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.

Showing 1-5 of 5 results.

A086708 Primes p such that p-1 and p+1 are both divisible by cubes (other than 1).

Original entry on oeis.org

271, 487, 593, 751, 809, 919, 1249, 1567, 1783, 1889, 1999, 2647, 2663, 2753, 2969, 3079, 3511, 3617, 3727, 3833, 3943, 4049, 4159, 4481, 4591, 4751, 4801, 5023, 6857, 6967, 7937, 8263, 8369, 9127, 9343, 10289, 10313, 10529, 10639, 11071, 11177
Offset: 1

Views

Author

Jason Earls and Amarnath Murthy, Jul 28 2003

Keywords

Links

Crossrefs

Cf. A162870 (subsequence).

Programs

  • Maple
    isA086708 := proc(n)
    if isprime(n) then
        isA046099(n-1) and isA046099(n+1) ;
    else
        false;
    end if;
end proc:
n := 1:
for c from 1 to 50000 do
    if isA086708(c) then
        printf("%d %d\n",n,c) ;
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Dec 08 2015
Res:= NULL: count:= 0:
p:= 1:
while count < 100 do
  p:= nextprime(p);
  if max(seq(t[2],t=ifactors(p-1)[2]))>=3 and max(seq(t[2],t=ifactors(p+1)[2]))>=3 then
    count:= count+1; Res:= Res, p;
  fi
od:
Res; # Robert Israel, Jul 11 2018
  • Mathematica
    f[n_]:=Max[Last/@FactorInteger[n]]; lst={};Do[p=Prime[n];If[f[p-1]>=3&&f[p+1]>=3,AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 03 2009 *)
dbcQ[p_]:=AnyTrue[Surd[#,3]&/@Rest[Divisors[p-1]],IntegerQ]&&AnyTrue[Surd[#,3]&/@Rest[ Divisors[ p+1]],IntegerQ]; Select[ Prime[Range[1500]],dbcQ] (* Harvey P. Dale, Sep 21 2024 *)
  • PARI
    \\ Input no. of iterations n, power p and number to subtract and add k.
powerfreep4(n,p,k) = { c=0; pc=0; forprime(x=2,n, pc++; if(!ispowerfree(x-k,p) && !ispowerfree(x+k,p), c++; print1(x","); ) ); print(); print(c","pc","c/pc+.0) }
ispowerfree(m,p1) = { flag=1; y=component(factor(m),2); for(i=1,length(y), if(y[i] >= p1,flag=0;break); ); return(flag) } \\ Cino Hilliard, Dec 08 2003

Formula

{p in A000040: p+1 in A046099 and p-1 in A046099}. - R. J. Mathar, Dec 08 2015
A089199 INTERSECT A089200. - R. J. Mathar, Dec 08 2015

Extensions

Definition clarified by Harvey P. Dale, Sep 21 2024

A007459 Higgs's primes: a(n+1) = smallest prime > a(n) such that a(n+1)-1 divides the product (a(1)...a(n))^2.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 79, 101, 107, 127, 131, 139, 149, 151, 157, 173, 181, 191, 197, 199, 211, 223, 229, 263, 269, 277, 283, 311, 317, 331, 347, 349, 367, 373, 383, 397, 419, 421, 431, 461, 463, 491, 509, 523, 547, 557, 571
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Named after the British mathematician Denis A. Higgs (1932-2011). - Amiram Eldar, Jun 05 2021
No prime of the form a*b^k + 1 (those in A089200) with a > 0, b > 1 and k > 2 is a Higgs's prime. - Mauro Fiorentini, Aug 08 2023

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A057447, A057448, A057459, A089200, A282027.

Programs

  • Haskell
    a007459 n = a007459_list !! (n-1)
a007459_list = f 1 a000040_list where
  f q (p:ps) = if mod q (p - 1) == 0 then p : f (q * p ^ 2) ps else f q ps
-- Reinhard Zumkeller, Apr 14 2013
  • Maple
    a:=[2]; P:=1; j:=1;
for n from 2 to 32 do
P:=P*a[n-1]^2;
  for i from j+1 to 250 do
  if (P mod (ithprime(i)-1)) = 0 then
  a:=[op(a),ithprime(i)]; j:=i; break; fi;
od:
od:
a; # N. J. A. Sloane, Feb 12 2017
  • Mathematica
    f[ n_List ] := (a = n; b = Apply[ Times, a^2 ]; d = NextPrime[ a[ [ -1 ] ] ]; While[ ! IntegerQ[ b/(d - 1) ] || d > b, d = NextPrime[ d ] ]; AppendTo[ a, d ]; Return[ a ]); Nest[ f, {2}, 75 ]
nxt[{p_,a_}]:=Module[{np=NextPrime[a]},While[PowerMod[p,2,np-1] != 0,np = NextPrime[np]];{p*np,np}]; NestList[nxt,{2,2},60][[All,2]] (* Harvey P. Dale, Jul 09 2021 *)
  • PARI
    step(v)=my(N=vecprod(v)^2);forprime(p=v[#v]+1,,if(N%(p-1)==0,return(concat(v,p))))
first(n)=my(v=[2]);for(i=2,n,v=step(v));v \\ Charles R Greathouse IV, Jun 11 2015

Extensions

More terms from David W. Wilson
Definition clarified by N. J. A. Sloane, Feb 12 2017

A166003 Primes p such that p+-1, p+-2 and p+-3 are not squarefree.

Original entry on oeis.org

47527, 186247, 218527, 245149, 269953, 377543, 390449, 432277, 447823, 453053, 469649, 518123, 568177, 584911, 589273, 606323, 632347, 661547, 761347, 831751, 848213, 897577, 913327, 925949, 952253, 1172351, 1205647, 1220347, 1241477
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Oct 03 2009

Keywords

Links

Crossrefs

Cf. A086708, A086709, A089199, A089200, A089212, A166000, A166001, A166002.

Programs

  • Mathematica
    f[n_]:=Max[Last/@FactorInteger[n]]; q=2;lst={};Do[p=Prime[n];If[f[p-3]>=q&&f[p-2]>=q&&f[p-1]>=q&&f[p+1]>=q&&f[p+2]>=q&&f[p+3]>=q,AppendTo[lst,p]],{n,6*8!}];lst
Select[Prime[Range[100000]],NoneTrue[#+{-3,-2,-1,1,2,3},SquareFreeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 17 2018 *)

Extensions

Edited by N. J. A. Sloane, Oct 04 2009

A166001 Primes p such that p-5, p-4, p+4, and p+5 are each divisible by a cube > 1.

Original entry on oeis.org

751379, 2414507, 2839621, 3170371, 4469629, 5736371, 21154909, 22556371, 22991629, 23313371, 23748629, 24338371, 28372621, 31628371, 32079757, 33009629, 41078371, 42270629, 43465307, 44446621, 49746667, 50579339
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Oct 03 2009

Keywords

Links

Crossrefs

Cf. A086708, A086709, A089199, A089200, A089212, A166000.

Programs

  • Mathematica
    f[n_]:=Max[Last/@FactorInteger[n]]; q=3;lst={};Do[p=Prime[n];If[f[p-5]>=q&&f[p-4]>=q&&f[p+4]>=q&&f[p+5]>=q,AppendTo[lst,p]],{n,4*8!}];lst

Extensions

Extended by Charles R Greathouse IV, Oct 09 2009

A166002 Primes p such that p-6, p-5, p+5, and p+6 are each divisible by a cube greater than 1.

Original entry on oeis.org

1934869, 6136619, 11195869, 11845499, 12385381, 33919619, 39139381, 39790381, 52937869, 53209381, 53631131, 54601619, 58690381, 62892131, 67951381, 77212381, 80224619, 88874869, 94544869, 95734381, 99936131, 103805869, 108827869
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Oct 03 2009

Keywords

Crossrefs

Cf. A086708, A086709, A089199, A089200, A089212, A166000, A166001

Programs

  • Mathematica
    f[n_]:=Max[Last/@FactorInteger[n]]; q=3;lst={};Do[p=Prime[n];If[f[p-6]>=q&&f[p-5]>=q&&f[p+5]>=q&&f[p+6]>=q,AppendTo[lst,p]],{n,5*9!}];lst

Extensions

Edited by N. J. A. Sloane, Oct 04 2009
Extended and edited by Charles R Greathouse IV, May 12 2010
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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