A317689
Largest nonrepunit base-n absolute prime (conjectured).
Original entry on oeis.org
7, 53, 3121, 211, 1999, 3803, 6469, 991, 161047, 19793, 16477, 24907, 683437, 3547, 67853, 80273, 94109, 72421
Offset: 3
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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
decimal(v, base) = my(w=[]); for(k=0, #v-1, w=concat(w, v[#v-k]*base^k)); sum(i=1, #w, w[i])
is_absolute_prime(n, base) = my(db=vecsort(digits(n, base))); if(vecmin(db)==0 || vecmax(db)==1, return(0)); while(1, my(dec=decimal(db, base)); if(!ispseudoprime(dec), return(0)); db=next_permutation(db); if(db==0, return(1)))
a(n) = my(absp=0, lim=10^7, i=0); forprime(p=n+1, , if(is_absolute_prime(p, n), absp=p); i++; if(i==lim, return(absp)))
A204844
Cyclic primes that are not absolute primes (A003459).
Original entry on oeis.org
197, 719, 971, 1193, 1931, 3119, 3779, 7793, 7937, 9311, 9377, 11939, 19391, 19937, 37199, 39119, 71993, 91193, 93719, 93911, 99371, 193939, 199933, 319993, 331999, 391939, 393919, 919393, 933199, 939193, 939391, 993319, 999331
Offset: 1
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
- J. L. Boal and J. H. Bevis, Permutable primes, Mathematics Magazine, Vol. 55, No. 1 (1982), 38-41.
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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 *)
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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
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