A004648 -id:A004648 - cp's OEIS frontend

A087611 a(n) = (prime(n) - 1) mod n.

0, 0, 1, 2, 0, 0, 2, 2, 4, 8, 8, 0, 1, 0, 1, 4, 7, 6, 9, 10, 9, 12, 13, 16, 21, 22, 21, 22, 21, 22, 2, 2, 4, 2, 8, 6, 8, 10, 10, 12, 14, 12, 18, 16, 16, 14, 22, 30, 30, 28, 28, 30, 28, 34, 36, 38, 40, 38, 40, 40, 38, 44, 54, 54, 52, 52, 62, 64, 1, 68, 68, 70, 1, 2, 3, 2, 3, 6, 5, 8

Offset: 1

Reinhard Zumkeller, Sep 11 2003

Cf. A023143 (indices of 0's).

Cf. A006093, A004648.

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

a(n) = (prime(n)-1) % n; \\ Kevin Ryde, Feb 03 2023

A099851 Numbers k such that A099850(k) is divisible by k.

Original entry on oeis.org

1, 3, 16, 57, 112, 160, 272, 404, 20924, 147153, 274617, 4409708, 24881389, 34850872, 39808233, 186610952, 980830465, 1777956414

Offset: 1

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

			A099850(1)=0, which is divisible by 1.
A099850(3)=3, which is divisible by 3.
A099850(16)=48, which is divisible by 16.

Cf. A099850, A004648.

a(12)-a(18) from Donovan Johnson, Dec 14 2009

A245071 a(n) = 12n - prime(n).

Original entry on oeis.org

10, 21, 31, 41, 49, 59, 67, 77, 85, 91, 101, 107, 115, 125, 133, 139, 145, 155, 161, 169, 179, 185, 193, 199, 203, 211, 221, 229, 239, 247, 245, 253, 259, 269, 271, 281, 287, 293, 301, 307, 313, 323, 325, 335, 343, 353, 353, 353, 361, 371, 379, 385, 395, 397, 403, 409, 415, 425

Offset: 1

Freimut Marschner, Jul 21 2014

Prime(n) > n for n > 0. Let prime(n) = k*n with k as an even integer constant, for example, k = 12; then a(n) = k*n - prime(n) is a sequence of odd integers that are positive as long as k*n > prime(n). This is the case up to a(40072) = 11. If k*n < prime(n) then a(n) < 0, a(40073) = -5 up to a(40083) = -5. From a(40084) = 5 up to a(40121) = 5, a(n) > 0 again, but a(n) < 0 for n >= 40122. For k = 12 the table shows this result compared with floor(prime(n)/n) and (prime(n) mod n) <= (prime(n+1) mod (n+1)) for n >= 1. Observations:
(1) If k > floor(prime(n)/n) then a(n) is positive.
(2) If k <= floor(prime(n)/n) and (prime(n) mod n) < (prime(n+1) mod (n+1)) and n > 1 then a(n) is negative.
(3) If k <= floor(prime(n)/n) and (prime(n) mod n) > (prime(n+1) mod (n+1)) then a(n) is positive.
.
n     prime(n) floor(prime(n)/n) (prime(n) mod n)  a(n)
40072 480853        12                 5            11
40073 480881        12                23            -5
40083 481001        11             40079            -5
40084 481003        11             40074             5
40121 481447        12                 5             5
40122 481469        12                13            -5

			a(3) = 12*3 - prime(3) = 36 - 5 = 31.

Freimut Marschner, Table of n, a(n) for n = 1..100000

A000040 (prime(n)), A038605 (floor(prime(n)/n)), A004648 (prime(n) mod n), A038606 (Least k such that k-th prime > n * k), A038607 (the smallest prime number k such that k > n*pi(k)), A102281 (the largest number m such that m = pi(n*m)).

Table[12n - Prime[n], {n, 60}] (* Alonso del Arte, Jul 27 2014 *)

PARI
```
vector(133, n, 12*n-prime(n) )
```

a(n) = 12*n - prime(n).

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

A363751 Numbers k such that prime(k) mod k is prime.

Original entry on oeis.org

3, 4, 7, 8, 9, 13, 15, 16, 18, 20, 22, 24, 26, 28, 30, 31, 32, 33, 34, 36, 38, 39, 40, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 55, 57, 59, 60, 65, 66, 69, 72, 73, 74, 76, 78, 82, 84, 86, 88, 90, 92, 96, 98, 100, 102, 106, 108, 112, 116, 120, 126, 128, 130

Offset: 1

Nicholas Leonard, Jun 19 2023

nonn

			9 is a term of this sequence as prime(9) mod 9 = 5, which is prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000720, A004648, A363752.

Table[If[PrimeQ[Mod[Prime[k], k]], k, Nothing], {k, 1, 100}]

isok(k) = isprime(prime(k) % k); \\ Michel Marcus, Jun 19 2023

from sympy import prime, isprime
a363751=[]
for k in range(1,101):
    if isprime(prime(k)%k):
        a363751.append(k)

a(n) = A000720(A363752(n)).

A363752 Primes prime(k) such that prime(k) mod k is prime.

Original entry on oeis.org

5, 7, 17, 19, 23, 41, 47, 53, 61, 71, 79, 89, 101, 107, 113, 127, 131, 137, 139, 151, 163, 167, 173, 181, 191, 193, 197, 211, 223, 227, 229, 233, 239, 241, 257, 269, 277, 281, 313, 317, 347, 359, 367, 373, 383, 397, 421, 433, 443, 457, 463, 479, 503, 521, 541

Offset: 1

Nicholas Leonard, Jun 19 2023

nonn

			The 9th prime is 23 and 23 mod 9 = 5, which is prime, so 23 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A004648, A363751.

Table[If[PrimeQ[Mod[Prime[k], k]], Prime[k], Nothing], {k, 1, 100}]

from sympy import prime, isprime
a363752=[]
for k in range(1, 101):
    if isprime(prime(k)%k):
        a363752.append(prime(k))

a(n) = A000040(A363751(n)).

A065521 a(n) = floor(prime(n) / n) * n - prime(n) mod n.

Original entry on oeis.org

2, 1, 1, 1, 9, 11, 11, 13, 13, 11, 13, 35, 37, 41, 43, 43, 43, 47, 47, 49, 53, 53, 55, 55, 53, 55, 59, 61, 65, 67, 121, 125, 127, 133, 131, 137, 139, 141, 145, 147, 149, 155, 153, 159, 163, 169, 165, 161, 165, 171, 175, 177, 183, 181, 183, 185, 187, 193, 195, 199

Offset: 1

Reinhard Zumkeller, Nov 27 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A038605, A004648.

Table[Floor[Prime[n]/n]n-Mod[Prime[n],n],{n,60}] (* Harvey P. Dale, Dec 27 2019 *)

{ for (n=1, 1000, a=floor(prime(n) / n) * n - prime(n) % n; write("b065521.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 20 2009

a(n) = n*(prime(n)\n) - (prime(n) % n); \\ Michel Marcus, Jun 18 2018

a(n) = A038605(n) * n - A004648(n).

A072623 Numbers n such that A065863(n) = 1, i.e., prime(n) mod (n - Pi(n)) = 1.

Original entry on oeis.org

4, 5, 6, 11, 19, 25, 34, 36, 75, 82, 87, 90, 94, 237, 604, 609, 614, 1583, 1592, 10466, 10467, 10498, 10504, 10505, 70501, 70511, 180227, 180294, 180358, 180443, 180447, 466078, 8103422, 21058343, 21058649, 143052872, 143052877, 143053068

Offset: 1

Labos Elemer, Jun 26 2002

nonn

A004648, A065134 and A065863 behave similarly; they grow relatively slowly and drop suddenly at unexpected values of n. Parity of A004648 behaves most regularly.

Each cluster of entries exceeds the previous cluster by a power of e.

			For the cluster started at n = 10466 the remainders of A065863(n) are as follows: {9089, 9092, 9117, 9127, 9148, 9159, 1, 1, 9180, 9183, 9182, 9179, 9172, 9169, 9168, 9177, 9176, 9178, 9183, 9192, 43}. It behaves like A004648 or A065134.

Cf. A065860-A065863, A065133-A065135, A072608-A072610, A004648, A057809.

Do[ If[ Mod[ Prime[n], n-PrimePi[n]] == 1, Print[n]], {n, 1, 150000000}]
(* Second program: *)
Position[Table[Mod[Prime[n], n - PrimePi[n]], {n, 10^6}], 1] // Flatten (* Michael De Vlieger, Jul 30 2017 *)

Edited by Robert G. Wilson v, Jun 27 2002

A072624 a(n) = prime(n^2) mod n^2.

Original entry on oeis.org

0, 3, 5, 5, 22, 7, 31, 55, 14, 41, 56, 107, 164, 17, 77, 83, 145, 199, 271, 341, 437, 73, 100, 179, 262, 319, 416, 519, 594, 697, 846, 993, 25, 93, 131, 259, 369, 497, 575, 699, 879, 989, 1085, 1259, 1409, 1533, 1799, 1961, 2183, 2307, 2519, 23, 188, 329, 514

Offset: 1

Labos Elemer, Jun 28 2002

nonn

			For n = 10: a(10) = prime(100) mod 100 = 541 mod 100 = 41.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A004648, A065863, A065134, A065875, A072623.

a[n_] := Mod[Prime[n^2], n^2]; Array[a, 100] (* Amiram Eldar, Mar 17 2025 *)

a(n) = prime(n^2) % (n^2); \\ Amiram Eldar, Mar 17 2025

a(n) = A004648(n^2).

A096197 a(n) = (1+prime(n)) mod n.

Original entry on oeis.org

0, 0, 0, 0, 2, 2, 4, 4, 6, 0, 10, 2, 3, 2, 3, 6, 9, 8, 11, 12, 11, 14, 15, 18, 23, 24, 23, 24, 23, 24, 4, 4, 6, 4, 10, 8, 10, 12, 12, 14, 16, 14, 20, 18, 18, 16, 24, 32, 32, 30, 30, 32, 30, 36, 38, 40, 42, 40, 42, 42, 40, 46, 56, 56, 54, 54, 64, 66, 3, 0, 70, 0, 3, 4, 5, 4, 5, 8, 7, 10, 15

Offset: 1

Labos Elemer, Jul 26 2004

Graph is similar to that of A004648.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A082495, A006521, A087611, A004648.

[(NthPrime(n)+1) mod(n): n in [1..90]]; // Vincenzo Librandi, Sep 11 2014

Table[Mod[Prime[n] + 1, n], {n, 100}] (* Vincenzo Librandi, Sep 11 2014 *)

lista(nn) = {forprime(p=2, n, print1((p+1) % primepi(p), ", "););} \\ Michel Marcus, Sep 11 2014

cp's OEIS Frontend

A087611 a(n) = (prime(n) - 1) mod n.

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A099851 Numbers k such that A099850(k) is divisible by k.

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A245071 a(n) = 12n - prime(n).

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

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Maple

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A363751 Numbers k such that prime(k) mod k is prime.

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A363752 Primes prime(k) such that prime(k) mod k is prime.

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A065521 a(n) = floor(prime(n) / n) * n - prime(n) mod n.

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A072623 Numbers n such that A065863(n) = 1, i.e., prime(n) mod (n - Pi(n)) = 1.

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