cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

Views

Author

Jud McCranie

Keywords

Comments

Based on problem by G. L. Honaker, Jr.
A subsequence of A073465. - Ivan N. Ianakiev, Aug 06 2019

Examples

			13 is the 6th prime and 13 == 1 mod 6.

Links

Crossrefs

Cf. A000040, A023143, A045924, A052013.

Programs

  • Mathematica
    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 *)
  • PARI
    lista(nn) = forprime(p=1, nn, if (Mod(p, primepi(p)) == 1, print1(p, ", "))); \\ Michel Marcus, Jan 08 2015; Aug 06 2019

Formula

A087611(a(n)) = 0. - Reinhard Zumkeller, Sep 11 2003
a(n) = A000040(A023143(n)). - Zak Seidov, Feb 19 2015

Extensions

More terms from Zak Seidov, Feb 19 2015
Terms a(33)-a(37) sorted into correct order by Giovanni Resta, Feb 23 2020

A023148 Numbers k such that prime(k) == 6 (mod k).

Original entry on oeis.org

1, 5, 79, 445, 16055, 27067045, 1208198549, 1208198609, 1208198615, 2636913002983, 885992692751479, 885992692751483, 885992692751833, 43525513764815035, 43525513764815057, 43525513764815173, 43525513764816367
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

a(18) > 2.8*10^20. - Giovanni Resta, Feb 23 2020

Crossrefs

Cf. A092048, A023143, A023144, A023145, A023146, A023147, A023149, A023150, A023151, A023152.

Programs

  • Mathematica
    NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 1; Do[ If[ Mod[p = NextPrim[p], n] == 6, Print[n]], {n, 1, 10^9}] (* Robert G. Wilson v, Feb 18 2004 *)
  • Sage
    def A023148(max) :
    terms = []
    p = 2
    for n in range(1, max+1) :
        if (p - 6) % n == 0 : terms.append(n)
        p = next_prime(p)
    return terms
# Eric M. Schmidt, Feb 05 2013

Extensions

Extended by Robert G. Wilson v, Feb 18 2004
a(7)-a(9) from Robert G. Wilson v, Feb 22 2006
First two terms inserted by Eric M. Schmidt, Feb 05 2013
a(10)-a(17) from Giovanni Resta, Feb 23 2020

A023151 Numbers k such that prime(k) == 9 (mod k).

Original entry on oeis.org

1, 2, 10, 11, 35, 37, 80, 100364, 251711, 251717, 251731, 251735, 251741, 251770, 4124456, 4124582, 27067096, 27067520, 69709706, 69709717, 69709723, 69709868, 69709931, 69709933, 465769825, 465769826, 465769831, 1208198548, 8179002130
Offset: 1

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Cf. A092051, A023143, A023144, A023145, A023146, A023147, A023148, A023149, A023150, A023152.

Programs

  • Mathematica
    NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 1; Do[ If[ Mod[p = NextPrim[p], n] == 9, Print[n]], {n, 1, 10^9}] (* Robert G. Wilson v, Feb 18 2004 *)
  • Sage
    def A023151(max) :
    terms = []
    p = 2
    for n in range(1, max+1) :
        if (p - 9) % n == 0 : terms.append(n)
        p = next_prime(p)
    return terms
# Eric M. Schmidt, Feb 05 2013

Extensions

More terms from Robert G. Wilson v, Feb 18 2004
a(25)-a(29) from Robert G. Wilson v, Feb 22 2006
First two terms inserted by Eric M. Schmidt, Feb 05 2013

A052013 Primes that are congruent to -1 mod n, where n is the index of the prime.

Original entry on oeis.org

2, 3, 5, 7, 29, 349, 359, 1091, 3079, 8423, 64579, 64609, 64709, 481043, 481067, 3524317, 3524387, 9559799, 9560009, 9560039, 25874767, 70115921, 189962009, 189962189, 189964241, 189964259, 189964331, 189964367, 189968741, 189968921
Offset: 1

Views

Author

Patrick De Geest, Nov 15 1999

Keywords

Examples

			29 is the tenth prime and 29 == -1 mod 10, so 29 is in the sequence.
31 is the eleventh prime but 31 == 9 mod 11, so 31 is not in the sequence.

Links

Crossrefs

Cf. A045924, A023143, A048891.
Subsequence of A162567.

Programs

  • Mathematica
    divbleQ[m_, n_] := Mod[m, n] == 0; A052013 = {}; Do[p = Prime[n]; If[divbleQ[p + 1, n], AppendTo[A052013, p]], {n, 10!}]; A052013 (* Vladimir Joseph Stephan Orlovsky, Dec 08 2009 *)
Select[Prime[Range[5000]], Divisible[# + 1, PrimePi[#]] &] (* Alonso del Arte, May 12 2017 *)
Select[Table[{n,Prime[n]},{n,1056*10^4}],Mod[#[[2]],#[[1]]]==#[[1]]-1&][[All,2]] (* Harvey P. Dale, Oct 29 2022 *)
  • PARI
    lista(nn) = forprime(p=2, nn, if (Mod(p,primepi(p)) + 1 == 0, print1(p, ", "))) \\ Michel Marcus, Jan 09 2015
  • PARI
    list(lim)=my(v=List(), n, t); forprime(p=2, lim, t=(p+1)/n++; if(denominator(t)==1, listput(v, p))); Vec(v) \\ Charles R Greathouse IV, Feb 18 2016

Formula

a(n) = prime(A045924(n)). - Michel Marcus, Jan 09 2015

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

Views

Author

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

Keywords

Crossrefs

Cf. A004648, A023143 - A023152, A116657, A116658, A116659: prime(n) == m (mod n), m=1..14.
Cf. A116678.

Programs

  • Mathematica
    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}]
  • Sage
    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

Extensions

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

Views

Author

Zak Seidov, Feb 21 2006

Keywords

Comments

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

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • Sage
    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

Extensions

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

Views

Author

Zak Seidov, Feb 21 2006

Keywords

Comments

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

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • Sage
    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

Extensions

a(24)-a(34) from Robert G. Wilson v, Feb 22 2006
First four terms inserted by Eric M. Schmidt, Feb 05 2013

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

Original entry on oeis.org

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

Views

Author

Zak Seidov, Feb 21 2006

Keywords

Comments

Starting with a(4), positions of 14 in A004648. - Robert G. Wilson v, Feb 22 2006, corrected by Eric M. Schmidt, Feb 05 2013

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • Sage
    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

Extensions

a(9)-a(13) from Robert G. Wilson v, Feb 22 2006
First three terms inserted by Eric M. Schmidt, Feb 05 2013

A077255 Numbers k such that prime(k)^k == 1 (mod k).

Original entry on oeis.org

2, 4, 5, 6, 8, 10, 12, 14, 16, 18, 20, 24, 27, 32, 36, 40, 42, 48, 50, 52, 54, 60, 64, 70, 72, 80, 84, 96, 100, 105, 108, 110, 114, 120, 121, 124, 125, 126, 128, 136, 144, 148, 156, 160, 162, 168, 180, 181, 182, 189, 192, 200, 210, 216, 220, 231, 234, 240, 243, 246
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 31 2002

Keywords

Comments

Contains A023143. All terms not in A023143 are in A060679. - Robert Israel, Oct 31 2016

Examples

			prime(16)^16 mod 16 = 53^16 mod 16 = 3876269050118516845397872321 mod 16 = 1, therefore 16 is a term.

Links

Crossrefs

Cf. A000040, A023143, A060679, A077254, A077256.

Programs

  • Maple
    select(n -> ithprime(n) &^ n mod n = 1, [$1..1000]); # Robert Israel, Oct 31 2016
  • Mathematica
    Select[Range[1000], PowerMod[Prime[#], #, #] == 1&] (* Jean-François Alcover, Dec 16 2021 *)
  • PARI
    isok(k) = lift(Mod(prime(k), k)^k) == 1; \\ Michel Marcus, Dec 16 2021

Formula

A077254(a(n)) = 1; A077256(n) = A000040(a(n)).

A078931 Numbers k that divide prime(k)+1 or prime(k)-1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 12, 14, 70, 72, 181, 182, 440, 1053, 6458, 6459, 6460, 6461, 6466, 6471, 40087, 40089, 100362, 251712, 251732, 251737, 251742, 637236, 637320, 637334, 637336, 1617173, 4124466, 10553445, 10553455, 10553504, 10553505, 10553547, 10553569
Offset: 1

Views

Author

Benoit Cloitre, Jan 12 2003

Keywords

Examples

			181 is in the sequence because the 181st prime is 1087, and 1086 is divisible by 181 (although 1088 is not so divisible).

Links

Crossrefs

Cf. A000720, A023143, A045924, A162567.

Programs

  • Mathematica
    ndpQ[n_]:=Module[{p=Prime[n]},Divisible[p-1,n]||Divisible[p+1,n]]; Select[Range[100000],ndpQ]  (* Harvey P. Dale, Apr 03 2011 *)

Formula

Equals A023143 union A045924.
a(n) = A000720(A162567(n)). - Alois P. Heinz, Feb 20 2023

Extensions

Corrected and example added by Harvey P. Dale, Apr 03 2011
Extended with terms from A023143 and A045924 by Michel Marcus, Nov 30 2013
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