A004648 -id:A004648 - cp's OEIS frontend

A072608 Parity of remainder Mod(prime(n),n) = A004648(n).

0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0

Offset: 1

Labos Elemer, Jun 24 2002

			n=25:p(25)=97,Mod[97,25]=22, a(25)=Mod[22,2]=0. With increasing n, a(n) alternates:...010101..,followed after by a range consisting only of 1-s. This secondary alternation also goes on.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A004648.

a072608 n = a000040 n `mod` n `mod` 2 -- Reinhard Zumkeller, Dec 16 2013

mm[x_] := Mod[Mod[Prime[x], x], 2] Table[mm[w], {w, 1, 256}]
Table[Mod[Mod[Prime[n],n],2],{n,110}] (* Harvey P. Dale, Apr 22 2016 *)

a(n)=prime(n)%n%2 \\ Charles R Greathouse IV, Feb 09 2017

a(n) = Mod(Mod(prime(n), n), 2) = Mod(A004648(n), 2).

A099850 Partial sums of A004648.

Original entry on oeis.org

0, 1, 3, 6, 7, 8, 11, 14, 19, 28, 37, 38, 40, 41, 43, 48, 56, 63, 73, 84, 94, 107, 121, 138, 160, 183, 205, 228, 250, 273, 276, 279, 284, 287, 296, 303, 312, 323, 334, 347, 362, 375, 394, 411, 428, 443, 466, 497, 528, 557, 586, 617, 646, 681, 718, 757, 798, 837

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Oct 27 2004

			A004648 begins: 0, 1, 2, 3, 1, 1, 3, 3, 5, 9, 9, ... so the partial sums are 0, 1, 3, 6, 7, 8, 11, 14, 19, 28, 37, ...

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A004648.

[(&+[(NthPrime(k) mod k): k in [1..n]]): n in [1..100]]; // G. C. Greubel, Apr 20 2023

Table[Sum[Mod[Prime[j], j], {j,n}], {n,100}] (* G. C. Greubel, Apr 20 2023 *)
Accumulate[Table[Mod[Prime[n],n],{n,100}]] (* Harvey P. Dale, Jun 14 2023 *)

s=vector(100):s[1]=prime(1)%1:for(n=2,100,s[n]=s[n-1]+prime(n)%n)

def A004648(n): return (nth_prime(n)%n)
def A099850(n): return sum(A004648(k) for k in range(1,n+1))
[A099850(n) for n in range(1,101)] # G. C. Greubel, Apr 20 2023

a(n) = Sum_{k=1..n} A004648(k).

A127149 Records in A004648.

Original entry on oeis.org

0, 1, 2, 3, 5, 9, 10, 11, 13, 14, 17, 22, 23, 31, 35, 37, 39, 41, 45, 55, 63, 65, 69, 71, 74, 79, 82, 83, 84, 86, 87, 89, 97, 102, 109, 111, 118, 122, 132, 134, 139, 142, 143, 152, 153, 156, 157, 164, 166, 169, 171, 181, 182, 183, 184, 185, 187, 193, 197, 203, 209, 217, 235, 245

Offset: 1

N. J. A. Sloane, Mar 25 2007

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A004648, A127150.

Module[{r = -1, v}, Table[If[(v = Mod[Prime[k], k]) > r, r = v, Nothing], {k, 500}]] (* Paolo Xausa, Jul 29 2025 *)

A127150 Where records occur in A004648.

Original entry on oeis.org

1, 2, 3, 4, 9, 10, 19, 20, 22, 23, 24, 25, 26, 48, 54, 55, 56, 57, 62, 63, 67, 68, 70, 72, 127, 128, 129, 130, 131, 133, 134, 136, 138, 139, 140, 142, 147, 151, 155, 157, 158, 159, 162, 163, 166, 167, 168, 169, 173, 176, 178, 182, 187, 188, 189, 298, 300, 310, 311, 313, 320

Offset: 1

N. J. A. Sloane, Mar 25 2007

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A004648, A127149.

Module[{r = -1, v}, Table[If[(v = Mod[Prime[k], k]) > r, r = v; k, Nothing], {k, 500}]] (* Paolo Xausa, Jul 29 2025 *)

A247254 a(1)=0; for n>1, a(n+1)= a(n)- A004648(n) if a(n)>= A004648(n) else a(n) + A004648(n).

Original entry on oeis.org

0, 0, 1, 3, 0, 1, 0, 3, 0, 5, 14, 5, 4, 2, 1, 3, 8, 0, 7, 17, 6, 16, 3, 17, 0, 22, 45, 23, 0, 22, 45, 42, 39, 34, 31, 22, 15, 6, 17, 6, 19, 4, 17, 36, 19, 2, 17, 40, 9, 40, 11, 40, 9, 38, 3, 40, 1, 42, 3, 44, 3, 42, 87, 32, 87, 34, 87, 24, 89, 87, 18, 87, 16, 14

Offset: 1

Michel Lagneau, Nov 30 2014

nonn

Michel Lagneau, Table of n, a(n) for n = 1..10000

Cf. A004648.

A247254 := proc(n)
    option remember;
    local a48 ;
    if n=1 then
        0;
    else
        a48 := A004648(n-1) ;
        if procname(n-1) >= a48 then
            procname(n-1)-a48 ;
        else
            procname(n-1)+a48 ;
        end if;
    end if;
end proc: # R. J. Mathar, Dec 02 2014

A099641 Number of solutions to x*frac[p(x)/x]<=Log[n] or A004648(n)<=Log[n].

Original entry on oeis.org

1, 5, 6, 12, 13, 14, 15, 31, 32, 34, 69, 73, 74, 75, 76, 77, 181, 445, 1052, 6455, 6456, 6457, 6459, 6460, 6466, 15928, 16055, 40073, 40078, 40080, 40081, 40082, 40083, 40122, 100362, 100364, 100365, 251707, 251711, 251712, 251717, 251719, 251721

Offset: 1

Labos Elemer, Nov 02 2004

nonn

Solutions appear in clusters because of features of diagram visible at A004648. Later clusters are introduced by 6455, 15928, 40073, 100362, 251707, 637235, 4124455, respectively.

Number of solutions in consecutive clusters seem to be as follows: 1,2,4,3,6,1,1,1,6,2,7,3 etc..

Cf. A004648.

Cf. A023143-A023152, A072610.

ta={{0}};Do[s=w*fra[Prime[w]/w];If[ !Greater[s, Log[n]], Print[w]; ta=Append[ta, w]], {w, 1, 1000000}];ta=Delete[ta, 1]

A099644 a[n]=Mod[q(n),PrimePi[q(n)]]=Mod[A001359(n), A000720(A001359(n))] where q(n) is the n-th lesser-twin-prime. A004648 restricted to lesser twins.

Original entry on oeis.org

1, 2, 1, 3, 9, 2, 8, 11, 23, 23, 5, 9, 15, 19, 17, 31, 31, 41, 41, 55, 2, 14, 16, 16, 31, 49, 54, 52, 61, 59, 109, 111, 107, 117, 121, 164, 166, 169, 171, 181, 11, 23, 41, 35, 29, 29, 77, 71, 77, 71, 89, 83, 95, 107, 113, 125, 155, 149, 167, 173, 185, 185, 203, 197, 203, 209

Offset: 1

Labos Elemer, Nov 04 2004

nonn

Sequence display diagram similar to that of A004648.

			n=9: p(26)=101 is the 9th lesser-twin-prime,
a(9)-Mod[p(26),26]=Mod[101,26]=23=a(9).

Cf. A001359, A004648.

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

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

A065134 Remainder when n is divided by the number of primes not exceeding n.

Original entry on oeis.org

0, 1, 0, 2, 0, 3, 0, 1, 2, 1, 2, 1, 2, 3, 4, 3, 4, 3, 4, 5, 6, 5, 6, 7, 8, 0, 1, 9, 0, 9, 10, 0, 1, 2, 3, 1, 2, 3, 4, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 6, 7, 5, 6, 7, 8, 9, 10, 8, 9, 7, 8, 9, 10, 11, 12, 10, 11, 12, 13, 11, 12, 10, 11, 12, 13, 14, 15, 13, 14, 15, 16, 14, 15, 16, 17, 18, 19, 17, 18, 19

Offset: 2

Labos Elemer, Oct 15 2001

nonn

Also remainder when the number of nonprimes is divided by the number of primes (not exceeding n).

			n = 2: Pi[2] = 1,Mod[1,1] = 0, the first term = a(2) = 0; n = 100: Pi[100] = 25, Mod[100,25] = 0 = a(100); n = 20: Pi[20] = 8, Mod[20,8] = 4 = a(20).

Harry J. Smith, Table of n, a(n) for n = 2..1000

Cf. A000720, A057809, A057810, A000040, A065134, A004648, A062298, A065858-A065864, A065133.

Table[Last@ QuotientRemainder[n, PrimePi[n]], {n, 2, 91}] (* Michael De Vlieger, Jul 04 2016 *)

{ for (n=2, 1000, write("b065134.txt", n, " ", n%primepi(n)) ) } \\ Harry J. Smith, Oct 11 2009

a(n) = n (mod pi(n)).

Term a(1) removed so OFFSET changed from 1,5 to 2,4 by Harry J. Smith, Oct 11 2009

Since OFFSET is 2,4; Term a(1) removed and a(91) added by Harry J. Smith, Oct 11 2009

cp's OEIS Frontend

A072608 Parity of remainder Mod(prime(n),n) = A004648(n).

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A099850 Partial sums of A004648.

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A127149 Records in A004648.

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A127150 Where records occur in A004648.

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A247254 a(1)=0; for n>1, a(n+1)= a(n)- A004648(n) if a(n)>= A004648(n) else a(n) + A004648(n).

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Maple

A099641 Number of solutions to x*frac[p(x)/x]<=Log[n] or A004648(n)<=Log[n].

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Mathematica

A099644 a[n]=Mod[q(n),PrimePi[q(n)]]=Mod[A001359(n), A000720(A001359(n))] where q(n) is the n-th lesser-twin-prime. A004648 restricted to lesser twins.

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

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