A004648 -id:A004648 - cp's OEIS frontend

A116659 Numbers k such that prime(k) == 14 (mod k).

1, 3, 9, 23, 81, 85, 87, 16057, 4124457, 27067011, 27067127, 1208198605, 1208198851

Offset: 1

Zak Seidov, Feb 21 2006

nonn

Starting with a(4), positions of 14 in A004648. - Robert G. Wilson v, Feb 22 2006, 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] == 14, Print[n]], {n, 10^9}] (* Robert G. Wilson v *)

def A116659(max) :
    terms = []
    p = 2
    for n in range(1, max+1) :
        if (p - 14) % n == 0 : terms.append(n)
        p = next_prime(p)
    return terms
# Eric M. Schmidt, Feb 05 2013

a(9)-a(13) from Robert G. Wilson v, Feb 22 2006

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

A072630 Values of n where A072629 switches from 01010.. into 0000.. or back.

Original entry on oeis.org

1, 7, 19, 53, 147, 403, 1095, 2979, 8103, 22025, 59873, 162753, 442413, 1202603, 3269017, 8886109, 24154951, 65659969, 178482299, 485165195, 1318815733, 3584912845, 9744803445, 26489122129, 72004899337, 195729609427

Offset: 1

Labos Elemer, Jun 28 2002

nonn

See also A004648, A072608, A072609, A072610, A072630.

m[x_] := Mod[x*Floor[Log[x]//N],2]; Do[s=m[n]+m[n+1]; s1=m[n+1]+m[n+2]; If[ !Equal[s1,s],Print[n]],{n,1,1000000}]

See program below.

a(n) = A000149(n) or A000149(n)-1 whichever is odd. [From Max Alekseyev, Feb 06 2010]

More terms from Max Alekseyev, Feb 06 2010

A065863 Remainder when n-th prime is divided by the number of nonprimes not exceeding n.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 3, 3, 5, 1, 2, 6, 3, 2, 3, 9, 6, 1, 11, 8, 9, 13, 14, 1, 16, 13, 12, 14, 13, 7, 5, 5, 1, 5, 1, 7, 7, 5, 5, 11, 7, 17, 13, 11, 7, 19, 25, 23, 19, 17, 17, 19, 23, 23, 23, 23, 19, 25, 23, 25, 29, 37, 35, 31, 29, 43, 43, 47, 43, 47, 47, 3, 2, 1, 53, 53, 55, 2, 3, 6, 1, 11, 6

Offset: 1

Labos Elemer, Nov 26 2001

nonn

			For n=25, prime(25)=97, n - pi(n) = 25 - 9 = 16, a(25)=1 because 97 = 6*16 + 1.

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

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

Table[Mod[Prime[n],n-PrimePi[n]],{n,90}] (* Harvey P. Dale, Aug 04 2015 *)

a(n) = { prime(n)%(n - primepi(n)) } \\ Harry J. Smith, Nov 02 2009

a(n) = prime(n) mod (n - pi(n)) = A000040(n) mod A062298(n).

A072631 Floor( n*log(n) ) mod n.

Original entry on oeis.org

0, 1, 0, 1, 3, 4, 6, 0, 1, 3, 4, 5, 7, 8, 10, 12, 14, 16, 17, 19, 0, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, 14, 16, 17, 19, 21, 22, 24, 25, 27, 29, 30, 32, 34, 36, 38, 39, 41, 43, 45, 47, 49, 51, 53, 0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29

Offset: 1

Labos Elemer, Jun 28 2002

nonn

Compare with A004648 because prime(n) ~ n * log(n).

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

Cf. A004648, A072608, A072609, A072610, A072630.

[Floor(n*Log(n)) mod n: n in [1..100]]; // Vincenzo Librandi, Jun 30 2014

Table[Floor[Mod[n*Log[n],n]],{n,80}] (* Harvey P. Dale, Jun 29 2014 *)

a(n)=floor(n*log(n))%n \\ Charles R Greathouse IV, Aug 01 2011

A073325 a(n) = least k > 0 such that prime(k) == n (mod k).

Original entry on oeis.org

1, 2, 3, 4, 75, 9, 79, 18, 17, 10, 19, 20, 91, 22, 23, 41, 83, 24, 16049, 43, 2711, 94, 25, 26, 95, 198, 449, 452, 99, 50, 451, 48, 453, 1072, 447, 54, 16043, 55, 2719, 56, 459, 57, 101, 472, 100371, 62, 105, 102, 103, 104, 467, 110, 107, 65, 109, 63, 115, 118, 117

Offset: 1

Labos Elemer, Jul 30 2002

nonn

First appearance of n-1 in A004648. Are all positive integers present in A004648 and hence in this sequence? - Zak Seidov, Sep 02 2012

			a(4) = 75 as prime(75) = 379 == 4 (mod 75).
a(44) = 100371 since prime(100371) = 1304867 == 44 (mod 100371) and prime(k) <> 44 (mod k) for k < 100371.

Zak Seidov, Table of n, a(n) for n = 1..301

Cf. A000040, A002808, A004648, A073324, A073326.

nn = 60; f[x_] := Mod[Prime[x], x]; t = Table[0, {nn}]; k = 0; While[Times @@ t == 0, k++; n = f[k]; If[n <= nn && t[[n]] == 0, t[[n]] = k]]; Join[{1}, t]
lk[n_]:=Module[{k=1},While[Mod[Prime[k],k]!=n,k++];k]; Array[lk,60,0] (* Harvey P. Dale, Nov 29 2013 *)

stop=110000; for(n=0,59,k=1; while(k

from sympy import prime, nextprime
def A073325(n):
    p, m = prime(n), n
    while p%m != n-1:
        p = nextprime(p)
        m += 1
    return m # Chai Wah Wu, Mar 18 2023

a(n) = Min{x; Mod[A000040(x), x]=n} = Min{x; A004648[x]=n}.

Definition revised by N. J. A. Sloane, Aug 12 2009

A216162 merged into this sequence by T. D. Noe, Sep 07 2012

A125718 a(1)=1. a(n) = the smallest positive integer not occurring earlier in the sequence such that the n-th prime is congruent to a(n) (mod n).

Original entry on oeis.org

1, 3, 2, 7, 6, 13, 10, 11, 5, 9, 20, 25, 15, 29, 17, 21, 8, 43, 48, 31, 52, 35, 14, 41, 22, 23, 49, 51, 80, 53, 34, 67, 38, 37, 44, 79, 46, 87, 50, 93, 56, 55, 19, 61, 62, 107, 70, 127, 129, 179, 131, 83, 82, 89, 92, 39, 98, 97, 100, 101, 161, 45, 118, 119, 183, 185, 63, 65

Offset: 1

Leroy Quet, Feb 01 2007

nonn

This sequence seems likely to be a permutation of the positive integers. It will be if every positive number appears in A004648 (cf. A127149, A127150).

If this is a permutation of the positive integers, then A249678 is the inverse permutation. - M. F. Hasler, Nov 03 2014

Ferenc Adorjan, Table of n,a(n) for n=1,10000
Ferenc Adorjan, Some characteristics of _Leroy Quet_'s permutation sequences
Ferenc Adorjan, More about the structure of _Leroy Quet_'s sequences A125715, A125717, A125718 & A125727

Cf. A004648.

f[l_List] := Block[{n = Length[l] + 1, k = Mod[Prime[n], n, 1]},While[MemberQ[l, k], k += n];Append[l, k]];Nest[f, {1}, 70] (* Ray Chandler, Feb 04 2007 *)

{Quet_p3(n)= /* Permutation sequence a'la Leroy Quet, A125718 */local(x=[1],k=0,w=1); for(i=2,n,if((k=prime(i)%i)==0,k=i);while(bittest(w,k-1)>0,k+=i);x=concat(x,k);w+=2^(k-1));return(x)}

A125718(n,show=0,u=1)={for(n=1,n,p=prime(n)%n;while(bittest(u,p),p+=n);u+=1<M. F. Hasler, Nov 03 2014

Extended by Ray Chandler, Feb 04 2007

A127064 a(0)=1. a(n) = a(prime(n)(mod n)) + 1, where prime(n) is the n-th prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 3, 5, 5, 4, 5, 5, 3, 4, 3, 4, 4, 6, 6, 6, 6, 6, 5, 4, 7, 6, 5, 6, 5, 6, 5, 5, 5, 4, 5, 5, 6, 5, 6, 6, 5, 5, 5, 7, 7, 7, 5, 5, 6, 6, 7, 7, 6, 7, 6, 6, 7, 6, 7, 6, 6, 7, 8, 7, 7, 8, 8, 8, 9, 4, 5, 5, 6, 4, 5, 6, 5, 6, 6, 4, 5, 4, 6, 5, 5, 4, 5, 4, 7, 5, 5, 4, 7, 6, 7, 8, 5, 8, 6, 6, 6, 6, 6, 7, 7

Offset: 0

Leroy Quet, Mar 21 2007

nonn

			The 7th prime, 17, is congruent to 3 (mod 7). So a(7) = a(3) + 1 = 4 + 1 = 5.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A004648, A127066.

a[0]:=1: for n from 1 to 125 do a[n]:=1+a[ithprime(n) mod n] od: seq(a[n],n=0..125); # Emeric Deutsch, Mar 25 2007

f[l_List] := Block[{n = Length[l]},Append[l, l[[Mod[Prime[n], n] + 1]] + 1]];Nest[f, {1}, 105] (* Ray Chandler, Mar 25 2007 *)

a(n)={k=1;if(n>0,k=a(prime(n)%n)+1);k;} \\ Jinyuan Wang, Feb 01 2019

Extended by Ray Chandler and Emeric Deutsch, Mar 25 2007

A004650 a(n) = prime(n) mod (n+1).

Original entry on oeis.org

0, 0, 1, 2, 5, 6, 1, 1, 3, 7, 7, 11, 13, 13, 15, 2, 5, 4, 7, 8, 7, 10, 11, 14, 19, 20, 19, 20, 19, 20, 31, 32, 1, 34, 5, 3, 5, 7, 7, 9, 11, 9, 15, 13, 13, 11, 19, 27, 27, 25, 25, 27, 25, 31, 33, 35, 37, 35, 37, 37, 35, 41, 51, 51, 49, 49, 59, 61, 67, 65, 65, 67

Offset: 1

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

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

Cf. A004648, A004649.

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

Table[Mod[Prime[n], n + 1], {n, 1, 90}] (* Vincenzo Librandi, Sep 11 2014 *)

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

A065859 Remainder when the n-th prime is divided by the n-th composite number.

Original entry on oeis.org

2, 3, 5, 7, 1, 1, 3, 4, 7, 11, 11, 16, 19, 19, 22, 1, 5, 5, 7, 7, 7, 11, 13, 17, 21, 23, 23, 23, 21, 23, 35, 35, 39, 39, 47, 47, 49, 53, 55, 2, 5, 1, 5, 4, 5, 4, 13, 19, 20, 19, 17, 17, 16, 23, 26, 29, 29, 28, 31, 29, 28, 35, 46, 47, 43, 44, 55, 58, 65, 64, 65, 65, 70, 73, 73, 71

Offset: 1

Labos Elemer, Nov 26 2001

nonn

			n=100, p(100)=541, c(100)=133, a(100)=9 because 541 = 4*133 + 9.

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

Cf. A000040, A002808, A004648, A065133, A065134, A065855-A065864, A065134.

a[n]=Mod[p(n), c(n)]=Mod[A000040(n), A002808(n)]
With[{nn=80},Module[{prs=Prime[Range[nn]],comps},comps=Take[Complement[ Range[2,Prime[nn]+1],prs],Length[prs]];Mod[#[[1]],#[[2]]]&/@ Thread[ {prs,comps}]]] (* Harvey P. Dale, Apr 18 2012 *)

Composite(n) = { local(k); k=n + primepi(n) + 1; while (k != n + primepi(k) + 1, k = n + primepi(k) + 1); return(k) } { for (n = 1, 1000, a=prime(n)%Composite(n); write("b065859.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 01 2009

A072629 Parity of n*floor(log n).

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Jun 28 2002

nonn

a(n)=1 for n: 3, 5, 7, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 149, 151, .... - Robert G. Wilson v, Feb 01 2015

			Parity either alternates or it is steadily 0. Intervals of such kind also change and return: 01010...0000....0101.., etc.

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for characteristic functions

Cf. A004648, A072608, A072609, A072610, A072630.

f[n_] := Mod[n*Floor@ Log@ n, 2]; Array[f, 105] (* Robert G. Wilson v, Feb 01 2015 *)

a(n) = (log(n)\1*n)%2; \\ Charles R Greathouse IV, Sep 04 2015, corrected by Antti Karttunen, Feb 06 2019

a(n) = n*floor(log(n)) mod 2.

cp's OEIS Frontend

A116659 Numbers k such that prime(k) == 14 (mod k).

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A072630 Values of n where A072629 switches from 01010.. into 0000.. or back.

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A065863 Remainder when n-th prime is divided by the number of nonprimes not exceeding n.

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A072631 Floor( n*log(n) ) mod n.

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Magma

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A073325 a(n) = least k > 0 such that prime(k) == n (mod k).

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A125718 a(1)=1. a(n) = the smallest positive integer not occurring earlier in the sequence such that the n-th prime is congruent to a(n) (mod n).

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A127064 a(0)=1. a(n) = a(prime(n)(mod n)) + 1, where prime(n) is the n-th prime.

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