeis.org

1, 439, 100349, 100361, 100363, 27066991, 27067117, 1208198633, 8179002133

Offset: 1

Robert G. Wilson v, Feb 18 2004

nonn

Cf. A023148, A045924, A092044, A092045, A092046, A092047, A092049, A092050, A092051, A092052.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 1; Do[ If[ Mod[p = NextPrim[p], n] == n - 6, Print[n]], {n, 1, 10^9}]

Corrected by Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Feb 20 2004

a(8)-a(9) from Robert G. Wilson v, Feb 22 2006

cp's OEIS Frontend

A049204 Duplicate of A045924.

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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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A092044 Numbers n such that prime(n) == -2 (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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A092045 Numbers n such that prime(n) == -3 (mod n).

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A092051 Numbers n such that prime(n) == -9 (mod n).

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A092046 Numbers n such that prime(n) == -4 (mod n).

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