A203897 Nonprime numbers k >= 1 such that k and phi(k) contain only digits 0 and 1.

1, 1111, 110111111, 1111011011, 11000111111, 110011011011, 110111101111, 111100011011, 111101010101, 111101111011, 1010011111111, 1010101001111, 1010101101101, 1010111100101, 1011111001111, 1100000111111, 1100110011011, 1111001011111, 1111001101111

Offset

1,2

Comments

From Robert Israel, Feb 26 2018: (Start)

Many terms are semiprimes of the form p*q where p = 11 or 101 and q is in A020449.

The first term after 1 that is not a semiprime is a(199) = 111100111111111111 = 11*101*100000100010001.

The first term after 1 that is not divisible by 11 or 101 is a(1023) = 1110010101001011111101 = 11111101 * 99901000000001. (End)

Links

Robert Israel, Table of n, a(n) for n = 1..2000

Example

a(3) = 110111111 = 11*10010101 is in the sequence because phi(110111111) = 100101000, which contains digits 0 and 1 only.
Remark: phi(11)=10, phi(10010101)=10010100, but 100101000 = 2*3*5*61*547.

Maple

with(numtheory): T:=array(1..23):k:=1:a:={0, 1}:b:={1}:for a12 from 0 to 1 do: for a11 from 0 to 1 do: for a10 from 0 to 1 do: for a9 from 0 to 1 do: for a8 from 0 to 1 do: for a7 from 0 to 1 do: for a6 from 0 to 1 do: for a5 from 0 to 1 do: for a4 from 0 to 1 do: for a3 from 0 to 1 do: for a2 from 0 to 1 do: for a1 from 0 to 1 do: for a0 from 0 to 1 do:n:=a0+a1*10+a2*10^2+ a3*10^3+ a4*10^4+ a5*10^5+ a6*10^6+ a7*10^7+ a8*10^8+ a9*10^9 + a10*10^10+ a11*10^11+ a12*10^12: m:=phi(n):x:=convert(convert(m, base, 10), set): if type(n, prime)=false and (a union x = a or  a union x = b) then T[k]:=n:k:=k+1:else fi:od: od: od: od: od: od: od: od: od:od:od:od:od: print(T):
Res:= NULL: count:= 0:
for q from 1 while count < 100 do
  L:= convert(q, base, 2);
  n:=add(L[i]*10^(i-1), i=1..nops(L));
  if isprime(n) then next fi;
  r:= numtheory:-phi(n);
  if max(convert(r, base, 10))=1 then
    Res:= Res, n;
    count:= count+1;
  fi
od:
Res; # Robert Israel, Feb 26 2018

Mathematica

d = Table[FromDigits[IntegerDigits[n, 2]], {n, 10000}]; Select[d, ! PrimeQ[#] && Max[IntegerDigits[EulerPhi[#]]] == 1 &] (* T. D. Noe, Jan 11 2012 *)
Rest[Select[FromDigits/@Tuples[{0, 1}, 13], !PrimeQ[#]&&Max[IntegerDigits[ EulerPhi[ #]]] <2&]] (* Harvey P. Dale, Jan 17 2023 *)

Prog

(PARI) lista(nn) = {for (n=1, nn, x = fromdigits(binary(n), 10); if (! isprime(x) && (vecmax(digits(eulerphi(x))) < 2), print1(x, ", ")); ); } \\ Michel Marcus, Jun 12 2017

Crossrefs

Cf. A000010, A020449, A203304.

Sequence in context: A015292 A274831 A154805 * A236050 A213979 A225351

Adjacent sequences: A203894 A203895 A203896 * A203898 A203899 A203900

Keyword

nonn,base

Author

Michel Lagneau, Jan 07 2012

Extensions

Incorrect comments deleted by Robert Israel, Feb 26 2018

Definition edited by N. J. A. Sloane, Jan 17 2023 at the suggestion of Harvey P. Dale.

Status

approved