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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 Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008. 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) Sum_{n>=1} 1/a(n) = 3.961674246441345455010500439753914974057344229353697593567607096540565407371... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 15 2024 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 A369254 A073085 Adjacent sequences: A001739 A001740 A001741 * A001743 A001744 A001745 KEYWORD base,nonn,easy,changed AUTHOR N. J. A. Sloane STATUS approved

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Last modified August 3 23:57 EDT 2024. Contains 374905 sequences. (Running on oeis4.)