A017042 -id:A017042 - cp's OEIS frontend

A046034 Numbers whose digits are primes.

2, 3, 5, 7, 22, 23, 25, 27, 32, 33, 35, 37, 52, 53, 55, 57, 72, 73, 75, 77, 222, 223, 225, 227, 232, 233, 235, 237, 252, 253, 255, 257, 272, 273, 275, 277, 322, 323, 325, 327, 332, 333, 335, 337, 352, 353, 355, 357, 372, 373, 375, 377, 522, 523, 525, 527, 532

Offset: 1

Eric W. Weisstein

If n is represented as a zerofree base-4 number (see A084544) 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)=2,3,5,7 for k=1..4. - Hieronymus Fischer, May 30 2012

According to A153025, it seems that 5, 235 and 72335 are the only terms whose square is also a term, i.e., which are also in the sequence A275971 of square roots of the terms which are squares, listed in A191486. - M. F. Hasler, Sep 16 2016

			a(100)   = 2277,
a(10^3)  = 55327,
a(9881)  = 3233232,
a(10^4)  = 3235757,
a(10922) = 3333333,
a(10^5)  = 227233257.

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.
Eric Weisstein's World of Mathematics, Smarandache Sequences.
Index entries for 10-automatic sequences.

Cf. A046035, A118950, A019546 (primes), A203263, A035232, A039996, A085823, A052382, A084544, A084984, A017042, A001743, A001744, A014261, A014263, A153025, A191486, A193238, A202267, A202268, A211681, A365471 (complement).

a046034 n = a046034_list !! (n-1)
a046034_list = filter (all (`elem` "2357") . show ) [0..]
-- Reinhard Zumkeller, Jul 19 2011

[n: n in [2..532] | Set(Intseq(n)) subset [2, 3, 5, 7]];  // Bruno Berselli, Jul 19 2011

Table[FromDigits /@ Tuples[{2, 3, 5, 7}, n], {n, 3}] // Flatten (* Michael De Vlieger, Sep 19 2016 *)

is_A046034(n)=Set(isprime(digits(n)))==[1] \\ M. F. Hasler, Oct 12 2013

def A046034(n):
    m = (3*n+1).bit_length()-1>>1
    return int(''.join(('2357'[(3*n+1-(1<<(m<<1)))//(3<<((m-1-j)<<1))&3] for j in range(m)))) # Chai Wah Wu, Feb 08 2023

A055642(a(n)) = A193238(a(n)). - Reinhard Zumkeller, Jul 19 2011
From Hieronymus Fischer, Apr 20, May 30 and Jun 25 2012: (Start)
a(n) = Sum_{j=0..m-1} ((2*b(j)+1) mod 8 + floor(b(j)/4) - floor((b(j)-1)/4))*10^j, where m = floor(log_4(3*n+1)), b(j) = floor((3*n+1-4^m)/(3*4^j)).
a(n) = Sum_{j=0..m-1} A010877(A005408(b(j)) + A002265(b(j)) - A002265(b(j)-1))*10^j.
Special values:
a(1*(4^n-1)/3) = 2*(10^n-1)/9.
a(2*(4^n-1)/3) = 1*(10^n-1)/3.
a(3*(4^n-1)/3) = 5*(10^n-1)/9.
a(4*(4^n-1)/3) = 7*(10^n-1)/9.
Inequalities:
a(n) <= 2*(10^log_4(3*n+1)-1)/9, equality holds for n = (4^k-1)/3, k>0.
a(n) <= 2*A084544(n), equality holds iff all digits of A084544(n) are 1.
a(n) > A084544(n).
Lower and upper limits:
lim inf a(n)/10^log_4(n) = (7/90)*10^log_4(3) = 0.48232167706987..., for n -> oo.
lim sup a(n)/10^log_4(n) = (2/9)*10^log_4(3) = 1.378061934485343..., for n -> oo.
where 10^log_4(n) = n^1.66096404744...
G.f.: g(x) = (x^(1/3)*(1-x))^(-1) Sum_{j>=0} 10^j*z(j)^(4/3)*(2 + z(j) + 2*z(j)^2 + 2*z(j)^3 - 7*z(j)^4)/(1-z(j)^4), where z(j) = x^4^j.
Also g(x) = (x^(1/3)*(1-x))^(-1) Sum_{j>=0} 10^j*z(j)^(4/3)*(1-z(j))*(2 + 3*z(j) + 5*z(j)^2 + 7*z(j)^3)/(1-z(j)^4), where z(j)=x^4^j.
Also: g(x) = (1/(1-x))*(2*h_(4,0)(x) + h_(4,1)(x) + 2*h_(4,2)(x) + 2*h_(4,3)(x) - 7*h_(4,4)(x)), where h_(4,k)(x) = Sum_{j>=0} 10^j*x^((4^(j+1)-1)/3)*x^(k*4^j)/(1-x^4^(j+1)). (End)
Sum_{n>=1} 1/a(n) = 1.857333779940977502574887651449435985318556794733869779170825138954093657197... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 15 2024

More terms from Cino Hilliard, Aug 06 2006
Typo in second formula corrected by Hieronymus Fischer, May 12 2012
Two typos in example section corrected by Hieronymus Fischer, May 30 2012

A014263 Numbers that contain even digits only.

Original entry on oeis.org

0, 2, 4, 6, 8, 20, 22, 24, 26, 28, 40, 42, 44, 46, 48, 60, 62, 64, 66, 68, 80, 82, 84, 86, 88, 200, 202, 204, 206, 208, 220, 222, 224, 226, 228, 240, 242, 244, 246, 248, 260, 262, 264, 266, 268, 280, 282, 284, 286, 288, 400, 402, 404, 406, 408, 420, 422, 424

Offset: 1

N. J. A. Sloane

The set of real numbers between 0 and 1 that contain no odd digits in their decimal expansion has Hausdorff dimension log 5 / log 10.
Integers written in base 5 and then doubled (in base 10). - Franklin T. Adams-Watters, Mar 15 2006
The carryless mod 10 "even" numbers (cf. A004529) sorted and duplicates removed. - N. J. A. Sloane, Aug 03 2010.
Complement of A007957; A196564(a(n)) = 0; A103181(a(n)) = 0. - Reinhard Zumkeller, Oct 04 2011
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)=0,2,4,6,8 for k=0..4. - Hieronymus Fischer, Jun 03 2012

			a(1000) = 24888.
a(10^4) = 60888.
a(10^5) = 22288888.
a(10^6) = 446888888.

K. J. Falconer, The Geometry of Fractal Sets, Cambridge, 1985; p. 19.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.
Index entries for sequences related to carryless arithmetic

Subsequence of A059708.

Cf. A061810, A061811, A007091, A014261, A046034, A052382, A084544, A089581, A084984, A017042, A001743, A202267, A202268, A194182, A196563.

a014263 n = a014263_list !! (n-1)
a014263_list = filter (all (`elem` "02468") . show) [0,2..]
-- Reinhard Zumkeller, Jul 05 2011

[n: n in [0..424] | Set(Intseq(n)) subset [0..8 by 2]];  // Bruno Berselli, Jul 19 2011

a:= proc(m) local L,i;
  L:= convert(m-1,base,5);
  2*add(L[i]*10^(i-1),i=1..nops(L))
end proc:
seq(a(i),i=1..100); # Robert Israel, Apr 07 2016

Select[Range[450], And@@EvenQ[IntegerDigits[#]]&] (* Harvey P. Dale, Jan 30 2011 *)
FromDigits/@Tuples[{0,2,4,6,8},3] (* Harvey P. Dale, Jul 07 2025 *)

a(n) = 2*fromdigits(digits(n-1, 5), 10); \\ Michel Marcus, Nov 04 2022

is(n)=#setminus(Set(digits(n)), [0,2,4,6,8])==0 \\ Charles R Greathouse IV, Mar 03 2025

from sympy.ntheory.digits import digits
def a(n): return int(''.join(str(2*d) for d in digits(n, 5)[1:]))
print([a(n) for n in range(58)]) # Michael S. Branicky, Jan 13 2022

from itertools import count, islice, product
def agen(): # generator of terms
    yield 0
    for d in count(1):
        for first in "2468":
            for rest in product("02468", repeat=d-1):
                yield int(first + "".join(rest))
print(list(islice(agen(), 58))) # Michael S. Branicky, Jan 13 2022

A045888(a(n)) = 0. - Reinhard Zumkeller, Aug 25 2009
a(n) = A179082(n) for n <= 25. - Reinhard Zumkeller, Jun 28 2010
From Hieronymus Fischer, Jun 06 2012: (Start)
a(n) = ((2*b_m(n)) mod 8 + 2)*10^m + Sum_{j=0..m-1} ((2*b_j(n)) mod 10)*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) = 2*10^n.
a(2*5^n+1) = 4*10^n.
a(3*5^n+1) = 6*10^n.
a(4*5^n+1) = 8*10^n.
a(n) = 2*10^log_5(n-1) for n=5^k+1,
a(n) < 2*10^log_5(n-1), else.
a(n) > (8/9)*10^log_5(n-1) n>1.
a(n) = 2*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)* (2+4x^5^j+ 6(x^2)^5^j+ 8(x^3)^5^j)/(1-x^5^(j+1)).
Also: g(x) = 2*(x/(1-x))*Sum_{j>=0} 10^j*x^5^j * (1-4x^(3*5^j)+3x^(4*5^j))/((1-x^5^j)(1-x^5^(j+1))).
Also: g(x) = 2*(x/(1-x))*(h_(5,1)(x) + h_(5,2)(x) + h_(5,3)(x) + h_(5,4)(x) - 4*h_(5,5)(x)), where h_(5,k)(x) = Sum_{j>=0} 10^j*(x^5^j)^k/(1-(x^5^j)^5). (End)
a(5*n+i-4) = 10*a(n) + 2*i for n >= 1, i=0..4. - Robert Israel, Apr 07 2016
Sum_{n>=2} 1/a(n) = A194182. - Bernard Schott, Jan 13 2022

Examples and crossrefs added by Hieronymus Fischer, Jun 06 2012

A084984 Numbers containing no prime digits.

Original entry on oeis.org

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

A029581 Numbers in which all digits are composite.

Original entry on oeis.org

4, 6, 8, 9, 44, 46, 48, 49, 64, 66, 68, 69, 84, 86, 88, 89, 94, 96, 98, 99, 444, 446, 448, 449, 464, 466, 468, 469, 484, 486, 488, 489, 494, 496, 498, 499, 644, 646, 648, 649, 664, 666, 668, 669, 684, 686, 688, 689, 694, 696, 698, 699, 844, 846, 848

Offset: 1

N. J. A. Sloane

If n is represented as a zerofree base-4 number (see A084544) 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)=4,6,8,9 for k=1..4. - Hieronymus Fischer, May 30 2012

			From _Hieronymus Fischer_, May 30 2012: (Start)
a(1000) = 88649.
a(10^4) = 6468989
a(10^5) = 449466489. (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. A002808, A001744, A046034, A084544, A084984, A017042, A001743, A014261, A014263, A202267, A202268.

[n: n in [1..1000] | Set(Intseq(n)) subset [4, 6, 8, 9]]; // Vincenzo Librandi, Dec 17 2018

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

From Hieronymus Fischer, May 30 and Jun 25 2012: (Start)
a(n) = Sum_{j=0..m-1} (2*b(j) mod 8 + 4 + floor(b(j)/4) - floor((b(j)+1)/4))*10^j, where m = floor(log_4(3*n+1)), b(j) = floor((3*n+1-4^m)/(3*4^j)).
Also: a(n) = Sum_{j=0..m-1} (A010877(2*b(j)) + 4 + A002265(b(j)) - A002265(b(j)+1))*10^j.
Special values:
a(1*(4^n-1)/3) = 4*(10^n-1)/9.
a(2*(4^n-1)/3) = 2*(10^n-1)/3.
a(3*(4^n-1)/3) = 8*(10^n-1)/9.
a(4*(4^n-1)/3) = 10^n-1.
a(n) < 4*(10^log_4(3*n+1)-1)/9, equality holds for n=(4^k-1)/3, k > 0.
a(n) < 4*A084544(n), equality holds iff all digits of A084544(n) are 1.
a(n) > 2*A084544(n).
Lower and upper limits:
lim inf a(n)/10^log_4(n) = 1/10*10^log_4(3) = 0.62127870, for n --> inf.
lim sup a(n)/10^log_4(n) = 4/9*10^log_4(3) = 2.756123868970, for n --> inf.
where 10^log_4(n) = n^1.66096404744...
G.f.: g(x) = (x^(1/3)*(1-x))^(-1) Sum_{j>=0} 10^j*z(j)^(4/3)*(1-z(j))*(4 + 6z(j) + 8*z(j)^2 + 9*z(j)^3)/(1-z(j)^4), where z(j) = x^4^j.
Also: g(x) = (1/(1-x))*(4*h_(4,0)(x) + 2*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+1)-1)/3)*(x^(k*4^j)/(1-x^4^(j+1)). (End)
Sum_{n>=1} 1/a(n) = 1.039691381254753739202528087006945643166147087095114911673083135126969046250... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 15 2024

Offset corrected by Arkadiusz Wesolowski, Oct 03 2011

A017222 a(n) = (9*n + 5)^2.

Original entry on oeis.org

25, 196, 529, 1024, 1681, 2500, 3481, 4624, 5929, 7396, 9025, 10816, 12769, 14884, 17161, 19600, 22201, 24964, 27889, 30976, 34225, 37636, 41209, 44944, 48841, 52900, 57121, 61504, 66049, 70756

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Sequences of the form (m*n+5)^2: A010864 (m=0), A000290 (m=1), A016754 (m=2), A016790 (m=3), A016814 (m=4), A016850 (m=5), A016970 (m=6), A017042 (m=7), A017126 (m=8), this sequence (m=9), A017330 (m=10), A017450 (m=11), A017582 (m=12).

Cf. A017221, A017246.

[(9*n+5)^2: n in [0..35]]; // Vincenzo Librandi, Jul 24 2011

(9Range[0,30]+5)^2 (* or *) LinearRecurrence[{3,-3,1},{25,196,529},30] (* Harvey P. Dale, May 22 2012 *)

a(n)=(9*n+5)^2 \\ Charles R Greathouse IV, Jun 17 2017

[(9*n+5)^2 for n in range(41)] # G. C. Greubel, Dec 29 2022

a(n) = A017221(n)^2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, May 22 2012
G.f.: (25 + 121*x + 16*x^2)/(1-x)^3. - R. J. Mathar, Mar 20 2018
From G. C. Greubel, Dec 29 2022: (Start)
a(2*n+1) = 4*A017246(n).
a(n) = a(n-1) + 9*(18*n + 1).
E.g.f.: (25 + 171*x + 81*x^2)*exp(x). (End)

A085557 Numbers that have more prime digits than nonprime digits.

Original entry on oeis.org

2, 3, 5, 7, 22, 23, 25, 27, 32, 33, 35, 37, 52, 53, 55, 57, 72, 73, 75, 77, 122, 123, 125, 127, 132, 133, 135, 137, 152, 153, 155, 157, 172, 173, 175, 177, 202, 203, 205, 207, 212, 213, 215, 217, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232

Offset: 1

Jason Earls, Jul 04 2003

Begins to differ from A046034 at the 21st term (which is the first 3-digit term).

			133 is in the sequence as the prime digits are 3 and 3 (those are two digits; counted with multiplicity) and one nonprime digit 1 and so there are more prime digits than nonprime digits. - _David A. Corneth_, Sep 06 2020

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A193238, A046034, A046035, A118950, A019546, A203263, A035232, A039996, A085823, A052382, A084544, A084984, A017042, A001743, A001744, A014261, A014263, A202267, A202268, A211681.

is(n) = my(d = digits(n), c = 0); for(i = 1, #d, if(isprime(d[i]), c++)); c<<1 > #d \\ David A. Corneth, Sep 06 2020

from itertools import count, islice
def A085557_gen(startvalue=1): # generator of terms
    return filter(lambda n:len(s:=str(n))<(sum(1 for d in s if d in {'2','3','5','7'})<<1),count(max(startvalue,1)))
A085557_list = list(islice(A085557_gen(),20)) # Chai Wah Wu, Feb 08 2023

cp's OEIS Frontend

A046034 Numbers whose digits are primes.

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A014263 Numbers that contain even digits only.

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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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A029581 Numbers in which all digits are composite.