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 1-3 of 3 results.

A285015 Primes of the form k * b^b - 1, with b > 1.

Original entry on oeis.org

3, 7, 11, 19, 23, 31, 43, 47, 53, 59, 67, 71, 79, 83, 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, 199, 211, 223, 227, 239, 251, 263, 269, 271, 283, 307, 311, 331, 347, 359, 367, 379, 383, 419, 431, 439, 443, 463, 467, 479, 487, 491, 499, 503, 523, 547
Offset: 1

Views

Author

Vincenzo Librandi, May 08 2017

Keywords

Comments

This sequence has relative density 1 - Prod_p 1-1/(p^p-p) = 0.52098749404893... in the primes, hence a(n) ~ kn log n where k = 1.91943... is the reciprocal of this quantity. - Charles R Greathouse IV, May 12 2017

Examples

			a(1) = 1*(2^2)-1 = 3.
a(2) = 2*(2^2)-1 = 7.
a(9) = 2*(3^3)-1 = 53.

Links

Crossrefs

Cf. A175768.

Programs

  • Maple
    N:= 1000: # to get all terms <= N
bmax:= floor(ln(N+1)/LambertW(ln(N+1))):
sort(convert(select(isprime, {seq(seq(k*b^b-1, k=1..(N+1)/b^b),b=2..bmax)}),list)); # Robert Israel, May 11 2017
  • Mathematica
    Take[Select[Union@Flatten@Table[k b^b - 1, {b, 2, 20}, {k, 148}], PrimeQ], 55]
  • PARI
    upto(n)=my(l=List([3]), b=2, s=1); n++; while(b^b < n, c = b^b; forstep(i=2, n\c, s, if(isprime(i*c-1), listput(l, i*c-1))); s=3-s; b++); listsort(l, 1); l \\ David A. Corneth, May 11 2017
  • PARI
    is(n)=if(!isprime(n), return(0)); my(t); forprime(p=2,, t=p^p; if((n+1)%t==0, return(1)); if(t>=n, return(0))) \\ Charles R Greathouse IV, May 11 2017

A235467 Primes whose base-4 representation also is the base-3 representation of a prime.

Original entry on oeis.org

2, 89, 137, 149, 281, 293, 353, 389, 409, 421, 593, 613, 661, 1097, 1109, 1289, 1301, 1321, 1381, 1409, 1601, 1609, 1669, 2069, 2129, 2309, 2377, 2389, 2729, 4133, 4229, 4373, 4441, 4513, 4673, 5153
Offset: 1

Views

Author

M. F. Hasler, Jan 12 2014

Keywords

Comments

This sequence is part of the two-dimensional array of sequences based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.
For further motivation and cross-references, see sequence A235265 which is the main entry for this whole family of sequences.
This is a subsequence of A002144, A002313, A003655, A050150, A062090, A141293, A175768, A192592, A226181 (conjectural).

Examples

			E.g., 89 = 1121_4 and 1121_3 = 43 both are prime.

Links

Crossrefs

Cf. A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235461 - A235482. See the LINK for further cross-references.

Programs

  • Mathematica
    b4b3Q[n_]:=Module[{b4=IntegerDigits[n,4]},Max[b4]<3&&PrimeQ[ FromDigits[ b4,3]]]; Select[Prime[Range[700]],b4b3Q] (* Harvey P. Dale, Dec 14 2021 *)
  • PARI
    is(p,b=3,c=4)=vecmax(d=digits(p,c))
  • PARI
    forprime(p=1,1e3,is(p,4,3)&&print1(vector(#d=digits(p,3),i,4^(#d-i))*d~,",")) \\ To produce the terms, this is more efficient than to select them using straightforwardly is(.)=is(.,3,4)

A286658 Primes of the form p*b^b + 1, where p is a prime and b>1.

Original entry on oeis.org

13, 29, 53, 149, 173, 269, 293, 317, 389, 509, 557, 653, 769, 773, 797, 1109, 1229, 1493, 1637, 1733, 1949, 1997, 2309, 2477, 2693, 2837, 2909, 2957, 3329, 3413, 3533, 3677, 3989, 4133, 4157, 4253, 4349, 4373, 4493, 4517, 5189, 5309, 5693, 5717, 5813, 6173
Offset: 1

Views

Author

Vincenzo Librandi, May 12 2017

Keywords

Examples

			a(1) = 3*(2^2)+1 = 13.
a(2) = 7*(2^2)+1 = 29.
a(3) = 13*(2^2)+1 = 53.

Links

Crossrefs

Cf. A175768, A285015.

Programs

  • Maple
    N:= 10000: # for all terms <= N
Res:= NULL:
P:= select(isprime, [2,seq(i,i=3..N/4,2)]):
for b from 2  do
  q:= b^b; if q > N/2 then break fi;
  for i from 1 to nops(P) do
     x:= P[i]*q+1;
     if x > N then break fi;
     if isprime(x) then Res:= Res, x fi;
od od:
sort(convert({Res},list)); # Robert Israel, Nov 12 2019
  • Mathematica
    nmax=10^4; pimax=PrimePi[nmax]; bmax=1;While[(bmax+1)^(bmax+1)<=nmax,bmax++];Select[Union@Flatten@Table[Prime[pi] b^b+1,{b,2,bmax},{pi,pimax}],PrimeQ[#]&&#<=nmax&]
  • PARI
    list(lim)=my(v=List()); lim\=1; for(b=2,oo, my(p=2*b^b+1); if(p>lim, break); if(isprime(p), listput(v,p))); forstep(b=2,oo,2, my(B=b^b); if(3*B+1>lim, break); forprime(q=3,(lim-1)\B, my(p=q*B+1); if(isprime(p), listput(v,p)))); Set(v) \\ Charles R Greathouse IV, Jun 16 2022
Showing 1-3 of 3 results.

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