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

Author

Michel Lagneau, Jan 07 2012

Keywords

• nonn
• base

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)

Examples

			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.
		

Links

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

Crossrefs

Cf. A000010, A020449, A203304.

Programs

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 *)

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

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.