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.
197, 719 and 971 are terms of the sequence, because all three numbers are prime, each number can be obtained by cyclically permuting the digits of one of the other numbers and there exist some composites, namely 791 and 917, that can be obtained from non-cyclic permutations of the digits of those three numbers. - _Felix Fröhlich_, Aug 10 2018
Select[Prime@ Range@ PrimePi[10^6], Union[d = IntegerDigits[#], {1,3,7,9}] == {1, 3, 7, 9} && AllTrue[ RotateLeft[d, #] & /@ Range@ IntegerLength@ #, PrimeQ@ FromDigits@ # &] && AnyTrue[ FromDigits /@ Permutations[d], CompositeQ] &] (* Giovanni Resta, Aug 10 2018 *)
eva(n) = subst(Pol(n), x, 10) rot(n) = if(#Str(n)==1, v=vector(1), v=vector(#n-1)); for(i=2, #n, v[i-1]=n[i]); u=vector(#n); for(i=1, #n, u[i]=n[i]); v=concat(v, u[1]); v is_circularprime(n) = my(d=digits(n), r=rot(d)); if(vecmin(d)==0, return(0), while(1, if(!ispseudoprime(eva(r)), return(0)); r=rot(r); if(r==d, return(1)))) find_index_a(vec) = my(r=#vec-1); while(1, if(vec[r] < vec[r+1], return(r)); r--; if(r==0, return(-1))) find_index_b(r, vec) = my(s=#vec); while(1, if(vec[r] < vec[s], return(s)); s--; if(s==r, return(-1))) switch_elements(vec, firstpos, secondpos) = my(g); g=vec[secondpos]; vec[secondpos]=vec[firstpos]; vec[firstpos] = g; vec reverse_order(vec, r) = my(v=[], w=[]); for(x=1, r, v=concat(v, vec[x])); for(y=r+1, #vec, w=concat(w, vec[y])); w=Vecrev(w); concat(v, w) next_permutation(vec) = my(r=find_index_a(vec)); if(r==-1, return(0), my(s=find_index_b(r, vec)); vec=switch_elements(vec, r, s); vec=reverse_order(vec, r)); vec is_permutable_prime(n) = if(n < 10, return(1)); my(d=vecsort(digits(n))); while(1, if(!ispseudoprime(eva(d)), return(0)); d=next_permutation(d); if(d==0, return(1))) forprime(p=1, , if(is_circularprime(p) && !is_permutable_prime(p), print1(p, ", "))) \\ Felix Fröhlich, Aug 05 2018
a(19) = 19 because the digits are already in increasing order. a(20) = 2 because the digits of 20 are 2 and 0, which in increasing order are 0 and 2, but since zero-padding is not allowed on the left, the zero digit is dropped and we are left with 2. a(21) = 12 because the digits of 21 are 2 and 1, which in increasing order are 1 and 2.
import Data.List (sort) a004185 n = read $ sort $ show n :: Integer -- Reinhard Zumkeller, Aug 10 2011
A004185:=func; [n eq 0 select 0 else A004185(n): n in [0..57]]; // Bruno Berselli, Apr 03 2012
A004185 := proc(n) local dgs; convert(n,base,10) ; dgs := sort(%,`>`) ; add( op(i,dgs)*10^(i-1),i=1..nops(dgs)) ; end proc: seq(A004185(n),n=0..20) ; # R. J. Mathar, Jul 26 2015
FromDigits[Sort[DeleteCases[IntegerDigits[#], 0]]]&/@Range[0, 60] (* Harvey P. Dale, Nov 29 2011 *)
a(n)=fromdigits(vecsort(digits(n))) \\ Charles R Greathouse IV, Feb 06 2017
def A004185(n): return int(''.join(sorted(str(n))).replace('0','')) if n > 0 else 0 # Chai Wah Wu, Nov 10 2015
a(19) = 91 because the digits of 19 being 1 and 9, arranged in decreasing order they are 9 and 1. a(20) = 20 because the digits are already in decreasing order.
import Data.List (sort) a004186 = read . reverse . sort . show :: Integer -> Integer -- Reinhard Zumkeller, Aug 19 2011
A004186 := proc(n) local dgs; convert(n,base,10) ; dgs := sort(%) ; add( op(i,dgs)*10^(i-1),i=1..nops(dgs)) ; end proc: seq(A004186(n),n=0..20) ; # R. J. Mathar, Jul 26 2015
sortDigitsDown[n_] := FromDigits@ Reverse@ Sort@ IntegerDigits@ n; Array[sortDigitsDown, 73, 0] (* Robert G. Wilson v, Aug 19 2011 *)
reconstruct(m) = {local(r); r=0; for(i=1,matsize(m)[2],r=r*10+m[i]); r}
A004186(n) = reconstruct(vecsort(digits(n),,4))
\\ Michael B. Porter, Nov 11 2009
a(n) = fromdigits(vecsort(digits(n), , 4)); \\ Joerg Arndt, Feb 24 2019
def a(n): return int("".join(sorted(str(n), reverse=True)))
print([a(n) for n in range(73)]) # Michael S. Branicky, Feb 21 2021
197 is a member as all the three cyclic permutations 197,971,719 are primes.
fQ[p_] := Module[{b = IntegerDigits[p]}, And @@ Table[PrimeQ[FromDigits[b = RotateLeft[b]]], {Length[b] - 1}]]; Select[Prime[Range[100000]], fQ] (* T. D. Noe, Mar 22 2012 *)
ecppQ[n_]:=AllTrue[FromDigits/@Table[RotateLeft[IntegerDigits[n],i],{i, IntegerLength[n]}],PrimeQ]; Select[Range[400000],ecppQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 25 2015 *)
circularPrimeQ[p_] := Module[{d = IntegerDigits[p], ps}, ps = Table[FromDigits[d = RotateLeft[d]], {Length[d]}]; If[p > Min[ps], False, And @@ PrimeQ[ps]]]; Select[Prime[Range[100000]], circularPrimeQ] (* T. D. Noe, Mar 22 2012 *)
Union[Select[Union/@((FromDigits/@Table[RotateRight[IntegerDigits[#],n],{n,IntegerLength[ #]}])&/@Prime[Range[20000]]),AllTrue[#,PrimeQ]&]][[All,1]] (* The program generates the first 19 terms of the sequence. *) (* Harvey P. Dale, Nov 14 2022 *)
167 is a member as the two cyclic permutations other than the number itself i.e. 671 and 716 are composite.
Select[Prime[Range[100]],Union[PrimeQ[FromDigits/@Table[ RotateRight[ IntegerDigits[#],i],{i,IntegerLength[#]-1}]]]=={False}&] (* Harvey P. Dale, Dec 08 2012 *)
import Data.List (permutations, (\\))
a258706 n = a258706_list !! (n-1)
a258706_list = f a000040_list where
f ps'@(p:ps) | any (== 0) (map a010051' dps) = f ps
| otherwise = p : f (ps' \\ dps)
where dps = map read $ permutations $ show p
-- Reinhard Zumkeller, Jun 10 2015
Flatten@{2, 3, 5, 7,
Table[Select[
Table @@
Prepend[Prepend[
Table[{A@k, A[k - 1], 4}, {k, 2, n}], {A[1], 4}],
Unevaluated[
Unevaluated[FromDigits[{1, 3, 7, 9}[[A /@ Range[n]]]]]]] //
Flatten,
Function[L,
And[PrimeQ[#],
And @@ PrimeQ[
FromDigits /@ (Permute[L, #] & /@
RandomPermutation[Length@L, 5])],
And @@ PrimeQ[FromDigits /@ Rest[Permutations[L]]]]]@
IntegerDigits@# &], {n, 2, 33}]}
(* Exhaustively searches thru 33 digits in ~7.5 sec, and up to 69 digits in 5 min, but cannot reach 317 digits. Not helpful in the light of Schroeppel's theorem that it's all repunits past 991. - Bill Gosper, Jan 06 2017 *)
{A=[2,5]; for(n=1, 317, my(D=[1,3,7,9], r=10^n\9); for(a=1,4, for(b=a^(n<3),4, for(j=0, if(b!=a,n-1), ispseudoprime(D[a]*r+(D[b]-D[a])*10^j)||next(2)); A=setunion(A, [r*D[a]+(D[b]-D[a])*10^if(b=n[1] && #select(d->d,n[^1]-n[^-1])<2 && !for(i=1,(#n)^(n[#n]>1), isprime(fromdigits(n=concat(n[^1],n[1])))||return)} \\ By Johnson's theorem and minimality required here, the number must be of the form ab...b or a...ab (=> first difference of digits has at most 1 nonzero component) and then is sufficient to consider rotations of the digits.
\\ M. F. Hasler, Jun 26 2018
176 is a term as the two cyclic shifts other than the number itself, 761 and 617, are primes.
LiQ[n_] := Module[{s=0}, li1=IntegerDigits[n]; k=Length[li1]; t={li1}; Do[li1=RotateLeft[li1]; AppendTo[t,li1], {i,k-1}]; If[Length[Select[Table[FromDigits[p],{p,t}], PrimeQ]] == k-1, s=1]; s]; t1={}; Do[If[!PrimeQ[i] && LiQ[i]==1, AppendTo[t1,i]], {i,10,3112}]; t1 (* Jayanta Basu, May 03 2013 *)
cppQ[n_]:=Module[{c=FromDigits/@NestList[RotateLeft[#]&,IntegerDigits[n], IntegerLength[ n]-1]},CompositeQ[c[[1]]]&&AllTrue[Rest[c],PrimeQ]]; Select[ Range[10,5000],cppQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 12 2014 *)
from itertools import product from sympy import isprime A068653_list = [] for l in range(1,9): for m in product(('1379' if l > 1 else '123579'),repeat=l): for d in '0123456789': s = ''.join(m)+d n = int(s) if not isprime(n): for k in range(len(s)-1): s = s[1:]+s[0] if not isprime(int(s)): break else: A068653_list.append(n) # Chai Wah Wu, May 06 2017
a(n)=1 means that rev(prime(n)) is prime i.e. prime(n) is in A007500; a(n)=2 means that rev(prime(n)^2) is prime but rev(prime(n)) is not, like n=8:p=19 and 91 is not a prime but rev[19^2]=rev[361]=163 is a prime; For n, the first k exponent providing rev(prime(n)^k) prime can be quite large, like at n=87: rev(p(87)^723)=rev(449^723) is the first [probably] prime has 1918 decimal digits: 948......573.
a:= proc(n) local k, p; p:= ithprime(n); for k while not isprime((s->
parse(cat(seq(s[-i], i=1..length(s)))))(""||(p^k))) do od; k
end:
seq(a(n), n=1..50); # Alois P. Heinz, Sep 04 2019
a[n_] := Block[{k = 1}, While[! PrimeQ@ FromDigits@ Reverse@ IntegerDigits[ Prime[n]^k], k++]; k]; Array[a, 87] (* Giovanni Resta, Sep 04 2019 *)
a(n) = {my(x=1, p=prime(n)); while (!ispseudoprime(fromdigits(Vecrev(digits(p^x)))), x++); x;} \\ Michel Marcus, Sep 04 2019
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