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
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
Programs
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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.
Comments