A004648 -id:A004648 - cp's OEIS frontend

A004649 a(n) = prime(n) mod (n-1).

0, 1, 1, 3, 3, 5, 5, 7, 2, 1, 4, 5, 4, 5, 8, 11, 10, 13, 14, 13, 16, 17, 20, 1, 1, 25, 26, 25, 26, 7, 7, 9, 7, 13, 11, 13, 15, 15, 17, 19, 17, 23, 21, 21, 19, 27, 35, 35, 33, 33, 35, 33, 39, 41, 43, 45, 43, 45, 45, 43, 49, 59, 59, 57, 57, 1, 2, 7, 4, 3, 4, 7, 8

Offset: 2

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

Vincenzo Librandi, Table of n, a(n) for n = 2..1000

Cf. A004648, A004650.

[NthPrime(n) mod (n-1): n in [2..90]]; // Vincenzo Librandi, Sep 11 2014

Table[Mod[Prime[n], n - 1], {n, 2, 100}] (* Vincenzo Librandi, Sep 11 2014 *)

a(n) = prime(n) % (n-1) \\ Michel Marcus, Jun 12 2013

A038606 Least k such that k-th prime > n * k.

Original entry on oeis.org

1, 5, 12, 31, 69, 181, 443, 1052, 2701, 6455, 15928, 40073, 100362, 251707, 637235, 1617175, 4124437, 10553415, 27066974, 69709680, 179992909, 465769803, 1208198526, 3140421716, 8179002096, 21338685407, 55762149030, 145935689361, 382465573483, 1003652347100

Offset: 1

Vasiliy Danilov (danilovv(AT)usa.net) 1998 Jul

nonn

Log(a(n)) =~ -1.295 + 0.964312n. - Robert G. Wilson v, Jan 25 2002
Numbers n such that prime(n) (mod n) begins the next cycle of terms in A004648. Generally prime(i) (mod i) exceeds prime(i-1) (mod i-1) but there are numerous times where for a short run prime(i) (mod i) is minimally less than its predecessor. Here n is substantially less. See Labos's graph.
A090973(a(n)) = n+1. [From Reinhard Zumkeller, Aug 16 2009]
With offset 2: Index j of prime p(j) such that ceiling[p(j)/j]=n is first satisfied. a(n) = A062742(n) = A038624(n) for n >= 3. [From Jaroslav Krizek, Dec 13 2009]

Giovanni Resta, Table of n, a(n) for n = 1..50
Labos, E. Graph of first 50000 terms of A004648
Andrew R. Booker, The Nth Prime Page

Cf. A038607, A004648, A038625.

A038606 := proc(n)
    for k from 1 do
        if ithprime(k)> n*k then
            return k;
        end if;
    end do:
end proc: # R. J. Mathar, Aug 24 2013

k = 1; Do[ While[ Floor[ Prime[k]/k] < n, k++ ]; Print[k]; k++, {n, 1, 30} ]

k=1;n=1;forprime(p=3,4e9,if(p/n++>k,print1(n", ");k++)) \\ Charles R Greathouse IV, Sep 06 2011

a(n) = pi(A038607(n)) = A000720(A038607(n)).

Edited by Robert G. Wilson v, Jan 25 2002
a(21)=179992909 corrected by Ray Chandler, Dec 01 2004
a(29)-a(30) from Charles R Greathouse IV, Sep 06 2011
a(31)-a(50) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Sep 01 2018

A065133 Remainder when n-th prime is divided by the number of primes not exceeding n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 3, 3, 1, 1, 2, 5, 1, 5, 5, 3, 5, 3, 7, 1, 7, 2, 8, 7, 2, 4, 8, 9, 3, 6, 10, 5, 7, 6, 8, 1, 7, 11, 5, 10, 12, 9, 11, 1, 3, 1, 13, 2, 4, 8, 14, 1, 11, 1, 7, 13, 15, 5, 9, 13, 5, 1, 5, 7, 11, 8, 14, 5, 7, 13, 19, 10, 16, 1, 5, 11, 19, 5, 13, 1, 3, 17, 19, 2, 6, 12, 20, 5, 7, 11, 23

Offset: 2

Labos Elemer, Oct 15 2001

nonn

			n = 2: pi(2) = 1, prime(2) = 3, 3 mod 1 = 0, the first term = a(2);
n = 100: pi(100) = 25, prime(100) = 541, 541 mod 25 = 16 = a(100). [corrected by _Jon E. Schoenfield_, Jun 18 2018]

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

Cf. A000720, A000040, A065134, A004648.

Table[Mod[Prime[n],PrimePi[n]],{n,2,100}] (* Harvey P. Dale, Nov 28 2013 *)

{ for (n=2, 1000, write("b065133.txt", n, " ", prime(n)%primepi(n)) ) } \\ Harry J. Smith, Oct 11 2009

a(n) = prime(n) mod pi(n) = A000040(n) mod A000720(n), n > 1.

A065864 Remainder when n is divided by the number of nonprimes not exceeding n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 21, 21, 21, 21, 21, 21

Offset: 1

Labos Elemer, Nov 26 2001

nonn

			For n=100, pi(100)=25, so a(100) = 100 mod (100-25) = 25.

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

Cf. A000040, A004648, A062298, A065133, A065134, A065858-A065864.

Table[Mod[n, n - PrimePi@ n], {n, 78}] (* or *)
Table[Mod[n, Count[Range@ n, k_ /; ! PrimeQ@ k]], {n, 78}] (* Michael De Vlieger, Jan 01 2017 *)

{ for (n = 1, 1000, a=n%(n - primepi(n)); write("b065864.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 02 2009

a(n) = n mod (n-pi(n)) = n mod (n-A000720(n)) = n mod A062298(n).

A069547 a(n) = n^2 mod n-th prime.

Original entry on oeis.org

1, 1, 4, 2, 3, 10, 15, 7, 12, 13, 28, 33, 5, 24, 37, 44, 53, 19, 26, 45, 3, 10, 31, 42, 43, 70, 8, 35, 78, 109, 72, 107, 130, 44, 33, 88, 113, 140, 18, 43, 70, 135, 130, 6, 55, 126, 99, 74, 131, 210, 38, 75, 158, 155, 198, 243, 21, 112, 157, 228, 42, 35, 285, 53, 156, 235

Offset: 1

Reinhard Zumkeller, Apr 17 2002

nonn

What is the origin of pattern at n~(6-7)*10^3 (see link to figure)? - Zak Seidov, Nov 22 2011

Zak Seidov, Table of n, a(n) for n = 1..10000
Zak Seidov, Figure of dependence a(n) for n=1..10000

Cf. A000040, A004648.

Table[ PowerMod[n, 2, Prime[n]], {n, 1, 70}]

vector(100, n, n^2 % prime(n)) \\ Michel Marcus, Jun 01 2015

Edited and extended by Robert G. Wilson v, Apr 22 2002

A072609 Changing of parity of remainder A072608(n) from alternation [..010101..] to steadily 1-range [...1111..]. AC-range corresponds to 0, while DC-range labeled by 1.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Jun 24 2002

			Take n = 11,12,13,14: A004648[n]=9,1,2,1. Parity A072608(n) = 1,1,0,1. So ..11.. transforms into 01 between n = 11 and n = 12: a(11) = 1, a(12)=0. With increasing n, A072609(n) changes from ..0000.. into ...1111. reflected by this sequence. by a range consisting only of 1-s. This secondary alternation also goes on.

Cf. A004648, A072608.

mm[x_] := Mod[Mod[Prime[x], x], 2] Table[mm[w]*mm[w+1], {w, 1, 256}]
Times@@@Partition[Table[Mod[Mod[Prime[n],n],2],{n,110}],2,1] (* Harvey P. Dale, Dec 21 2014 *)

a(n)=Mod[A004648(n), 2]*Mod[A004648(n+1), 2]= A072608(n)*A072608(n+1)

A072610 Values of transition of A072608(n) from alternating behavior (0,1,0,1,..) into steadily-1 (1,1,1,..) behavior or changing back. Expressing in terms of A072609(n): at n values it switches from steadily 0 into steadily 1 successive values or back.

Original entry on oeis.org

3, 11, 29, 67, 69, 71, 179, 181, 189, 441, 1059, 2699, 6453, 6459, 6471, 15927, 40071, 40083, 40121, 100363, 251705, 251707, 251709, 251721, 251723, 251735, 251737, 251741, 251761, 637233, 637235, 637319, 637321, 637323, 637325, 637329, 637333

Offset: 1

Labos Elemer, Jun 24 2002

nonn

Values of n such that A072609(n) != A072609(n+1).

Switching zones appear in clusters of n. Remainder A004648 either drops or starts to increase at these values of n.

			At n=637330...637370 the change of remainder A004648 is as follows: {..637323, 4, 13, 4, 637333, 637327, 637335, 637323, 637319, 637325, 637321, 4, 11, 4, 7, 34, 29, 26, 17, 44, 43, 46, 41, 38, 49, 52, 49, 44, 37, 28, 37..}

Cf. A004648, A072608, A072609.

mm[x_] := Mod[Mod[Prime[x], x], 2];
pm[x_] := mm[x]*mm[x+1];
Do[s1=pm[n]; s2=pm[n+1]; If[ !Equal[s1, s2], Print[n]], {n, 10^9}]

A116677 Numbers k such that prime(k) == 12 (mod k).

Original entry on oeis.org

1, 91, 4124467, 27067043, 27067229, 27067261, 27067523, 1208198857, 8179002137, 8179002191

Offset: 1

Giovanni Resta and Robert G. Wilson v, Feb 22 2006

nonn

Cf. A004648, A023143 - A023152, A116657, A116658, A116659: prime(n) == m (mod n), m=1..14.

Cf. A116678.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 1; Do[ If[ Mod[p = NextPrim[p], n] == 12, Print[n]], {n, 10^9}]

def A116677(max) :
    terms = []
    p = 2
    for n in range(1, max+1) :
        if (p - 12) % n == 0 : terms.append(n)
        p = next_prime(p)
    return terms
# Eric M. Schmidt, Feb 05 2013

First term inserted by Eric M. Schmidt, Feb 05 2013

A116657 Numbers k such that prime(k) == 11 (mod k).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 20, 38, 39, 82, 190, 192, 444, 2702, 40079, 40156, 251719, 251725, 251733, 251740, 251788, 637322, 637342, 10553424, 10553571, 10553575, 10553646, 10553824, 27066990, 69709708, 69709870, 69709881, 69709941, 179992918, 179993010

Offset: 1

Zak Seidov, Feb 21 2006

nonn

Starting with a(7), positions of 11 in A004648. - corrected by Eric M. Schmidt, Feb 05 2013

Cf. A004648; A023143 - A023152, A116657, A116677, A116658, A116659: prime(n) == m (mod n), m=1..14.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 1; Do[ If[ Mod[p = NextPrim[p], n] == 11, Print[n]], {n, 10^9}] (* Robert G. Wilson v, Feb 22 2006 *)

def A116657(max) :
    terms = []
    p = 2
    for n in range(1, max+1) :
        if (p - 11) % n == 0 : terms.append(n)
        p = next_prime(p)
    return terms
# Eric M. Schmidt, Feb 05 2013

a(17)-a(35) from Robert G. Wilson v, Feb 22 2006

First six terms inserted by Eric M. Schmidt, Feb 05 2013

A116658 Numbers k such that prime(k) == 13 (mod k).

Original entry on oeis.org

1, 2, 6, 12, 22, 40, 42, 84, 86, 90, 193, 2712, 16056, 16058, 40077, 40123, 40124, 40125, 251720, 251766, 251769, 251787, 637332, 10553432, 10553435, 10553501, 10553568, 10553817, 10553826, 27067042, 27067132, 69709722, 179993160, 465769803

Offset: 1

Zak Seidov, Feb 21 2006

nonn

Starting with a(5), positions of 13 in A004648. - corrected by Eric M. Schmidt, Feb 05 2013

Cf. A004648; A023143 - A023152, A116657, A116677, A116658, A116659: prime(n) == m (mod n), m=1..14.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 1; Do[ If[ Mod[p = NextPrim[p], n] == 13, Print[n]], {n, 10^9}] (* Robert G. Wilson v, Feb 22 2006 *)

def A116658(max) :
    terms = []
    p = 2
    for n in range(1, max+1) :
        if (p - 13) % n == 0 : terms.append(n)
        p = next_prime(p)
    return terms
# Eric M. Schmidt, Feb 05 2013

a(24)-a(34) from Robert G. Wilson v, Feb 22 2006

First four terms inserted by Eric M. Schmidt, Feb 05 2013

cp's OEIS Frontend

A004649 a(n) = prime(n) mod (n-1).

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Magma

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A038606 Least k such that k-th prime > n * k.

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Maple

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A065133 Remainder when n-th prime is divided by the number of primes not exceeding n.

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A065864 Remainder when n is divided by the number of nonprimes not exceeding n.

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A069547 a(n) = n^2 mod n-th prime.

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A072609 Changing of parity of remainder A072608(n) from alternation [..010101..] to steadily 1-range [...1111..]. AC-range corresponds to 0, while DC-range labeled by 1.

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A072610 Values of transition of A072608(n) from alternating behavior (0,1,0,1,..) into steadily-1 (1,1,1,..) behavior or changing back. Expressing in terms of A072609(n): at n values it switches from steadily 0 into steadily 1 successive values or back.

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Mathematica

A116677 Numbers k such that prime(k) == 12 (mod k).

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