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A001742 Numbers whose digits contain no loops (version 2). 15
1, 2, 3, 5, 7, 11, 12, 13, 15, 17, 21, 22, 23, 25, 27, 31, 32, 33, 35, 37, 51, 52, 53, 55, 57, 71, 72, 73, 75, 77, 111, 112, 113, 115, 117, 121, 122, 123, 125, 127, 131, 132, 133, 135, 137, 151, 152, 153, 155, 157, 171, 172, 173, 175, 177, 211, 212, 213, 215 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Numbers all of whose decimal digits are in {1,2,3,5,7}.

If n is represented as a zerofree base-5 number (see A084545) according to n = d(m)d(m-1)...d(3)d(2)d(1)d(0) then a(n) = Sum_{j=0..m} c(d(j))*10^j, where c(k)=1,2,3,5,7 for k=1..5. - Hieronymus Fischer, May 30 2012

LINKS

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Index entries for 10-automatic sequences.

FORMULA

From Hieronymus Fischer, May 30 2012: (Start)

a(n) = Sum_{j=0..m-1} ((2*b_j(n)+1) mod 10 + 2*floor(b_j(n)/5) - floor((b_j(n)+3)/5) - floor((b_j(n)+4)/5))*10^j, where b_j(n)) = floor((4*n+1-5^m)/(4*5^j)), m = floor(log_5(4*n+1)).

a(1*(5^n-1)/4) = 1*(10^n-1)/9.

a(2*(5^n-1)/4) = 2*(10^n-1)/9.

a(3*(5^n-1)/4) = 1*(10^n-1)/3.

a(4*(5^n-1)/4) = 5*(10^n-1)/9.

a(5*(5^n-1)/4) = 7*(10^n-1)/9.

a(n) = (10^log_5(4*n+1)-1)/9 for n=(5^k-1)/4, k > 0.

a(n) < (10^log_5(4*n+1)-1)/9 for (5^k-1)/4 < n < (5^(k+1)-1)/4, k > 0.

a(n) <= A202268(n), equality holds for n=(5^k-1)/4, k > 0.

a(n) = A084545(n) iff all digits of A084545(n) are <= 3, a(n) > A084545(n), otherwise.

G.f.: g(x) = (x^(1/4)*(1-x))^(-1) Sum_{j>=0} 10^j*z(j)^(5/4)*(1 + z(j) + z(j)^2 + 2*z(j)^3 + 2*z(j)^4 - 7*z(j)^5)/(1-z(j)^5), where z(j) = x^5^j.

Also g(x) = (x^(1/4)*(1-x))^(-1) Sum_{j>=0} 10^j*z(j)^(5/4)*(1-z(j))*(1 + 2z(j) + 3*z(j)^2 + 5*z(j)^3 + 7*z(j)^4)/(1-z(j)^5), where z(j) = x^5^j.

Also: g(x)=(1/(1-x))*(h_(5,0)(x) + h_(5,1)(x) + h_(5,2)(x) + 2*h_(5,3)(x) + 2*h_(5,4)(x) - 7*h_(5,5)(x)), where h_(5,k)(x) = Sum_{j>=0} 10^j*x^((5^(j+1)-1)/4)*(x^5^j)^k/(1-(x^5^j)^5). (End)

EXAMPLE

From Hieronymus Fischer, May 30 2012: (Start)

a(10^3) = 12557.

a(10^4) = 275557.

a(10^5) = 11155557.

a(10^6) = 223555557. (End)

MATHEMATICA

nlQ[n_]:=And@@(MemberQ[{1, 2, 3, 5, 7}, #]&/@IntegerDigits[n]); Select[Range[ 160], nlQ] (* Harvey P. Dale, Mar 23 2012 *)

Table[FromDigits/@Tuples[{1, 2, 3, 5, 7}, n], {n, 3}] // Flatten (* Vincenzo Librandi, Dec 17 2018 *)

PROG

(Perl) for (my $k = 1; $k < 1000; $k++) {print "$k, " if ($k =~ m/^[12357]+$/)} # Charles R Greathouse IV, Jun 10 2011

(MAGMA) [n: n in [1..500] |  Set(Intseq(n)) subset [1, 2, 3, 5, 7]]; // Vincenzo Librandi, Dec 17 2018

CROSSREFS

Cf. A001729 (version 1), A190222 (noncomposite terms), A190223 (n with all divisors in this sequence).

Cf. A046034, A084545, A029581, A084984, A001743, A001744, A014261, A014263, A202267, A202268.

Sequence in context: A163753 A131930 A230918 * A307714 A073085 A316793

Adjacent sequences:  A001739 A001740 A001741 * A001743 A001744 A001745

KEYWORD

base,nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified January 23 22:16 EST 2020. Contains 331177 sequences. (Running on oeis4.)