A029581 -id:A029581 - cp's OEIS frontend

A084984 Numbers containing no prime digits.

0, 1, 4, 6, 8, 9, 10, 11, 14, 16, 18, 19, 40, 41, 44, 46, 48, 49, 60, 61, 64, 66, 68, 69, 80, 81, 84, 86, 88, 89, 90, 91, 94, 96, 98, 99, 100, 101, 104, 106, 108, 109, 110, 111, 114, 116, 118, 119, 140, 141, 144, 146, 148, 149, 160, 161, 164, 166, 168, 169

Offset: 1

Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 27 2003

Complement of A118950. - Reinhard Zumkeller, Jul 19 2011

If n-1 is represented as a base-6 number (see A007092) according to n-1=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)=0,1,4,6,8,9 for k=0..5. - Hieronymus Fischer, May 30 2012

			166 has digits 1 and 6 and they are nonprime digits.
a(1000) = 8686.
a(10^4) = 118186
a(10^5) = 4090986.
a(10^6) = 66466686.

Reinhard Zumkeller, 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.

Cf. A046034, A034844 (primes), A118950, A193238.

Cf. A007092, A029581, A017042, A001743, A001744, A014263, A202267, A202268.

a084984 n = a084984_list !! (n-1)
a084984_list = filter (not . any (`elem` "2357") . show ) [0..]
-- Reinhard Zumkeller, Jul 19 2011

[n: n in [0..169] | forall{d: d in [2,3,5,7] | d notin Set(Intseq(n))}];  // Bruno Berselli, Jul 19 2011

npdQ[n_]:=And@@Table[FreeQ[IntegerDigits[n],i],{i,{2,3,5,7}}]; Select[ Range[ 0,200],npdQ] (* Harvey P. Dale, Jul 22 2013 *)

is(n)=isprime(eval(Vec(Str(n))))==0 \\ Charles R Greathouse IV, Feb 20 2012

my(table=[0,1,4,6,8,9]); \
a(n) = fromdigits([table[d+1] |d<-digits(n-1,6)]); \\ Kevin Ryde, May 27 2025

A193238(a(n)) = 0. - Reinhard Zumkeller, Jul 19 2011
a(n) >> n^1.285. - Charles R Greathouse IV, Feb 20 2012
From Hieronymus Fischer, May 30 and Jun 25 2012: (Start)
a(n) = ((2*b_m(n)+1) mod 10 + floor((b_m(n)+4)/5) - floor((b_m(n)+1)/5))*10^m + sum_{j=0..m-1} ((2*b_j(n))) mod 12 + floor(b_j(n)/6) - floor((b_j(n)+1)/6) + floor((b_j(n)+4)/6) - floor((b_j(n)+5)/6)))*10^j, where n>1, b_j(n)) = floor((n-1-6^m)/6^j), m = floor(log_6(n-1)).
Special values:
a(1*6^n+1) = 1*10^n.
a(2*6^n+1) = 4*10^n.
a(3*6^n+1) = 6*10^n.
a(4*6^n+1) = 8*10^n.
a(5*6^n+1) = 9*10^n.
a(2*6^n) = 2*10^n - 1.
a(n) = 10^log_6(n-1) for n=6^k+1, k>0.
Inequalities:
a(n) < 10^log_6(n-1) for 6^k+10.
a(n) > 10^log_6(n-1) for 2*6^k=0.
a(n) <= 4*10^(log_6(n-1)-log_6(2)) = 1.641372618*10^(log_6(n-1)), equality holds for n=2*6^k+1, k>=0.
a(n) > 2*10^(log_6(n-1)-log_6(2)) = 0.820686309*10^(log_6(n-1)).
a(n) = A007092(n-1) iff the digits of A007092(n-1) are 0 or 1, a(n)>A007092(n-1), else.
a(n) >= A202267(n), equality holds if the representation of n-1 as a base-6 number has only digits 0 or 1.
Lower and upper limits:
lim inf a(n)/10^log_6(n) = 2/10^log_6(2) = 0.820686309, for n --> inf.
lim sup a(n)/10^log_6(n) = 4/10^log_6(2) = 1.641372618, for n --> inf.
where 10^log_6(n) = n^1.2850972089...
G.f.: g(x) = (x/(1-x))*sum_{j>=0} 10^j*x^6^j * (1-x^6^j)*((1+x^6^j)^4 + 4(1+2x^6^j) * x^(3*6^j))/(1-x^6^(j+1)).
Also: g(x) = (x/(1-x))*(h_(6,1)(x) + 3*h_(6,2)(x) + 2*h_(6,3)(x) + 2*h_(6,4)(x) + h_(6,5)(x) - 9*h_(6,6)(x)), where h_(6,k)(x) = sum_{j>=0} 10^j*x^(k*6^j)/(1-x^6^(j+1)). (End)
Sum_{n>=2} 1/a(n) = 3.614028405471074989720026361356036456697082276983705341077940360653303099111... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 15 2024

0 added by N. J. A. Sloane, Feb 02 2009
100 added by Arkadiusz Wesolowski, Mar 10 2011
Examples for n>=10^3 added by Hieronymus Fischer, May 30 2012

A001744 Numbers n such that every digit contains a loop (version 2).

Original entry on oeis.org

0, 4, 6, 8, 9, 40, 44, 46, 48, 49, 60, 64, 66, 68, 69, 80, 84, 86, 88, 89, 90, 94, 96, 98, 99, 400, 404, 406, 408, 409, 440, 444, 446, 448, 449, 460, 464, 466, 468, 469, 480, 484, 486, 488, 489, 490, 494, 496, 498, 499, 600, 604, 606, 608, 609, 640, 644, 646

Offset: 1

N. J. A. Sloane

See A001743 for the other version.

If n-1 is represented as a base-5 number (see A007091) according to n-1 = 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)=0,4,6,8,9 for k=0..4. - Hieronymus Fischer, May 30 2012

			a(1000) = 46999.
a(10^4) = 809999.
a(10^5) = 44499999.
a(10^6) = 668999999.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Cf. A061371, A029581, A007091, A046034, A084544, A084984, A017042, A001743, A014261, A014263, A202267, A202268.

FromDigits/@Tuples[{0,4,6,8,9},3] (* Harvey P. Dale, Aug 16 2018 *)

is(n) = #setintersect(vecsort(digits(n), , 8), [1, 2, 3, 5, 7])==0 \\ Felix Fröhlich, Sep 09 2019

From Hieronymus Fischer, May 30 2012: (Start)
a(n) = ((2*b_m(n)) mod 8 + 4 + floor(b_m(n)/4) - floor((b_m(n)+1)/4))*10^m + sum_{j=0..m-1} ((2*b_j(n))) mod 10 + 2*floor((b_j(n)+4)/5) - floor((b_j(n)+1)/5) -floor(b_j(n)/5)))*10^j, where n>1, b_j(n)) = floor((n-1-5^m)/5^j), m = floor(log_5(n-1)).
a(1*5^n+1) = 4*10^n.
a(2*5^n+1) = 6*10^n.
a(3*5^n+1) = 8*10^n.
a(4*5^n+1) = 9*10^n.
a(n) = 4*10^log_5(n-1) for n=5^k+1,
a(n) < 4*10^log_5(n-1), otherwise.
a(n) > 10^log_5(n-1) n>1.
a(n) = 4*A007091(n-1), iff the digits of A007091(n-1) are 0 or 1.
G.f.: g(x) = (x/(1-x))*sum_{j>=0} 10^j*x^5^j*(1-x^5^j)*(4 + 6x^5^j + 8(x^2)^5^j + 9(x^3)^5^j)/(1-x^5^(j+1)).
Also: g(x) = (x/(1-x))*(4*h_(5,1)(x) + 2*h_(5,2)(x) + 2*h_(5,3)(x) + h_(5,4)(x) - 9*h_(5,5)(x)), where h_(5,k)(x) = sum_{j>=0} 10^j*(x^5^j)^k/(1-(x^5^j)^5). (End)

Ambiguous comment deleted by Zak Seidov, May 25 2010

Examples added by Hieronymus Fischer, May 30 2012

A001743 Numbers in which every digit contains at least one loop (version 1).

Original entry on oeis.org

0, 6, 8, 9, 60, 66, 68, 69, 80, 86, 88, 89, 90, 96, 98, 99, 600, 606, 608, 609, 660, 666, 668, 669, 680, 686, 688, 689, 690, 696, 698, 699, 800, 806, 808, 809, 860, 866, 868, 869, 880, 886, 888, 889, 890, 896, 898, 899, 900, 906, 908, 909, 960, 966, 968, 969

Offset: 1

N. J. A. Sloane

See A001744 for the other version.

If n-1 is represented as a base-4 number (see A007090) according to n-1 = 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)=0,6,8,9 for k=0..3. - Hieronymus Fischer, May 30 2012

			a(1000) = 99896.
a(10^4) = 8690099.
a(10^5) = 680688699.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Cf. A007090, A046034, A029581, A084984, A017042, A001744, A014261, A014263, A202267, A202268.

Union[Flatten[Table[FromDigits/@Tuples[{0,6,8,9},n],{n,3}]]] (* Harvey P. Dale, Sep 04 2013 *)

is(n) = #setintersect(vecsort(digits(n), , 8), [1, 2, 3, 4, 5, 7])==0 \\ Felix Fröhlich, Sep 09 2019

From Hieronymus Fischer, May 30 2012: (Start)
a(n) = ((b_m(n)+6) mod 9 + floor((b_m(n)+2)/3) - floor(b_m(n)/3))*10^m + Sum_{j=0..m-1} (b_j(n) mod 4 +5*floor((b_j(n)+3)/4) +floor((b_j(n)+2)/4)- 6*floor(b_j(n)/4)))*10^j, where n>1, b_j(n)) = floor((n-1-4^m)/4^j), m = floor(log_4(n-1)).
a(1*4^n+1) = 6*10^n.
a(2*4^n+1) = 8*10^n.
a(3*4^n+1) = 9*10^n.
a(n) = 6*10^log_4(n-1) for n=4^k+1,
a(n) < 6*10^log_4(n-1), otherwise.
a(n) > 10^log_4(n-1) for n>1.
a(n) = 6*A007090(n-1), iff the digits of A007090(n-1) are 0 or 1.
G.f.: g(x) = (x/(1-x))*Sum_{j>=0} 10^j*x^4^j *(1-x^4^j)* (6 + 8x^4^j + 9(x^2)^4^j)/(1-x^4^(j+1)).
Also: g(x) = (x/(1-x))*(6*h_(4,1)(x) + 2*h_(4,2)(x) + h_(4,3)(x) - 9*h_(4,4)(x)), where h_(4,k)(x) = Sum_{j>=0} 10^j*(x^4^j)^k/(1-(x^4^j)^4). (End)

Examples added by Hieronymus Fischer, May 30 2012

A202268 Numbers in which all digits are nonprimes (1, 4, 6, 8, 9).

Original entry on oeis.org

1, 4, 6, 8, 9, 11, 14, 16, 18, 19, 41, 44, 46, 48, 49, 61, 64, 66, 68, 69, 81, 84, 86, 88, 89, 91, 94, 96, 98, 99, 111, 114, 116, 118, 119, 141, 144, 146, 148, 149, 161, 164, 166, 168, 169, 181, 184, 186, 188, 189, 191, 194, 196, 198, 199, 411, 414, 416, 418, 419

Offset: 1

Jaroslav Krizek, Dec 25 2011

Supersequence of A029581.
Subsequence of A084984.
If n-1 is represented as a zerofree base-5 number (see A084545) according to n-1=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)=1,4,6,8,9 for k=1..5. - Hieronymus Fischer, May 30 2012

			From _Hieronymus Fischer_, May 30 2012: (Start)
a(1000) = 14889.
a(10^4) = 498889
a(10^5) = 11188889.
a(10^6) = 446888889. (End)

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.

Cf. A046034 (numbers in which all digits are primes), A001742 (numbers in which all digits are noncomposites excluding 0), A202267 (numbers in which all digits are noncomposites), A084984 (numbers in which all digits are nonprimes), A029581 (numbers in which all digits are composites).

Cf. A084545, A001743, A001744, A014261, A014263.

[n: n in [1..500] | Set(Intseq(n)) subset [1, 4, 6, 8, 9]]; // Vincenzo Librandi, Dec 17 2018

Table[FromDigits/@Tuples[{1, 4, 6, 8, 9}, n], {n, 3}] // Flatten (* Vincenzo Librandi, Dec 17 2018 *)

From Hieronymus Fischer, May 30 2012: (Start)
a(n) = Sum_{j=0..m-1} ((2*b_j(n)+1) mod 10 + floor((b_j(n)+4)/5) - floor((b_j(n)+1)/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) = 4*(10^n-1)/9.
a(3*(5^n-1)/4) = 6*(10^n-1)/9.
a(4*(5^n-1)/4) = 8*(10^n-1)/9.
a(5*(5^n-1)/4) = 10^n-1.
a(n) = (10^log_5(4*n+1)-1)/9 for n=(5^k-1)/4, k>0.
a(n) <= 36/(9*2^log_5(9)-1)*(10^log_5(4*n+1)-1)/9 for n>0, equality holds for n=2.
a(n) > 0.776*10^log_5(4*n+1)-1)/9 for n>0.
a(n) >= A001742(n), equality holds for n=(5^k-1)/4, k>0.
a(n) = A084545(n) iff all digits of A084545(n) are 1, a(n)>A084545(n), else.
G.f.: g(x) = (x^(1/4)*(1-x))^(-1) Sum_{j>=0} 10^j*z(j)^(5/4)*(1-z(j))*(1 + 4z(j) + 6*z(j)^2 + 8*z(j)^3 + 9*z(j)^4)/(1-z(j)^5), where z(j)=x^5^j.
Also: g(x) = (1/(1-x))*(h_(5,0)(x) + 3h_(5,1)(x) + 2h_(5,2)(x) + 2h_(5,3)(x) + h_(5,4)(x) - 9*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) = 2.897648425695540438556738520657902585305276107220152307051361916356295164643... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 15 2024

A001742 Numbers whose digits contain no loops (version 2).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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

			From _Hieronymus Fischer_, May 30 2012: (Start)
a(10^3) = 12557.
a(10^4) = 275557.
a(10^5) = 11155557.
a(10^6) = 223555557. (End)

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.

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.

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

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 *)

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

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

A107342 Semiprimes with semiprime digits (digits 4, 6, 9 only).

Original entry on oeis.org

4, 6, 9, 46, 49, 69, 94, 446, 466, 469, 649, 669, 694, 699, 949, 4449, 4469, 4499, 4666, 4694, 4699, 4946, 6499, 6646, 6649, 6694, 6999, 9446, 9449, 9466, 9469, 9946, 9969, 44494, 44669, 44949, 44966, 44969, 44999, 46469, 46666, 46946, 46969, 46994

Offset: 1

Jonathan Vos Post, May 22 2005

Numbers n such that all digits of n are elements of A001358 and n is an element of A001358.
Numbers n such that n is an element of A107665 and n is an element of A001358.
Conjecture: almost all terms (asymptotic density 1) end with 9 and have either 3k+1 or 3k+2 occurrences of the digit 4 for some nonnegative k. (Otherwise they'd be divisible by 2 or 3 and these semiprimes would be expected to be rare; the sequence is too thin to prove this directly.) - Charles R Greathouse IV, Nov 12 2021

			4 = 2^2
6 = 2 * 3
9 = 3^2
46 = 2 * 23
49 = 7^2
69 = 3 * 23
94 = 2 * 47

Vincenzo Librandi, Table of n, a(n) for n = 1..600

Intersection of A001358 and A107665.

Cf. A029581, A107666, A111494, A111496, A111697, A254715.

fQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2 && Union[ Join[{4, 6, 9}, IntegerDigits[n]]] == {4, 6, 9}; Select[ Range[ 47000], fQ[ # ] &] (* Robert G. Wilson v, May 27 2005 *)
Flatten[Table[Select[FromDigits/@Tuples[{4,6,9},n],PrimeOmega[#]==2&],{n,5}]] (* Harvey P. Dale, Jun 14 2015 *)

is(n)=bigomega(n)==2 && #setminus(Set(digits(n)),[4,6,9])==0 \\ Charles R Greathouse IV, Nov 12 2021

More terms from Robert G. Wilson v, May 27 2005

A202267 Numbers in which all digits are noncomposites (1, 2, 3, 5, 7) or 0.

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 10, 11, 12, 13, 15, 17, 20, 21, 22, 23, 25, 27, 30, 31, 32, 33, 35, 37, 50, 51, 52, 53, 55, 57, 70, 71, 72, 73, 75, 77, 100, 101, 102, 103, 105, 107, 110, 111, 112, 113, 115, 117, 120, 121, 122, 123, 125, 127, 130, 131, 132, 133, 135, 137, 150

Offset: 1

Jaroslav Krizek, Dec 25 2011

If n-1 is represented as a base-6 number (see A007092) according to n-1=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)=0,1,2,3,5,7 for k=0..5. - Hieronymus Fischer, May 30 2012

			a(1000) = 5353.
a(10^4) = 115153
a(10^5) = 2070753.
a(10^6) = 33233353.

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.

Supersequence of A001742 and A046034.
Cf. A046034 (numbers in which all digits are primes), A001742 (numbers in which all digits are noncomposites excluding 0), A202268 (numbers in which all digits are nonprimes excluding 0), A084984 (numbers in which all digits are nonprimes), A029581 (numbers in which all digits are composites).
Cf. A007092, A001743, A001744, A193238.

Union[Flatten[FromDigits/@Tuples[{0,1,2,3,5,7},3]]] (* Harvey P. Dale, Mar 11 2015 *)

From Hieronymus Fischer, May 30 2012: (Start)
a(n) = (b_m(n)+1) mod 10 + floor((b_m(n)+2)/5) + floor((b_m(n)+1)/5) - 2*floor(b_m(n)/5))*10^m + sum_{j=0..m-1} (b_j(n) mod 6 + floor((b_j(n)+1)/6) + floor((b_j(n)+2)/6) - 2*floor(b_j(n)/6)))*10^j, where n>1, b_j(n)) = floor((n-1-6^m)/6^j), m = floor(log_6(n-1)).
a(1*6^n+1) = 1*10^n.
a(2*6^n+1) = 2*10^n.
a(3*6^n+1) = 3*10^n.
a(4*6^n+1) = 5*10^n.
a(5*6^n+1) = 7*10^n.
a(n) = 10^log_6(n-1) for n=6^k+1, k>0,
a(n) < 10^log_6(n-1) else.
a(n) = A007092(n-1) iff the digits of A007092(n-1) are <= 3, a(n)>A007092(n-1), else.
a(n) <= A084984(n), equality holds if the representation of n-1 as a base-6 number only has digits 0 or 1.
G.f.: g(x) = (x/(1-x))*sum_{j>=0} 10^j*x^6^j *(1-x^6^j)* (1 + 2x^6^j + 3(x^2)^6^j + 5(x^3)^6^j + 7(x^4)^6^j)/(1-x^6^(j+1)).
Also: g(x) = (x/(1-x))*(h_(6,1)(x) + h_(6,2)(x) + h_(6,3)(x) + 2*h_(6,4)(x) + 2*h_(6,5)(x) - 7*h_(6,6)(x)), where h_(6,k)(x) = sum_{j>=0} 10^j*x^(k*6^j)/(1-x^6^(j+1)). (End)
Sum_{n>=2} 1/a(n) = 4.945325883472729555972742252181522711968119529132581193614012706741310832798... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 15 2024

Examples added by Hieronymus Fischer, May 30 2012

A107665 Numbers with semiprime digits (digits 4, 6, 9 only).

Original entry on oeis.org

4, 6, 9, 44, 46, 49, 64, 66, 69, 94, 96, 99, 444, 446, 449, 464, 466, 469, 494, 496, 499, 644, 646, 649, 664, 666, 669, 694, 696, 699, 944, 946, 949, 964, 966, 969, 994, 996, 999, 4444, 4446, 4449, 4464, 4466, 4469, 4494, 4496, 4499, 4644, 4646, 4649, 4664

Offset: 1

Rick L. Shepherd, May 19 2005

Cf. A107666 (primes in this sequence), A001358 (semiprimes), A029581 (all digits are composite).

Cf. A107342, A111494, A111496, A111697.

Select[Range[5000],Union[Pick[DigitCount[#],{1,1,1,0,1,0,1,1,0,1},1]] == {0}&] (* Harvey P. Dale, Oct 21 2011 *)
Flatten[Table[FromDigits/@Tuples[{4,6,9},n],{n,4}]] (* Harvey P. Dale, Oct 21 2014 *)

A202262 Composite numbers in which all substrings are composite.

Original entry on oeis.org

4, 6, 8, 9, 44, 46, 48, 49, 64, 66, 68, 69, 84, 86, 88, 94, 96, 98, 99, 444, 446, 448, 464, 466, 468, 469, 484, 486, 488, 494, 496, 498, 644, 646, 648, 649, 664, 666, 668, 669, 684, 686, 688, 694, 696, 698, 699, 844, 846, 848, 849, 864, 866, 868, 869, 884, 886

Offset: 1

Jaroslav Krizek, Dec 25 2011

Subsequence of A062115, A202260, A029581.
Supersequence of A202265.
This is a 10-automatic sequence, see A071062. - Charles R Greathouse IV, Jan 01 2012

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A202262
Index entries for 10-automatic sequences

Cf. A085823 (primes in which all substrings are primes), A068669 (noncomposite numbers in which all substrings are noncomposite), A062115 (nonprimes in which all substrings are nonprimes).

Cf. A010051.

sub[n_] := Block[{d = IntegerDigits[n]}, Union@ Reap[ Do[Sow@ FromDigits@ Take[d, {i, j}], {j, Length@ d}, {i, j}]][[2, 1]]]; Select[ Range@ 900, Union[{4, 6, 8, 9}, IntegerDigits[#]] == {4, 6, 8, 9} && AllTrue[sub[#], CompositeQ] &] (* Giovanni Resta, Dec 20 2019 *)

PARI
```
See Links section.
```

Data corrected by Reinhard Zumkeller, May 05 2012
Data corrected by Rémy Sigrist, Dec 19 2019
Incorrect Haskell program deleted by M. F. Hasler, Dec 20 2019

cp's OEIS Frontend

A084984 Numbers containing no prime digits.

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A001744 Numbers n such that every digit contains a loop (version 2).

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A001743 Numbers in which every digit contains at least one loop (version 1).

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A202268 Numbers in which all digits are nonprimes (1, 4, 6, 8, 9).

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A001742 Numbers whose digits contain no loops (version 2).

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A107342 Semiprimes with semiprime digits (digits 4, 6, 9 only).

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